The dynamics of the stochastic multi-molecule biochemical reaction model

The paper introduces the dynamics of a stochastic multi-molecule biochemical reaction model.First, we show that there is a unique positive solution of the stochastic model. Furthermore, we deduce the conditions when the reaction will end and when the reaction being proceed. At last, we derive that the solution of (1.5) oscillates around the endemic proportion equilibrium $P^*(x^*,y^*)$ , and the intensity of fluctuation is proportional to white noise. The key to the analysis in this paper is choosing appropriate Lyapunov function. The outcomes are illustrated by computer simulations throughout this paper.     

Dynamics of a stochastic model for continuous flow bioreactor with Contois growth rate

In the modelling of the continuous flow bioreactor, due to uncertainties in the environment the growth rate parameter is under perturbation of white noise, which results in a mathematical model governed by a set of stochastic differential equations. In this paper, assume the Contois growth rate is used and then we first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout and non-washout equilibria are stochastically stable. At the end, we carry out some numerical simulation, which supports our theoretical conclusion well. Also, by the quantities introduced in the last section, both residence time and intensity of the noise have significant effect on the performance of the reactor.     

Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, due to the effect of singularities in the interactions between fields involving several replicas below dimension eight. In the zero-replica limit, several coupling constants have thus to be considered, instead of just one. As a result we found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, which does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin t₀ at which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complete correspondence between the equilibrium singularities in the limit at n = 0, and the singularities in the dynamics when the initial time t₀ goes to minus infinity, with n replaced by −1/t₀. There is thus complete coherence between the two approaches.     

Threshold dynamics of a stochastic Keizer’s model with stochastic incidence

In this paper, we incorporate stochastic incidence of a chemical reaction into the standard Keizer’s open chemical reaction. We prove that a positive stationary distribution (PSD) for the associated chemical master equation exists and is globally asymptotically stable. We present threshold dynamics of the stochastic Keizer’s model in term of the profile of the PSD for both finite and infinite volume size V. This establishes a sharp link between deterministic Keizer’s model and the stochastic model. In this way, we resolve Keizer’s paradox from a new perspective. This simple model reveals that such stochastic incidence incorporated, though negligible when V goes to infinity, may play an indispensable role in the stochastic formulation for irreversible biochemical reactions.     

Studies of C<Subscript>8</Subscript> aromatics adsorption in BaY and mordenite molecular sieves using the headspace technique

A headspace technique, that consists in analyzing the composition of the vapor phase in equilibrium with the condensed phase of a mixture in a sealed vial containing the adsorbent sample, has been recently applied to acquire equilibrium data for adsorption of xylenes in liquid phase. In this study, we used this technique to measure experimental binary equilibrium data for C₈ aromatics in Y and mordenite zeolitic molecular sieves. For the Y zeolite, we also measured C₈ aromatics quaternary equilibrium data. Measurements were made at temperatures between C 40–80 °C. A more tedious, but traditional, chromatographic pulses method was also used to validate some of the results.     

Mean residence time of Markov processes for particle transport in fludized bed reactors

A new approach for studying the particle dynamics and RTD (residence time distribution) in processes is to formulate stochastic models. A common question to all models for RTD is whether Danckwerts’ law for mean residence time holds. In this paper we revisit a Markov process that has been proposed by Dehling et al. (1999) as a stochastic model for particle transport in fluidized bed reactors. Under the volumetric flow balance conditions, we deduce different boundary conditions at the entrance and the exit of the reactor, and in both discrete model and continuous model we show that processes satisfy Danckwerts’ law, stating that the mean residence time of particle transport in fluidized bed reactors equals V/v, where V denotes the volume of the reactor occupied by the fluid and v the volumetric inflow rate.     

Pseudomolecular atoms: geometry of two-electron intrashell excited states

Supermultiplet patterns indicating pronounced collective motion of electrons are prominent among intrashell excited states of helium or alkaline earth atoms. Many aspects of these patterns have previously been discussed in terms of an empirical rovibrator model, e-core-e, analogous to alinear triatomic molecule. However, this has appeared incompatible with a dimensional scaling treatment that predicts pseudomolecular features and relates properties of some excited states to the ground state, but corresponds to abent equilibrium geometry. We examine both the full two-electron Hamiltonian, including angular momentum, and a prototype model for angular correlation which restricts the two particles to the surface of a sphere. By virtue of the boundary conditions imposed by the Jacobian volume element, we find that alinear equilibrium geometry is not allowed. If the Hamiltonian is transformed to reduce the Jacobian to unity, an infinite barrier appears in the effective potential when the electrons are 180° apart and equidistant from the nucleus. We find the supermultiplet patterns, including some “antimolecular” features, are consistent with a floppy butbent geometry, anasymmetric rotor model. The most probable interelectron angle ? _m varies markedly with the principal quantum number and also with the space-quantization of the angular momentum with respect to body-fixed axes.     

