Statistical Mechanical Procedure Research Articles

Statistical mechanical determination of nanocluster size distributions in the phase coexistence region of a first order phase transition from the isotherms of DMPC monolayers at the air-water interface.

A statistical mechanical deconvolution procedure for the experimentally measured surface pressure-area isotherms has been presented to obtain the surface pressure dependence of the liquid expanded (LE) and liquid condensed (LC) nanocluster size distributions in the LE-LC phase coexistence region of the first order phase transition of Dimyristoyl phosphatidylcholine (DMPC) monolayers at the air-water interface. This study presents the application of the deconvolution formulation introduced originally by Freire and Biltonen for the experimentally measured specific heat to calculate the submicroscopic lipid cluster distribution function in the phase coexistence region [E. Freire, R. L. Biltonen, Biopolymers, 1978, 17, 481-496] and extends their formulation to surface pressure isotherms. The present procedure involves the extraction of the pressure partition function calculated from the isotherm and utilizes the general relation between molecular density fluctuations and macroscopic lateral compressibility. In this procedure the high-density LC phase boundary has been determined uniquely. The average nanoscopic cluster sizes obtained in this study have been compared with the results from previous experimental studies. The cause of the finite difference between the values of the LC phase boundary area obtained from the present deconvolution procedure and the conventional extrapolation method on the same isotherm has been discussed from the viewpoint of slow hierarchical growth from nanoscopic clusters to macroscopic domains in the coexistence region.

Estimates of the optimal density of sphere packings in high dimensions

The problem of finding the asymptotic behavior of the maximal density ϕmax of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound on ϕmax that is controlled asymptotically by 1∕2d, where d is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski’s bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a “test” pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Exp. Math. 15, 307 (2006)], the putative exponential improvement on ϕmax was found with an asymptotic behavior controlled by 1∕2(0.77865⋯)d. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions corresponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same putative exponential improvement and therefore the asymptotic form 1∕2(0.77865⋯)d is much more general than previously surmised. This class of test functions leads to an optimized average kissing number that is controlled by the same asymptotic behavior as the one found in the aforementioned paper.

Is it a compound or cluster energy formalism?

Abstract The Compound Energy Formalism was not developed by using standard statistical mechanical procedures. If it had been, the model would have been based on cluster solution members instead of on compound end members. In its present form, using the Bragg – Williams approximation, the distinction between these two types of entity has no impact on the model's application, with one notable exception – the case of vacancy properties in intermediate phases. The distinction between the two viewpoints becomes more significant, however, for the future development of the model. It is not clear how short range order or the effects of atomic size mismatch can be explicitly taken into account in its compound end member form. Both of these features are readily absorbed into the model, however, when cluster solution members are considered to be the important entities.

