Simple Statistical Mechanical Model Research Articles

How ions affect the structure of water.

We model ion solvation in water. We use the MB model of water, a simple two-dimensional statistical mechanical model in which waters are represented as Lennard-Jones disks having Gaussian hydrogen-bonding arms. We introduce a charge dipole into MB waters. We perform (NPT) Monte Carlo simulations to explore how water molecules are organized around ions and around nonpolar solutes in salt solutions. The model gives good qualitative agreement with experiments, including Jones-Dole viscosity B coefficients, Samoilov and Hirata ion hydration activation energies, ion solvation thermodynamics, and Setschenow coefficients for Hofmeister series ions, which describe the salt concentration dependence of the solubilities of hydrophobic solutes. The two main ideas captured here are (1) that charge densities govern the interactions of ions with water, and (2) that a balance of forces determines water structure: electrostatics (water's dipole interacting with ions) and hydrogen bonding (water interacting with neighboring waters). Small ions (kosmotropes) have high charge densities so they cause strong electrostatic ordering of nearby waters, breaking hydrogen bonds. In contrast, large ions (chaotropes) have low charge densities, and surrounding water molecules are largely hydrogen bonded.

A study ofviscoelasticity in gelling systems

We propose a simple statistical mechanics model for the study of thedynamics of gelling systems. It is based on percolation andbond-fluctuation dynamics for the bond vectors: we study thecritical viscoelastic properties and the relaxation patterns in thecase of irreversible gelation and the results obtained are discussedby means of scaling arguments. By introducing the idea of a finitelifetime for the bonds and by simply tuning this timescale, themodel can be made to present very different dynamics and relaxationpatterns corresponding to different gelling phenomenology.

Effects of torsional strain on thermal denaturation of DNA.

A class of simple statistical mechanical models for DNA melting, first proposed by Poland and Scheraga, has been demonstrated to exhibit a first or second order thermodynamic singularity, notwithstanding the intrinsic one-dimensional nature of the problem. In the present paper we consider the case of circular DNA and show that the inclusion of twist elastic energy in the Poland-Scheraga models leads either to suppression of the thermodynamic singularity or to a weak, third order singularity. Such behavior may also be present in linear DNA under mechanical influences that preclude the release of torsional strain.

Thermodynamics and kinetics of downhill protein folding investigated with a simple statistical mechanical model

AbstractThe possibility of downhill protein folding is one of the most intriguing predictions of the energy landscape approach. Finding examples of downhill folding has important practical implications because in the absence of a free energy barrier the mechanism of folding is amenable to experimental observation. Here, a simple statistical mechanical model of protein folding is used to analyze the thermodynamic and kinetic properties of downhill and two‐state folding. Folding free energy surfaces with and without a barrier separating denatured and native states can be generated with this model by simply modifying the curvature of the energy as a function of the order parameter, i.e., number of amino acids in incorrect conformation. Thermodynamic and kinetic analysis of surfaces with a barrier show the typical properties of two‐state folding: a first‐order equilibrium unfolding transition and exponential biphasic folding kinetics. In contrast, in downhill surfaces the unfolding process is noncooperative, and the apparent transition depends on the structural probe used to measure it. Therefore, this inherent behavior can be used as an experimental criterion for the identification of downhill folding. Surprisingly, the kinetics of downhill folding do not greatly deviate from exponential behavior unless the effective diffusion coefficient is assumed to decrese as folding progresses, exemplifying the increase in the roughness of the landscape. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Structural transitions in DNA driven by external force and torque

Experiments on single DNA molecules have shown that abrupt transitions between states of different extensions can be driven by stretching and twisting. Here we show how a simple statistical-mechanical model can be used to globally fit experimental force-extension data of Léger et al. [Phys. Rev. Lett. 83, 1066 (1999)], over a wide range of DNA molecule twisting. We obtain the mean twists, extensions, and free energies of the five DNA states found experimentally. We also predict global force-torque and force-linking number phase diagrams for DNA. At zero force, the unwinding torque for zero-force structural transition from the double helix to an unwound structure is found to be approximately -2kBT, while the right-handed torque needed to drive DNA to a highly overwound state approximately 7kBT.

Interplay between hydrophobic cluster and loop propensity in β-hairpin formation

Autonomously folding β-hairpins have recently emerged as powerful tools for elucidating the origins of antiparallel β-sheet folding preferences. Analysis of such model systems has suggested four potential sources of β-sheet stability: (1) the conformational propensity of the loop segment that connects adjacent strands; (2) favorable contacts between side-chains on adjacent strands; (3) interstrand hydrogen bonds; and (4) the intrinsic β-sheet propensities of the strand residues. We describe the design and analysis of a series of isomeric 20 residue peptides in which factors (1)-(4) are identical. Differences in β-hairpin formation within this series demonstrate that these four factors, individually, are not sufficient to explain β-sheet stability. In agreement with the prediction of a simple statistical mechanical model for β-hairpin formation, our results show that the separation between the loop segment and an interstrand cluster of hydrophobic side-chains strongly influences β-hairpin size and stability, with a smaller separation leading to greater stability.