A comparative study of particle swarm optimization and its variants for phase stability and equilibrium calculations in multicomponent reactive and non-reactive systems

Particle swarm optimization is a novel evolutionary stochastic global optimization method that has gained popularity in the chemical engineering community. This optimization strategy has been successfully used for several applications including thermodynamic calculations. To the best of our knowledge, the performance of PSO in phase stability and equilibrium calculations for both multicomponent reactive and non-reactive mixtures has not yet been reported. This study introduces the application of particle swarm optimization and several of its variants for solving phase stability and equilibrium problems in multicomponent systems with or without chemical equilibrium. The reliability and efficiency of a number of particle swarm optimization algorithms are tested and compared using multicomponent systems with vapor–liquid and liquid–liquid equilibrium. Our results indicate that the classical particle swarm optimization with constant cognitive and social parameters is a reliable method and offers the best performance for global minimization of the tangent plane distance function and the Gibbs energy function in both reactive and non-reactive systems.     

Simulation studies of phase inversion in agitated vessels using a Monte Carlo technique

A speculative study on the conditions under which phase inversion occurs in agitated liquid-liquid dispersions is conducted using a Monte Carlo technique. The simulation is based on a stochastic model, which accounts for fundamental physical processes such as drop deformation, breakup, and coalescence, and utilizes the minimization of interfacial energy as a criterion for phase inversion. Profiles of the interfacial energy indicate that a steady-state equilibrium is reached after a sufficiently large number of random moves and that predictions are insensitive to initial drop conditions. The calculated phase inversion holdup is observed to increase with increasing density and viscosity ratio, and to decrease with increasing agitation speed for a fixed viscosity ratio. It is also observed that, for a fixed viscosity ratio, the phase inversion holdup remains constant for large enough agitation speeds. The proposed model is therefore capable of achieving reasonable qualitative agreement with general experimental trends and of reproducing key features observed experimentally. The results of this investigation indicate that this simple stochastic method could be the basis upon which more advanced models for predicting phase inversion behavior can be developed.     

Electron transfer networks

In this paper, we study electron transfer networks. These are generalisations of electron transport chains, and consist of a set of substrates which can exist in reduced and oxidised forms. The reduced forms can transfer electrons to the oxidised forms, and there are some electron inflow and outflow processes. We show that under mild assumptions, such systems can have only very simple behaviour, with a single globally stable equilibrium. To prove this we show that the Jacobian of the system has negative logarithmic norm in an appropriate norm. From this result, uniqueness and global stability of any equilibrium follows. The results extend, with only minor modifications, to binary interconversion networks, where the only allowed reactions are interconversions between substrates, and inflow/outflow processes.      

Quantification of the Interaction of SmI2 with Substrates and Ligands

The method developed and introduced here enables for the first time (to the authors’ knowledge), a quantitative assessment of the interaction of SmI₂ with substrates prior to the electron transfer stage. As a proof of concept, equilibrium constants for some model substrates including carbonyl compounds and aromatic nuclei are reported here. In addition, the first equilibrium constants with some common ligands were also determined. The equilibrium constants range from approximately 0.07 m ⁻¹ for diisopropyl ketone to 2500 m ⁻¹ for hexamethylphosphoramide (HMPA). It is shown that the data acquired by this method, which is based on the concept of shift reagents, can shed light on the most intimate details of the reaction mechanism, and this method is a useful tool for planning a synthetic process.     

Quasielastic scattering of neutrons by a liquid crystal substance with flexible molecules

Abstract

When interpreting quasielastic neutron scattering (QNS) spectra of liquid crystals with complicated and flexible molecules, it seems absolutely necessary to take into consideration the internal reorientations. The present QNS study concerns TCDCBPh/di-(4-n-butyloxyphenyl) trans-cyclohexane-1,4-dicarboxylate). It contains a cyclohexane ring which possesses a high internal mobility. As a probing parameter we use the ‘excess elasticity’ p which in a way represents the elastic form-factor. We make drastic assumptions: we equalize all partial correlation times, we allow each hydrogen atom to participate in one reorientation only, and we ignore possible couplings between reorientations. Then we analyse the behaviour with temperature of the p parameter for eleven models in which either all specified molecular fragments reorient or some of them reorient and others are at rest. In the crystal, far below melting, all internal stochastic mobilities seem to be at rest. However, still 30°C below melting, the cyclohexane ring starts large angle internal deformations which have a stochastic character. It seems that some fragments in the molecules (e.g., benzene rings and/or alkyl terminal groups) start to reorient a few degrees before melting. This is connected with the formation of a ‘mobile’ crystal phase, just before melting. Then all internal reorientations become free, and one cannot observe much difference between the smectic A phase and the nematic phase. This study shows that the cyclohexane ring is the most mobile sub-entity in the TCDCBPh molecule. Observations by polarized microscopy seem to corroborate our claim for the existence of a “mobile” phase just before melting. The literature information about the phase sequence is C 107°C S_A 156°C N 209°C I.     