Selectivity of 12-Crown-4 to Cations in CH3OH: A Monte Carlo Simulation Study

Crown ethers display the ability to selectively bind specific metals cations in the presence of complex aqueous mixtures of chemically similar ions. For example, 15-crown5 binds Na and 18-crown-6 binds K selectively, although the configuration of crown depends on the solvents used. That ability is thought to be dependent on the size of the crown cavity, a balance between cation-ether and cationwater interactions, the polarity of solvent and the nature of the electron donor atoms in the ring. These characteristics have been used in the design of novel materials for such processes as isotope separations, ion transport through membrane, and transport of therapeutic doses of radiations to tumor sites. Theoretical calculations based on molecular mechanics (MM) and molecular orbital (MO) theories give structures in a vacuum, however, those results cannot be directly compared with those of experiments. In the hostguest interaction, factors in the selective extraction of any host species include the relative free energy of desolvation of the guest molecules and the free energy of organizing the host into a suitable conformation having remote substitution for binding. Because of the large number of particles forming these systems and the variety of different interactions established, molecular dynamics (MD) and Monte Carlo (MC) statistical mechanics computer simulations are wellsuited theoretical tools for understanding and predicting the physicochemical properties of those solutions at the microscopic level. Several statistical mechanical procedures have evolved for computing free energy differences. It is known that solvent effects often play an important role in determining equilibrium constants, transition states and rates of reactions, π-facial selectivity, conformations, and other quantities of chemical, chemical physics and biochemical interest. Therefore, computational chemists have focused their interests on the crown ethers as those molecules are the simplest that show enzyme-like specificity. However, few molecular dynamics (MD) and Monte Carlo (MC) statistical mechanics computer simulation studies of both the selectivity of 12-crown-4 (1,4,7,10-tetraoxacyclododecane) to univalent cations and ∆log Ks, are available. This prompted us to study the selectivity of 12-crown-4 to univalent cations and ∆log Ks using Monte Carlo simulations of statistical perturbation theory (SPT) in CH3OH. Experimental studies of the selectivity of 12-crown-4 to univalent cations in methanol and ab initio studies of 12-crown-4 with univalent cations have been reported. In this note, we present the first calculation to computing on the selectivity of 12-crown-4 to entire series of univalent cations and ∆log Ks of 12-crown-4 to entire series of univalent cations in CH3OH using a Monte Carlo simulation of SPT. We found 12-crown-4/cation complexes with the 12-crown-4 of almost Cs symmetry never found in any crystal structure of the cation complexes of 12-crown-4. Those are the reasons why this study is communicated as note. The calculated relative binding Gibbs free energies (selectivity index) of 12-crown-4 complexes and the published data of the relative free energies and Dlog Ks (the difference of stability constant of binding) obtained using by (1) shown in Scheme 1, are listed in Table 1. We have noticed that 12-crown-4 binds K more tightly than the other cation in CH3OH, i.e., the selectivity of 12-crown-4 to K is more favorable than to other cations in CH3OH. A similar trend has been observed in the study of alkali cation complexes of 18-crown-6 and its derivatives in CCl4 solutions and in the study of alkali cation complexes of 18-crown-6 in diverse solutions. Binding selectivity is often associated with the ionic radius of the cation and the size of the crown ether cavity that it will occupy. The larger mismatch is existed between the ionic radius of the cation and the size of the crown ether cavity, the smaller chance that the cation binds favorably to the crown ether cavity. 12-crown-4 has distances between diagonal oxygen atoms of 4.0 A and selectively binds K over the other cations in CH3OH where the cationic radius of K is 1.38 and the others are 0.74, 1.02, 1.49, and 1.7 A, respectively. Selectivity is apparently the result of a delicate balance of the forces that the cation experiences as the crown ether and solvent molecules compete for the cation in solution. In this study, the cations

Ideal-Gas Heat Capacities and Virial Coefficients of HFC Refrigerants

Thermodynamic properties of HFC (hydrofluorocarbon) compounds have been extensively studied with worldwide interest as alternative refrigerants. Both quality and quantity in the experimental data far exceed those for the CFC and HCFC refrigerants. These data now provide a great opportunity to examine the validity of theoretical models, and vice versa. Among them, the ideal-gas heat capacity C p 0 and virial coefficients derived from the experimental data are of particular interest, since they are directly related to the intramolecular and intermolecular potentials through the statistical mechanical procedure. There have been some discrepancies reported in the observed and theoretical C p 0 for HFC compounds. We have performed new calculations of C p 0 for several HFCs. The present results are consistent with the selected experimental values. The second (B) and third (C) virial coefficients have been reported for these HFC refrigerants from speed of sound data and Burnett PVT data. Often, a square well-type intermolecular potential is employed to correlate the data. However, the model potential cannot account consistently for both B and C coefficients with the same potential parameters. We have analyzed the data with the Stockmayer potential and obtained self-consistent results for various HFC (R-23, R-32, R-125, R-134a, R-143a, and R-152a) compounds with physically reasonable potential parameters.