Statistical modeling of a medium in amorphous and crystalline states

A simple statistical mechanical model of condensed molecular medium has been developed on the basis of the thermodynamic-functional method and the Rott correlation function method of conditional distributions. The short-range and long-range orders, pressure, and chemical potential are investigated with due regard for the correlation in the filling of microvolumes by particles of the medium (in the presence of vacant microcells). Two solutions are found in the range of volumes typical of the crystalline state. These solutions correspond to a splitting of the pressure isotherm and the chemical potential of the system into crystalline and amorphous branches.

Fast kinetics and mechanisms in protein folding.

This review describes how kinetic experiments using techniques with dramatically improved time resolution have contributed to understanding mechanisms in protein folding. Optical triggering with nanosecond laser pulses has made it possible to study the fastest-folding proteins as well as fundamental processes in folding for the first time. These include formation of alpha-helices, beta-sheets, and contacts between residues distant in sequence, as well as overall collapse of the polypeptide chain. Improvements in the time resolution of mixing experiments and the use of dynamic nuclear magnetic resonance methods have also allowed kinetic studies of proteins that fold too fast (greater than approximately 10(3) s-1) to be observed by conventional methods. Simple statistical mechanical models have been extremely useful in interpreting the experimental results. One of the surprises is that models originally developed for explaining the fast kinetics of secondary structure formation in isolated peptides are also successful in calculating folding rates of single domain proteins from their native three-dimensional structure.

Molecular model of hydrophobic solvation

The physical basis for the “hydrophobic effect” is studied using a simple statistical mechanical model of water, the “MB” model, in which water molecules are represented as Lennard-Jones disks with hydrogen bonding arms. Using a four-state framework developed by Muller [Acc. Chem. Res. 23, 23 (1990)], and extended by Lee and Graziano [J. Am. Chem. Soc. 118, 5163 (1996)], we find the model reproduces the fingerprints of hydrophobicity, namely, the large positive heat capacity, and temperatures TH and TS at which the enthalpy and entropy of transfer, respectively, are zero. Further, the behavior can be interpreted readily in terms of hydrogen bonds that are either made or broken in the bulk or in the first solvation shell around a nonpolar solute. We find that inserting a nonpolar solute into cold water causes ordering and strengthening of the H bonds in the first shell, but that the reverse applies in hot water. This provides a physical interpretation for the crossover temperatures TH and TS.

Temperature Dependence of Nuclear Magnetic Resonance Chemical Shifts of 129Xe in the α-Cages of NaY Zeolite

One- and two-dimensional NMR spectra of 129Xe sorbed in NaY zeolite at a loading of approximately 1 Xe atom/4 α-cages were obtained as a function of temperature. One-dimensional spectra were measured over the range of 60−300 K, by far the largest range yet investigated with this method. The data were fitted to the simple statistical mechanical cylindrical pore model of Cheung yielding a van der Waals adsorption energy of 3.3 kJ/mol. 2D-NOESY spectra were obtained at 300 and 355 K. The two-dimensional data show that the intercrystalline diffusion is slow compared to intracrystalline diffusion. The results indicate that the Xe atoms spend most of their time in the α-cages and at the lower temperatures, near the cage walls.

Structural properties of methanol-polyamidoamine dendrimer solutions

We used small angle x-ray scattering to study interparticle structural properties of polyamidoamine starburst dendrimers (PAMAM, with an ethylenediamine central core) in methanol solutions. We have explored the intensity profiles for solutions of starbursts of different generations $(g=2,3.5,4)$ at various concentrations. We find from the measured spectra that the half-integer generation, $g=3.5,$ is characterized by pronounced interparticle correlations also in the very dilute regime. By means of a simple statistical-mechanical model, widely used in charged complex fluids (proteins and ionic micellar solutions), we calculate in this latter case the dendrimer-dendrimer structure factor $S(q).$ The results obtained give clear indication of a partial ionization process which involves the carboxylate $({\mathrm{COO}}^{\mathrm{\ensuremath{-}}}{\mathrm{Na}}^{+})$ terminal groups of the dendrimer. The calculated macroion charge number, independent of the solution concentration, is ${Z}_{0}\ensuremath{\simeq}6.$

A statistical mechanical model for beta-hairpin kinetics.

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a beta-hairpin, the minimal structural element of the antiparallel beta-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, beta-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 2(15) possible conformations and 15 x 2(15) possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the beta-hairpin peptide.

Monte Carlo Simulation of the Concentration Dependence of Segregation at Vicinal Grain Boundaries

Using the Metropolis algorithm Monte Carlo technique solute-atom segregation is studied at two vicinal grain boundaries (GBs)—the Σ = 5/(002)/θ = 36.89° symmetrical twist and the Σ = 5/(310)/θ = 53.13° symmetrical tilt—at 850 K on the Ni-rich side of the Ni-Pt phase diagram, over the concentration range 0–10 at.% Pt. Unlike the Pt-rich side of the phase diagram the structures of both GBs remain stable in this concentration range. The dilute limit behavior for most GB sites extends to at least 0.1 at. %. At higher concentrations the effective segregation energies steadily decrease with increasing solute concentrations, due to solute-solute interactions between segregated atoms, until saturation occurs. It is argued that simple statistical mechanical models, e.g., the Fowler-Guggenheim model do not work well, even in the case of simple vicinal GBs.

