The statistical theory of linear capillary chromatography with uniform stationary phase

Based on the random walk model and probability theory, general relations between the moments of column residence time and the moments of step sojourn time and step displacement are established. And starting from the mass-balances principle of solute molecules in the mobile and stationary phases, the moments of step sojourn time and step displacement are derived and expressed in terms of the basic parameters. Substituting the step moments into the general relations, the moments of column residence time are then obtained. The expression of retention time is completely identical to the well-known, the expressions of second moment or HETP unite and generalize various expressions of stochastic theory and mass balance theory, and the third and forth moments are given in more exact form.     

Transition event statistics in genetics and disordered kinetics. Theoretical approaches for extracting rate distributions from experimental data

We study the analogies between the theory of rate processes in disordered systems and the overdispersed molecular clocks in evolutionary biology. A biological "molecular clock" expresses the statistics of the number of amino acid or nucleotide substitutions during evolution. Random variations of the evolution rates lead to statistical (overdispersed) molecular clocks which are described by random point processes with random substitution rates. We find that the models for overdispersed molecular clocks are equivalent to those of the random-rate or random channel models used in disordered kinetics. The number of transport (reaction) events in disordered kinetics plays the same role as the number of substitution events in molecular biology. We study the connections between the (observed) statistics of the transition events and the statistics of random rate coefficients and random channels; a unified approach is developed which is valid both in molecular biology and in disordered kinetics. We develop methods for extracting statistical information about the variations of rate coefficients from experimental or observed data regarding the fluctuations of the numbers of substitution, reaction, or transport events. For systems with static disorder, the observed statistics of the number of reaction events, expressed in terms of probabilities at a given time or by the cumulants of the number of transition events at a given time, contains the information necessary for evaluating the cumulants or the probability density of the rate coefficients or the density of states for random channel kinetics. For dynamic disorder this is not possible; further information about multitime probability distributions of the reaction events is needed.     

Theory of non-Markovian rate processes

Starting from a multidimensional reaction kinetic equation with a general time evolution operator and a reaction sink function K, we derive formally exact expressions for the survival probability, reaction time distribution, and mean reaction time by using the projection operator technique. These rate expressions are given in the rational function form, with the irreducible memory function Omegairr as the key ingredient. This approach has an advantage over the direct perturbation approaches that use the reaction term as the small parameter, in that Omegairr has a structure that can be perturbatively treated with (K - K) as the small parameter. The well-known Wilemski-Fixman-type rate expressions are reproduced as the zeroth-order approximation from the present theory. Practical methods for evaluating the formal rate expressions are presented, and the results calculated for a model of electron transfer in non-Debye solvents are compared with computer simulations. It is found that the present approach is very promising for the study of non-Markovian dispersive kinetics.     

Life time of nonthinning liquid films - influence of the surface waves spatial correlations

On the basis of the theory of stochastic processes the evolution of the spatial correlation function of the surface waves on a thin liquid film as well as the corresponding root mean square amplitude A(t) and number of uncorrelated subdomains N(t) are obtained. A formulation of the life time t of unstable nonthinning films is proposed, based on the evolution of A and N. It is shown that the presence of uncorrelated subdomains shortens the life time of the film. Some numerical results for A(t), N(t) and t at different film thicknesses h and areas S, are demonstrated, taking into account only van der Waals and capillary forces.     

A macroscopic kinetic model for DNA polymerase elongation and high-fidelity nucleotide selection

The enzymatically catalyzed template-directed extension of ssDNA/primer complex is an important reaction of extraordinary complexity. The DNA polymerase does not merely facilitate the insertion of dNMP, but it also performs rapid screening of substrates to ensure a high degree of fidelity. Several kinetic studies have determined rate constants and equilibrium constants for the elementary steps that make up the overall pathway. The information is used to develop a macroscopic kinetic model, using an approach described by Ninio [Ninio J., 1987. Alternative to the steady-state method: derivation of reaction rates from first-passage times and pathway probabilities. Proc. Natl. Acad. Sci. U.S.A. 84, 663-667]. The principle idea of the Ninio approach is to track a single template/primer complex over time and to identify the expected behavior. The average time to insert a single nucleotide is a weighted sum of several terms, including the actual time to insert a nucleotide plus delays due to polymerase detachment from either the ternary (template-primer-polymerase) or quaternary (+nucleotide) complexes and time delays associated with the identification and ultimate rejection of an incorrect nucleotide from the binding site. The passage times of all events and their probability of occurrence are expressed in terms of the rate constants of the elementary steps of the reaction pathway. The model accounts for variations in the average insertion time with different nucleotides as well as the influence of G + C content of the sequence in the vicinity of the insertion site. Furthermore the model provides estimates of error frequencies. If nucleotide extension is recognized as a competition between successful insertions and time delaying events, it can be described as a binomial process with a probability distribution. The distribution gives the probability to extend a primer/template complex with a certain number of base pairs and in general it maps annealed complexes into extension products.     