Stress-induced duplex DNA destabilization in scaffold/matrix attachment regions

S/MARs are DNA elements 300 to several thousand base-pairs long, which are operationally defined by their affinity for the nuclear scaffold or matrix. S/MARs occur exclusively in eukaryotic genomes, where they mediate several functions. Because S/MARs do not have a clearcut consensus sequence, the characteristics that define their activity are thought to be structural. Ubiquitous S/MAR binding proteins have been identified, but to date no unique binding sequence or structural motif has been found. Here we show by computational analysis that S/MARs conform to a specific design whose essential attribute is the presence of stress-induced base-unpairing regions (BURs). Stress-induced destabilization (SIDD) profiles are calculated using a previously developed statistical mechanical procedure in which the superhelical deformation is partitioned between strand separation, twisting within denatured regions, and residual superhelicity. The results of these calculations show that BURs exhibit a succession of evenly spaced destabilized sites that would render part or all of the S/MAR sequence single stranded at sufficient superhelicity. These analyses are performed for a range of sequenced S/MAR elements from the borders of eukaryotic gene domains, from centromeres, and from positions where S/MARs are known to support the action of an enhancer. The results reported here are in excellent agreement with earlier in vitro chemical reactivity studies. This approach demonstrates the potential for computational analysis to predict the points of division of the eukaryotic genome into functional units (domains), and also to locate certain cis-regulatory sequences.

Relation between volume correction and the standard state

The suggestion by Sharp et al. (Biochemistry 30 (1991) 9686) that a volume correction should be made to the standard chemical potential of a non-polar solute molecule in water is examined by re-deriving their results using an exact statistical mechanical procedure and comparing them with those of a couple of other similar schemes. It is shown that their suggestion is equivalent to choosing a standard state based on the ideal gas behavior.

Influence of short range interactions on the standard Gibbs energy of adsorption

The purpose of this communication is to suggest a general statistical mechanical procedure for analysing this question and to indicate the influence of particle-particle interactions in altering the standard Gibbs energy of adsorption, in the case of adsorption at electrode/electrolyte interface

General features of lattices of point dipoles against a charged wall

Some general properties of monolayers and multilayers of point dipoles against a charged wall, which do not depend on the discrete or infinite number of allowed orientations nor on the statistical-mechanical procedure employed, are derived. These properties stem directly from purely electrostatic considerations on the regular lattice structure of the layer and from the very definition of a statistical-mechanical average. The ‘‘effective coordination number’’ ce, frequently employed in monolayer models, is shown not to be an adjustable parameter, but rather a well-defined lattice sum. The implications of the use of erroneously high ce values are pointed out. Finally, the general features of monolayer models of point dipoles with an infinite number of orientations in the framework of the mean-field approximation are stressed, together with the limitations of this widely employed approximate statistical-mechanical procedure.

Conformational Energy Calculations on Polypeptides and Proteins: Use of a Statistical Mechanical Procedure for Evaluating Structure and Propertiesa

We are using a variety of theoretical and computational techniques to study protein structure, protein folding, and higher-order structures. Our earlier work involved treatments of liquid water and aqueous solutions of nonpolar and polar solutes, computations of the stabilities of the fundamental structures of proteins and their packing arrangements, conformations of small cyclic and open-chain peptides, structures of fibrous proteins (collagen), structures of homologous globular proteins, introduction of special procedures as constraints during energy minimization of globular proteins, and structures of enzyme-substrate complexes. Recently, we presented a new methodology for predicting polypeptide structure (described here); the method is based on the calculation of the probable and average conformation of a polypeptide chain by the application of equilibrium statistical mechanics in conjunction with an adaptive, importance sampling Monte Carlo algorithm. As a test, it was applied to Met-enkephalin.

Prediction of the native conformation of a polypeptide by a statistical-mechanical procedure. II. Average backbone structure of enkephalin.