Stochastic dynamics of complexation reaction in the limit of small numbers

We study stochastic dynamics of the non-linear bimolecular reaction A + B?AB. These reactions are common in several bio-molecular systems such as binding, complexation, protein multimerization to name a few. We use master equation to compute the full distribution of several stochastic equilibrium properties such as number of complexes formed (N(c)), equilibrium constant (K). We provide exact analytical and simpler approximate expression for equilibrium fluctuation quantities to quickly estimate the amount of noise as a function of reactant molecules and rates. We construct the phase diagram for a fluctuational quantity f, defined as the ratio of standard deviation to average (f=√(ΔN(c))(2)/N(c)), as a function of different number of reactant molecules and reaction rates. One of the striking result is, it is possible to have f as high as 45% or higher in significant regions of the phase diagram even when number of reactants involved are around 20-40, typical in biology. Our finding indicates studying averages alone using mass action law needs careful scrutiny. We also outline possible application of our findings in gene expression. Furthermore, we compute average and fluctuation properties of time dependent quantities and derive equations of motion for different moments such as N(c)(t) and N(c)(t)(2). While mean-field mass action law fails to reproduce the exact time dependence, approximate solutions of coupled equations of motions for different moments, capturing fluctuation, is in good agreement with exact results. This may be a way to compute time development of averages and fluctuations in such non-linear systems where mass action law breaks down. Moreover, for this reaction, we outline connection to variational principle of maximum caliber and other more traditional approaches such as chemical Langevin equation. We derive noise statistics for the equivalent Langevin equation and show possible departure from Gaussian white noise. We believe quantitative estimates of phase diagrams for noise, time dependent quantities, and simple analytical expression for equilibrium quantities will be particularly useful to guide experiments involving such non-linear reactions with small numbers of reactants that are often encountered in biology.     

Pt(100)/NO+CO体系随机共振的理论研究

在远离平衡的条件下,NO在Pt(100)表面进行的催化还原反应表现出化学振荡和化学多稳态等复杂的动力学行为.利用Fink等提出的反应模型,通过计算机模拟对体系的控制参量进行周期调制和随机调制,在简单双稳区以及振荡与稳态的共存区,观察到随机共振(StochasticResonance:SR)现象.     

Machine learning method for simulation of adsorption separation: Comparisons of model’s performance in predicting equilibrium concentrations

In this work, we implemented different models for predicting adsorption separation of a dye from aqueous solution using porous materials. The equilibrium data of solute concentrations were collected from resources and used in the models for training and verification purposes to develop the models. For prediction of the equilibrium solute concentrations (Ce), we used tree models: Multi-layer Perceptron (MLP), Passive aggressive regression, and Decision Tree (DT) Regressor. In the modeling, we considered the adsorbent dosage as well as solution pH as the input parameters to the model, and the model was able to generate the output values, i.e., equilibrium concentrations based on the input variables. The evaluation of the models’ performances revelated that the final R² scores are 0.99, 0.98, 0.93 for DT, MLP and Passive-Aggressive, respectively and a very low RMSE of 0.055 for decision tree that shows this model is the best among models used in this study. Indeed, decision tree model is recommended among the other three models to be employed for correlation of adsorption equilibrium data.     

Effective thermostat induced by coarse graining of simple point charge water

We investigate how the transport properties of a united atom fluid with a dissipative particle dynamics thermostat depend on the functional form and magnitude of both the conservative and the stochastic interactions. We demonstrate how the thermostat strongly affects the hydrodynamics, especially diffusion, viscosity, and local escape times. As model system we use simple point charge (SPC) water, from which projected trajectories are used to determine the effective interactions in the united atom model. The simulation results support our argument that the thermostat should be viewed as an integral part of the coarse-grained dynamics rather than a tool for approaching thermal equilibrium. As our main result we show that the united atom model with the adjusted effective interactions approximately reproduces the diffusion constant and the viscosity of the underlying detailed SPC water model.     

