Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

On the oscillatory phenomenon in an oil/water interface

A simple theoretical model is presented for simulating the self-sustained oscillations of electric potential and pH at an oil/water interface appearing in a two-phase system composed of 2-nitropropane solution containing picrate acid and an aqueous solution of cetyltrimethylammonium bromide. In the present model, a well-known condition necessary for the occurrence of self-sustained oscillations, i.e., the presence of a positive feedback process far from equilibrium, is taken into account in a set of kinetic equations to describe simplified characters of the following two processes: (i) a cooperative formation of ion pair complexes at the interface, and (ii) supply of picrate anions and cetyltrimethylammonium cations to the interface accompanied by release of ion pair complexes to the organic phase. The numerical solutions of the present equations are shown to reproduce fairly well the characteristic properties of the oscillation of electric potential and pH such as wave forms and frequencies.     

Weak acid transport across bilayer lipid membrane in the presence of buffers. Theoretical and experimental pH profiles in the unstirred layers.

This paper presents a simple model to describe experimental data on weak acid transport across planar bilayer lipid membrane separating two buffered solutions. The model takes into account multiple proton-transfer reactions occurring in the unstirred layers (ULs) adjacent to the membrane. Differential equations of the model are shown to be reduced to a set of nonlinear algebraic equations. Since the latter equations depend monotonically on unknown variables, they can be easily solved numerically, using bisection method. For the particular system studied experimentally (with acetate as the weak acid and TRIS+MES as the buffer mixture) pH profiles in the ULs are calculated from the model. These results are compared with experimental data obtained using pH microelectrode. The agreement between theoretical and experimental pH profiles is found to be satisfactory. The most pronounced deviations are observed at the UL/bulk solution boundary. To obtain a better correlation between the theoretical and experimental results, two other, less idealized models are considered. They take into account, respectively, (a) the electric field arising in the ULs from ion diffusion and (b) finiteness of the rates of proton-transfer reactions. However, both acetate membrane fluxes and pH profiles in the ULs computed from these models are found to be close to those of the simple model. One can thus conclude that the difference between experimental and theoretical pH profiles is due to the inconsistency of the generally accepted model of the "unstirred layer", assuming the existence of a strict boundary between the regions of "pure diffusion" and "ideal stirring".     

A functional model of the rat kidney

Summary A mathematical model of the nephron was developed by writing a set of material balance equations for the flow of urea, salt and water along the length of the nephron. The geometric proportions have been elaborated in a foregoing study and are taken here as a basis, in particular the model configuration of the collecting duct system. The medullary interstitial solute concentration profiles are taken to increase linearly in outer and inner zone. The several transepithelial fluxes are driven by diffusion, osmosis, solvent drag and active transport. The development of osmotic gradient in the inner medulla is taken here to be caused by active secretion of salt into the descending limb of Henle's loop. The parameters in the flux equations for all parts of the nephron and the concentration values at the end of each tubular section are determined by collecting and averaging the values given in literature and by extrapolating the measurement data.The simulation of the model equations with these averaged parameters resulted in concentration profiles which at the ends of the several tubular sections were consistent with the values observed in experimental investigations.This work was supported by the Deutsche Forschungsgemeinschaft.     

Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity

A computational fluid dynamics (CFD) approach was presented to model the blood flows in the carotid bifurcation and the brain arteries under altered gravity. Physical models required for CFD simulation were introduced including a model for arterial wall motion due to fluid-wall interactions, a shear thinning fluid model of blood, a vascular bed model for outflow boundary conditions, and a model for autoregulation mechanism. The three-dimensional unsteady incompressible Navier-Stokes equations coupled with these models were solved iteratively using the pseudocompressibility method and dual time stepping. Gravity source terms were added to the Navier-Stokes equations to take the effect of gravity into account. For the treatment of complex geometry, a chimera overset grid technique was adopted to obtain connectivity between arterial branches. For code validation, computed results were compared with experimental data for both steady-state and time-dependent flows. This computational approach was then applied to blood flows through a realistic carotid bifurcation and two Circle of Willis models, one using an idealized geometry and the other using an anatomical data set. A three-dimensional Circle of Willis configuration was reconstructed from subject-specific magnetic resonance images using an image segmentation method. Through the numerical simulation of blood flow in two model problems, namely, the carotid bifurcation and the brain arteries, it was observed that the altered gravity has considerable effects on arterial contraction/dilatation and consequent changes in flow conditions.     