AbstractThe average conformation of Met‐enkephalin was determined by using an adaptive, importance‐sampling Monte Carlo algorithm (SMAPPS—Statistical Mechanical Algorithm for Predicting Protein Structure). In the calculation, only the backbone dihedral angles (ϕ and ψ) were allowed to vary; i.e., all side‐chain (χ) and peptide‐bond (ω) dihedral angles were kept fixed at the values corresponding to a low‐energy structure of the pentapeptide. The total conformational energy for each randomly generated structure of the polypeptide was obtained by summing over the interaction energies of all pairs of nonbonded atoms of the whole molecule. The interaction energies were computed by the program ECEPP/2 (Empirical Conformational Energy Program for Peptides). Solvent effects were not included in the computation. The calculation was repeated until a total of 10 independent average conformations were established. The regions of conformational space occupied by the average structures were compared with the regions of low conditional free energy obtained by SMAPPS in the first paper of this series. Such a comparison provides an analysis of the capacity of SMAPPS to adjust the Monte Carlo search to regions of highest probability. The results demonstrate that the ability of SMAPPS to focus the Monte Carlo search is excellent. Finally, the 10 independent average conformations and the mean of the 10 average structures were utilized as the initial conformations for a direct energy minimization of the pentapeptide. Of the 11 final energy‐minimized structures, three of the conformations were found to be equivalent to the conformation of lowest energy determined previously. In addition, all but two of the remaining energy‐minimized structures were found to correspond to one of the two other conformations of high probability obtained in the first paper of this series. These results indicate that a set of independent average conformations can provide a rational, unbiased choice for the initial conformation, to be used in a direct energy minimization of a polypeptide. The final energy‐minimized structures consequently constitute a set of low‐energy conformations, which include the global energy minimum.

Chain flexibility and configurational dimensions of left handed Z- DNA and right handed B-DNA helices

The configurational behaviour of flexible helices of right handed B- and left handed Z-types have been analysed using statistical mechanical procedures. The configuration-dependent parameter, most importantly, the persistence length has been computed, using the heminucleotide scheme of treating polynucleotide chains under the approximation that perturbations in the backbone torsions produce sufficient flexibility in these helices. The values of persistence lengths obtained for Z-helices are very much higher than that of B-helices indicating that former is less flexible compared to the latter. These are in accordance with the results obtained recently on B- and Z-forms of poly(dG-dC). (dG-dC) using light scattering studies. Also the persistence lengths of BII-DNA helices characterised by a skew 3'-hemiucleotide (e-270°), and also when they coexist with B-DNA have been computed and the values lie within the range of experimentally reported values on B-helices. It is argued that the decrease in the persistence length values of B-DNA at higher salt concentration is due to additional small fluctuations in sugar residue torsions induced due to neutralisation of electrostatic repulsions between adjacent phosphates of the nucleotide. Noteworthy is that these are correlated to winding angle variations and the consequent bending of the helix.

Extremal f-trees and embedding spaces for molecular graphs

Problems concerning embedding trees in lattice-graph or Euclidean spaces are considered. A tree is defined to be ‘almost-embeddable’ in a lattice-graph if a sequence derived from the distance degree sequence of the lattice-graph and a corresponding sequence for the tree satisfy a specified inequality. This inequality is such that every tree that is embeddable in the lattice-graph is in the set of almost-embeddable trees. For Euclidean space embeddings the lattice-graph sequence is replaced by a sequence defined in terms of sphere packing numbers. This work has two practical objectives: Firstly, to furnish a framework within which intuitive chemical and physical notions about embedding spaces can be made explicit and self-consistent. Secondly, to obtain useable criteria which will exclude from statistical mechanical averaging procedures those molecular species which are inconsistent with a postulated embedding space. The inequality proposed here meets these objectives for molecular trees and its implications for chemical and physical theory are discussed in some detail.

Polymer conformation and NMR chemical shifts, 4. Contrast of 13C NMR spectra between poly(methyl acrylate‐alt‐styrene) and poly(methyl methacrylate‐alt‐styrene)