A non-local pseudopotential in the FSGO model: Study of some organometallic systems

A non-local core pseudopotential has been used in the framework of floating spherical Gaussian orbital (FSGO ) model to study the equilibrium geometries and valence electronic structures of some organolithium and organoberyllium systems. The calculated equilibrium geometries, heats of hydrogenation, average electric polarizabilities, and magnetic susceptibilities are in good agreement with the results of the all-electron FSGO model calculations. Valence electron wave functions obtained here have been used to predict the valence electron Compton profiles (CP ) and electron momentum distributions (EMD ) of the systems studied. A good correlation has been shown among the peak height of the CP (J(0)), valence electron energy (E_v), and number of valence electrons (N_v).     

非恒稳含时电极过程中涨落-耗散的电化学效应——恒电势滴汞电极体系的随机热力学模型

以滴汞电极体系为模型,对非恒稳恒电势动态不可逆电极过程中的耗散-涨落效应进行了系统的研究.基于滴汞电极体系的电化学特征,提出了一个简化的含时随机热力学模型,从而可能对这类重要的含时物理化学过程进行涨落和耗散的定量分析.借助该简化的模型,成功地建立了恒电势滴汞电极过程的基本随机热力学公式,由此推出耗散-涨落效应的理论极谱曲线.在滴汞电极生长缓慢及扩散步骤严重滞后情况下,含时的滴汞电极过程将趋于在有效扩散层厚度演化的慢流型上的准定态过程.在这种准定态近似下,具体分析了涨落对极谱曲线的影响.结果表明,在涨落影响可以忽略的近平衡区,从耗散-涨落导出的极谱方程与从平衡态Nernst公式导出的极谱方程完全吻合.还计算了一个涨落诱导的极谱曲线偏离的典型范例.     

A Monte Carlo simulation of energy transfer processes in thermal unimolecular reactions. II. An applications to polyatomic dissociation

The Monte Carlo method has been used to provide a numerical solution to the ro-vibrational master equation for the low pressure unimolecular decomposition of a polyatomic molecule. This type of solution is made possible through the use of a simple exponential transition probability function, that represents the efficiency with which energy transfer takes place between the reactant molecule and an unspecified heat bath gas. The Monte Carlo technique is used to generate random variables that are distributed in a manner prescribed by the transition probability function. In the case of the present simulation, these variables correspond to random energy jumps induced in the molecule through single collision events. In order to account for the energy dependence of the vibrational state densities, we have proposed that vibrational relaxation in the polyatomic takes place from a single vibrational mode. Under equilibrium conditions we are able to show that with this assumption, the Monte Carlo model is capable of reproducing molecular quantities, such as the average vibrational energy per molecule and the vibrational specific heat, that compare favourable with the corresponding values calculated from equilibrium statistical mechanics. The model has been applied to a study of the low pressure unimolecular decomposition of a series of polyatomics. For three of the molecules, CH₄, CD₄, and C₂H₆ the agreement between the calculated and the high temperature experimental rate constants is very good. The calculations indicate that a significant proportion of the molecules that dissociate are rotationally as well as vibrationally excited. Very few of the reactive molecules have a vibrational energy content equal to or greater than E₀, the dissociation energy. The extent of rotational excitation is found to be temperature dependent.     

Study of Interfacial Charge Transfer from an Electron Rich Organic Molecule to CdTe Quantum Dot by using Stern-Volmer and Stochastic Kinetic Models

Photoinduced electron transfer (PET) from N-methylaniline (NMA) to a photoexcited CdTe quantum dot (QD*) is studied in toluene. The PET mechanism at low to moderate quencher (NMA) concentrations (<0.08 M) remains mostly collisional with some contributions from QD-NMA complex formation. However, at high quencher concentrations (>0.10 M), QDs form larger numbers of static complexes with NMA molecules leading to a steep positive deviation in the steady-state Stern–Volmer curves. An isothermal titration calorimetry (ITC) study confirms the formation of QD-NMA complexes (K∼150 M⁻¹) at high quencher concentrations. Fitting our experimental data using a stochastic kinetic model indicates that the number of NMA molecules attached per QD at highest NMA concentration (∼0.16 M) used in this study decreases from ∼0.76 to ∼0.47 with reducing the QD size from ∼5.2 nm to ∼3.2 nm. However, the PET rate increases with decreasing QD size, which is commensurate with the observation that the chemical driving force (ΔG) increases with decreasing the QD particle size. We have analyzed the PET kinetics mainly by using Stern-Volmer fittings. However, in some cases Tachiya's stochastic kinetic model is used for stoichiometric analysis, which seems to be useful only at high quencher concentrations. The measured PET rate coefficients in all the cases are found to be at least an order of magnitude lower when compared to the diffusion-controlled rate of the reaction medium.