Theory of the ICR heating of a plasma column: Methodological note

A set of wave equations is derived that describes electromagnetic waves at frequencies on the order of the ion gyrofrequency in a plasma column with an arbitrary electron temperature. This set takes into account, in particular, the resonant interaction of electrons with waves in the transit-time magnetic pumping regime. The effect of the amplification of the electromagnetic fields of current-carrying antennas by the plasma is analyzed. The evolution of the fields with an increase of plasma density from a zero value (vacuum) is considered. The main parameters are determined for minority ion cyclotron resonance heating in the planned EPSILON system.     

Transport Effects on Multiple-Component Reactions in Optical Biosensors

Optical biosensors are often used to measure kinetic rate constants associated with chemical reactions. Such instruments operate in the surface–volume configuration, in which ligand molecules are convected through a fluid-filled volume over a surface to which receptors are confined. Currently, scientists are using optical biosensors to measure the kinetic rate constants associated with DNA translesion synthesis—a process critical to DNA damage repair. Biosensor experiments to study this process involve multiple interacting components on the sensor surface. This multiple-component biosensor experiment is modeled with a set of nonlinear integrodifferential equations (IDEs). It is shown that in physically relevant asymptotic limits these equations reduce to a much simpler set of ordinary differential equations (ODEs). To verify the validity of our ODE approximation, a numerical method for the IDE system is developed and studied. Results from the ODE model agree with simulations of the IDE model, rendering our ODE model useful for parameter estimation.     

Structural determinations by circular dichroism spectra analysis using coupled oscillator methods: an update of the applications of the DeVoe polarizability model

The exciton (coupled oscillator) model for optical activity is a very useful and powerful method which allows to analyze a circular dichroism (CD) spectrum in a nonempirical way, arriving at a safe assignment of the absolute configuration of organic and inorganic compounds. Usually in this model only the exciton coupling of two electrically allowed transitions (oscillators) is taken into account. This approach has the important advantage of an easy application but, sometimes, it may lead to wrong results. Thus, in this review article a more general treatment, which allows considering the simultaneous coupling of several oscillators, i.e., the DeVoe model, is presented and critically analyzed, discussing in detail the latest applications reported in the literature. In the authors opinion, since the DeVoe model joins generality and reliability requiring an almost negligible computational effort, it represents the method of choice for stereochemical assignments, even by nonspecialists.     

DNA的振动—内激发—转子模型

本文全面考察了DNA分子链的各种运动自由度,从B型DNA的双螺旋结构出发,建立了DNA的振动动-内激发-转子模型,并由此给出了相应的哈密顿函数、运动方程和孤子解,较为全面地阐明了DNA的动力学特性．     

Self-consistent modeling of radio-frequency plasma generation in stellarators

A self-consistent model of radio-frequency (RF) plasma generation in stellarators in the ion cyclotron frequency range is described. The model includes equations for the particle and energy balance and boundary conditions for Maxwell’s equations. The equation of charged particle balance takes into account the influx of particles due to ionization and their loss via diffusion and convection. The equation of electron energy balance takes into account the RF heating power source, as well as energy losses due to the excitation and electron-impact ionization of gas atoms, energy exchange via Coulomb collisions, and plasma heat conduction. The deposited RF power is calculated by solving the boundary problem for Maxwell’s equations. When describing the dissipation of the energy of the RF field, collisional absorption and Landau damping are taken into account. At each time step, Maxwell’s equations are solved for the current profiles of the plasma density and plasma temperature. The calculations are performed for a cylindrical plasma. The plasma is assumed to be axisymmetric and homogeneous along the plasma column. The system of balance equations is solved using the Crank-Nicholson scheme. Maxwell’s equations are solved in a one-dimensional approximation by using the Fourier transformation along the azimuthal and longitudinal coordinates. Results of simulations of RF plasma generation in the Uragan-2M stellarator by using a frame antenna operating at frequencies lower than the ion cyclotron frequency are presented. The calculations show that the slow wave generated by the antenna is efficiently absorbed at the periphery of the plasma column, due to which only a small fraction of the input power reaches the confinement region. As a result, the temperature on the axis of the plasma column remains low, whereas at the periphery it is substantially higher. This leads to strong absorption of the RF field at the periphery via the Landau mechanism.     