Folding dynamics and mechanism of beta-hairpin formation.

Protein chains coil into alpha-helices and beta-sheet structures. Knowing the timescales and mechanism of formation of these basic structural elements is essential for understanding how proteins fold. For the past 40 years, alpha-helix formation has been extensively investigated in synthetic and natural peptides, including by nanosecond kinetic studies. In contrast, the mechanism of formation of beta structures has not been studied experimentally. The minimal beta-structure element is the beta-hairpin, which is also the basic component of antiparallel beta-sheets. Here we use a nanosecond laser temperature-jump apparatus to study the kinetics of folding a beta-hairpin consisting of 16 amino-acid residues. Folding of the hairpin occurs in 6 micros at room temperature, which is about 30 times slower than the rate of alpha-helix formation. We have developed a simple statistical mechanical model that provides a structural explanation for this result. Our analysis also shows that folding of a beta-hairpin captures much of the basic physics of protein folding, including stabilization by hydrogen bonding and hydrophobic interactions, two-state behaviour, and a funnel-like, partially rugged energy landscape.

Local signals in the energy landscape of collapsed helical proteins

Using simple statistical mechanical models, we present a theory based on a free energy function that combines classical models of the helix-coil transition in polymers with approximate treatments of the effects of excluded volume, confinement, and helical packing. The theory includes “local signals” for folding in the form of stabilization energies for three types of local structure. Randomness in the energies of the conformational states is also considered. The thermal behavior of the model is presented for realistic estimates of the signal energies. Estimates of the relative contributions of local signals and specific tertiary interactions to the folding stability gap are obtained.

Structural homogeneity of liquid selenium tellurium alloys

Tight binding Monte Carlo simulations of the structure of liquid selenium tellurium alloys have shown a clustering tendency between species with different coordination numbers. These results are in agreement with the conclusions that can be drawn from a simple statistical mechanical model previously developed for this system.

Effect of vacancies and surfactants on the dynamics of ordering processes in multi-component systems

The dynamics of far-from-equilibrium ordering processes in multi-component systems, e.g. crystalline solids or fluid mixtures, that are being quenched in temperature is studied by computer-simulation techniques involving Monte Carlo (MC) as well as molecular dynamics (MD) methods. The systems are modeled by using simple two-dimensional statistical mechanical models such as multi-state Potts lattice models and models of particle systems interacting via Lennard-Jones-like potentials. The ordering dynamics is investigated under the conditions of both conservation and non-conservation of the order parameter as well as with and without the presence of additional ‘foreign’ components, such as vacancies and surfactants, that couple to the interfaces which develop during the ordering process. The present paper reviews recent progress in this field with an emphasis on a possible universal description of ordering dynamics.

Local Conformational Signals and the Statistical Thermodynamics of Collapsed Helical Proteins

We investigate the role that local conformational tendencies can have in guiding the folding of helical proteins, using simple statistical mechanical models. The theory provides a synthesis of classical models of the helix-coil transition in polymers with an approximate treatment of the effects of excluded volume, confinement, and packing alignment of the helices based on a free energy function. The theory studies the consequences of signals encoded locally in the sequence as stabilization energies associated with three types of local structure: native helical conformations, native non-helical conformations, and native helix caps or start-stop signals. The role of randomness in the energies of conformations due to tertiary interactions is also studied vis-à-visthe difficulty of conformational search. The thermal behavior of the model is presented for realistic values of the conformational signal energies, which can be estimated from experimental studies on peptide fragments. Estimates are made for the relative contribution of local signals and specific tertiary interactions to the folding stability gap.

Simulation Studies of the Adsorption of Xenon on the (110) Face of Rutile

Monte Carlo simulations of xenon adsorbed on the (110) face of rutile at various degrees of coverage are reported. At low coverage adsorption is restricted to the rows along the coordinatively unsaturated Ti 4+ ions. It is demonstrated that the adsorbate structure is mainly determined by the adsorbate-adsorbate interaction. As the adsorbate density increases additional adsorption sites appear above the superficial oxygen atoms between the rows. The nonspecific adsorption within the rows is described by a simple statistical-mechanical model of one dimensional nonlocalized adsorption of interacting particles. By use of Widom's test particle insertion method, isotherms of adsorption are calculated and compared with experimental adsorption isotherms on rutile powder at different temperatures between 196.2 and 273.2 K.

Statistical mechanics of membrane adhesion by reversible molecular bonds.

We present a simple statistical-mechanical model for membranes adhering by thermally fluctuating, reversible molecular bonds. The free energy of the molecular bonds, which interact through the membrane tension, can be related to that of the one-component two-dimensional Coulomb plasma in the grand-canonical ensemble. Using the known properties of the Coulomb plasma, we compute the macroscopic membrane-membrane adhesive potential. We find that, unlike existing theories of membrane adhesion, thermal fluctuations assist adhesion and that the weak-adhesion potential has an unexpectedly long, temperature-controlled range.