AbstractThe presence of the α‐methyl group in the alternating methyl methacrylate‐styrene copolymer produces a remarkable difference in its 13C NMR spectra as compared with the spectra of the alternating methyl acrylate‐styrene copolymer; cotacticity‐dependent splittings occur only in the former spectrum. A theoretical interpretation was attempted for this characteristic spectral variation in terms of the calculated time‐averaged chemical shifts for carbons in the dyads constituting the alternating copolymers. Each (meso or racemic) dyad sequence is considered to exist in several conformational states which are mutually exchanging rapidly and the probabilities of which are determined by a statistical mechanical procedure. The calculation of cotacticity‐dependent chemical shifts is conducted for the side chain carbonyl, aromatic C1 and methoxyl carbons as well as for some main chain carbons to give good agreement with the observed peak splittings, especially with respect to the side chain carbons. In accordance with the reported assignment, the calculated differences in shielding factors between meso and racemic dyads for the side chain carbons of poly(methyl methacrylate‐alt‐styrene) are positive, i.e. the meso or isotactic peak appears at higher field than the racemic or syndiotactic peak. Furthermore, the resulting values are relatively large (≈0,63 ppm) in conformity with the experiment. The calculation for the carbons in poly(methyl acrylate‐alt‐styrene) gives a very small shielding difference (−0,04 to 0,06 ppm) by the dyad cotacticity, being consistent with the observation, i. e. absence of peak splittings.

Ideal gas thermodynamic properties of propanone and 2-butanone

The ideal gas thermodynamic properties (Cp0, S0, H0−H00, (H0−H00)/T, −(G′−H00)/T,ΔHf0, ΔGf0, and log Kf) for propanone and 2-butanone in the temperature range 0 to 1500 K and at 1 atm were calculated by statistical mechanical procedures, using rigid-rotor and harmonic-oscillator approximations. The internal rotation contributions to thermodynamic properties were evaluated by use of a partition function formed by summation of internal rotation energy levels. The calculated heat capacities and entropies are in agreement with the available experimental values.

Equilimbrium Thermodynamics of Nonideal Reacting Gases.

view Abstract Citations (2) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Equilimbrium Thermodynamics of Nonideal Reacting Gases. Graboske, Harold C., Jr. Abstract An analysis is made of the equilibrium thermodynamic properties of ionizing and dissociating gas mixtures in a system where many-body interactions are significant. The use of certain approximations, such as single-level partition functions or hound-state cutoff methods to remove divergences in the atomic partition functions, lead to thermodynamic inconsistencies. A statistical-mechanical procedure is outlined which, for an appropriate many-body potential, includes a complete set of perturbed eigenstates, a nondivergent bound-state sum, and continuous and differentiable thermodynamic properties. Several theoretical models are applied to a solar-composition mixture to evaluate the region of validity of idealgas theory and the importance of various interaction effects. Publication: The Astrophysical Journal Pub Date: March 1972 DOI: 10.1086/151386 Bibcode: 1972ApJ...172..689G full text sources ADS |

Thermodynamics of Crystallization in High Polymers. I. Crystallization Induced by Stretching

A theory of oriented crystallization in elongated polymers having network structures (e.g., in vulcanized rubber) is developed through the application of statistical mechanical procedures similar to those employed in rubber elasticity theory. Expressions are derived which, within the limitation imposed by the simplifying assumptions, relate the incipient crystallization temperature with the elongation, the degree of crystallinity with the elongation and temperature, and the retractive force at crystallization equilibrium with the elongation at constant temperature. The reciprocal of the absolute temperature for incipient crystallization is predicted to decrease linearly with a simple function of the elongation and the average number of chain segments between cross linkages. Only moderate degrees of crystallinity are predicted at equilibrium. In conformity with requirements of the second law of thermodynamics, equilibrium crystallization decreases the tension in the stretched specimen. Apparent discrepancies between some of these predictions and various observations are attributed to severe departure from equilibrium crystallization when the polymer undergoes crystallization during the stretching process. A better approach to equilibrium should be achieved by stretching under conditions which prevent crystallization (e.g., at elevated temperature), then allowing crystallization to proceed at fixed elongation. Few experiments have been performed in this manner, but such results as are available confirm qualitatively the predictions of the theory. Reasons for the rapid increase in tension which is observed when crystallization occurs during ordinary stretching of rubber are discussed. It is pointed out that crystalline and amorphous regions preferably should not be regarded as separate phases. Likewise, the conversion of amorphous to crystalline polymer does not conform to the definition of a phase transition.