Mathematical modeling of living cell metabolism using the method of steady-state stoichiometric flux balance

This approach uses a set of algebraic linear equations for reaction rates (the method of steady-state stoichiometric flux balance) to model the purposeful metabolism of the living self-reproducing biochemical system (i.e. cell), which persists in steady-state growth. Linear programming (SIMPLEX method) is used to derive the solution for the model equations set (determining reaction rates which provide flux balance at given conditions). Here, we demonstrate the approach through the mathematical modeling of steady-state metabolism in Saccharomyces cerevisiae mitochondria.     

A triphasic analysis of corneal swelling and hydration control.

Physiological studies strongly support the view that hydration control in the cornea is dependent on active ion transport at the corneal endothelium. However, the mechanism by which endothelial ion transport regulates corneal thickness has not been elaborated in detail. In this study, the corneal stroma is modeled as a triphasic material under steady-state conditions. An ion flux boundary condition is developed to represent active transport at the endothelium. The equations are solved in cylindrical coordinates for confined compression and in spherical coordinates to represent an intact cornea. The model provides a mechanism by which active ion transport at the endothelium regulates corneal hydration and provides a basis for explaining the origin of the "imbibition pressure" and stromal "swelling pressure." The model encapsulates the Donnan view of corneal swelling as well as the "pump-leak hypothesis."     

Periodicity and boundedness of solutions of generalized differential equations of growth

Sufficient conditions are given for the existence of periodic solutions of differential equations, having as special cases the equations used to describe the competition between two species. The Poincaré bifurcation theory is used to secure one set of conditions, and another set of conditions is secured through a generalization of a method of V. Volterra. The question of boundedness is considered and conditions implying boundedness and conditions implying that populations are bounded away from zero are given. Several integrable classes of systems are discovered and a particular example having periodic solutions is examined in detail. This research was supported by the Air Force Office of Scientific Research under Grant 62-207.     

Mathematical modeling of an aqueous film coating process in a Bohle Lab-Coater, Part 1: Development of the model

The purpose of this study was to develop a model to predict (1) air and product temperatures, (2) product moisture, and (3) air humidity during an aqueous coating process using a Bohle Lab-Coater. Because of the geometrical properties and the airflow, the drum of the Bohle Lab-Coater can in principle be divided into 2 zones of equal size—the drying and the spraying zones. For each zone, 4 balance equations could be set up describing the change of the air humidity, the product moisture, the enthalpy of the air, and the enthalpy of the product in each zone. For this purpose, knowledge regarding heat and mass transfer and also the motion of the tablets in drums was used. Based on the considerations of the heat and mass transfer, a set of first-order coupled ordinary differential equations (ODEs) was developed. This set of ODEs can be solved numerically. In this part, the development of the model is described in detail, whereas the application of the model can be found in part 2.     

Nonlinear Electrical Effects in Lipid Bilayer Membranes: II. Integration of the Generalized Nernst-Planck Equations

In this paper the ion transport across a thin lipid membrane is treated using a generalized form of the Nernst-Planck equations. An additional term is introduced into the flux equations to account for the image force acting on the ion. As the membrane thickness is of the same order of magnitude as the range of the image forces, the potential energy of the ion in the membrane is strongly dependent on position. The integration of the flux equations leads to a general expression for the integral membrane conductance lambda as a function of the voltage u. The ratio lambda(u)/lambda(0) (lambda(0) = membrane conductance in the limit u --> 0) depends on the dielectric constant and the thickness of the membrane, but is independent of the ionic radius. When the numerical values of the potential energy function, as calculated by the method of electrical images, are inserted into the expression for lambda(u)/lambda(0), a strongly non-linear current-voltage characteristic is obtained. The theoretical current-voltage curve agrees satisfactorily with the experimental data at a low ionic strength and at low voltages; at higher voltages the observed membrane conductance exceeds the predicted value.     

One-dimensional hydrodynamic model of the atom and ion dynamics in a stationary plasma thruster

A one-dimensional hydrodynamic model of the atom, ion, and electron dynamics in the channel of a stationary plasma thruster is developed. The relevant set of integrodifferential equations is derived and investigated both analytically (steady-state solutions) and numerically (dynamic regimes). It is shown that adjusting only one parameter (the channel resistivity) makes it possible to achieve a good agreement between the calculated global parameters and experimental data. The general features of oscillations revealed with the help of the model are also found to agree fairly well with the experiment.     

Mechanisms of dust grain charging in plasma with allowance for electron emission processes

The process of dust grain charging is described with allowance for secondary, ion-induced, photoelectric, and thermal electron emission from the grain surface. The roughness of the grain surface is taken into account. An intermediate charging regime involving ion–atom collisions and electron ionization in the perturbed plasma region is analyzed using the moment equations and Poisson’s equation. A calculation method is proposed that allows one to take into account the influence of all the above effects and determine the radius of the plasma region perturbed by the dust grain.     

Formation of a charge-exchange target for fast ions in the plasma of large-scale toroidal devices under NBI conditions

High-energy (E>0.2 MeV) charge-exchange diagnostics allow the determination of the distribution function of fast atoms produced via the neutralization of hydrogen isotope ions by target hydrogen-like impurity ions. To derive the distribution function from the experimental data requires knowledge of the composition and spatial distribution of the target ions in a tokamak plasma. A charge-exchange target forms as a result of the interaction between the main impurity nuclei and the heating neutral beams. In different devices, the heating beams are arranged in different ways with respect to the diagnostics; hence, in order to accurately estimate the contribution of the secondary ions to the detected signal, it is necessary to calculate their trajectories for every particular case. A model is proposed that takes into account elementary processes resulting in the ionization equilibrium of the ions of different impurities with allowance for ion motion in a specific tokamak configuration. As an example, the model is applied to the plasma of the JT-60U tokamak. Mechanisms for the formation of charge-exchange atomic flows in various energy ranges are considered. The relative contributions of different heating injectors to the charge-exchange flow are estimated. Based on the calculated results, a method is proposed for local measurements of the ion distribution function with the help of a stationary analyzer.     

A Quantitative Model for Nonrandom Generalized Transduction, Applied to the Phage P22–SALMONELLA TYPHIMURIUM System

A mathematical model for nonrandom generalized transduction is proposed and analyzed. The model takes into account the finite number of transducing particle classes for any given marker. The equations for estimation of the distance between markers from contransduction frequency data are derived and standard errors of the estimates are given. The obtained relationships depend significantly on the number of classes of transducing fragments. The model was applied to estimate the number of transducing fragment classes for a given marker in transduction with phage P22 of Salmonella typhimurium. It was found that the literature data on frequencies of contransduction in crosses with mutual substitution of selective and nonselective markers can be rationalized most accurately by assuming that the mean number of classes is equal to 2. An improved method for analysis of cotransduction data is proposed on the basis of our model and the results of calculation. The method relies on solving a set of algebraic equations for cotransduction frequencies of markers located within one phage length. The method allows a relatively precise determination of distances between markers, positions of transducing particle ends and deletion or insertion lengths. The approach is applied to the trp-cysB-pyrF and aroC-hisT-purF-dhuA regions of the Salmonella typhimurium chromosome.     

Improved 3D continuum calculations of ion flux through membrane channels

A continuum model, based on the Poisson–Nernst–Planck (PNP) theory, is applied to simulate steady-state ion flux through protein channels. The PNP equations are modified to explicitly account (1) for the desolvation of mobile ions in the membrane pore and (2) for effects related to ion sizes. The proposed algorithm for a three-dimensional self-consistent solution of PNP equations, in which final results are refined by a focusing technique, is shown to be suitable for arbitrary channel geometry and arbitrary protein charge distribution. The role of the pore shape and protein charge distribution in formation of basic electrodiffusion properties, such as channel conductivity and selectivity, as well as concentration distributions of mobile ions in the pore region, are illustrated by simulations on model channels. The influence of the ionic strength in the bulk solution and of the externally applied electric field on channel properties are also discussed.