Time-dependent quantum Monte Carlo and the stochastic quantization

We examine the relation between the recently proposed time-dependent quantum Monte Carlo (TDQMC) method and the principles of stochastic quantization. In both TDQMC and stochastic quantization, particle motion obeys stochastic guidance equations to preserve quantum equilibrium. In this way the probability density of the Monte Carlo particles corresponds to the modulus square of the many-body wave function at all times. However, in TDQMC, the motion of particles and guide waves occurs in physical space unlike in stochastic quantization where it occurs in configuration space. Hence, the practical calculation of time evolution of many-body fully correlated quantum systems becomes feasible within the TDQMC methodology. We illustrate the TDQMC technique by calculating the symmetric and antisymmetric ground state of a model one-dimensional helium atom, and the time evolution of the dipole moment when the atom is irradiated by a strong ultrashort laser pulse.     

Stochastic hybrid modeling of intracellular calcium dynamics

Deterministic models of biochemical processes at the subcellular level might become inadequate when a cascade of chemical reactions is induced by a few molecules. Inherent randomness of such phenomena calls for the use of stochastic simulations. However, being computationally intensive, such simulations become infeasible for large and complex reaction networks. To improve their computational efficiency in handling these networks, we present a hybrid approach, in which slow reactions and fluxes are handled through exact stochastic simulation and their fast counterparts are treated partially deterministically through chemical Langevin equation. The classification of reactions as fast or slow is accompanied by the assumption that in the time-scale of fast reactions, slow reactions do not occur and hence do not affect the probability of the state. Our new approach also handles reactions with complex rate expressions such as Michaelis-Menten kinetics. Fluxes which cannot be modeled explicitly through reactions, such as flux of Ca(2+) from endoplasmic reticulum to the cytosol through inositol 1,4,5-trisphosphate receptor channels, are handled deterministically. The proposed hybrid algorithm is used to model the regulation of the dynamics of cytosolic calcium ions in mouse macrophage RAW 264.7 cells. At relatively large number of molecules, the response characteristics obtained with the stochastic and deterministic simulations coincide, which validates our approach in the limit of large numbers. At low doses, the response characteristics of some key chemical species, such as levels of cytosolic calcium, predicted with stochastic simulations, differ quantitatively from their deterministic counterparts. These observations are ubiquitous throughout dose response, sensitivity, and gene-knockdown response analyses. While the relative differences between the peak-heights of the cytosolic [Ca(2+)] time-courses obtained from stochastic (mean of 16 realizations) and deterministic simulations are merely 1%-4% for most perturbations, it is specially sensitive to levels of G(βγ) (relative difference as large as 90% at very low G(βγ)).     

A master equation investigation of coagulation reactions: Sol-gel transition

The kinetics of irreversible coagulation phenomena in spatially homogeneous systems is formulated in terms of a multivariate stochastic process. The latter is governed by a master equation for the joint probability distribution of the numbers of reacting species. An efficient numerical algorithm is used to simulate the complete time evolution of the stochastic process. The method is illustrated by simulating the coagulation reaction with configuration-dependent reaction kernels, K_ij = (ij)^ω, for clusters of mass i and j with 1/2 < ω ⩽ 1, which are designed to model gelation phenomena. It is demonstrated that the stochastic simulation allows the determination of critical exponents and the gel point directly from the master equation. The results are compared to predictions of the rate equation approach to the sol-gel transition.     

Energy level alignment at metal/organic semiconductor interfaces: "pillow" effect, induced density of interface states, and charge neutrality level

A unified model, embodying the "pillow" effect and the induced density of interface states (IDIS) model, is presented for describing the level alignment at a metal/organic interface. The pillow effect, which originates from the orthogonalization of the metal and organic wave functions, is calculated using a many-body linear combination of atomic orbitals Hamiltonian, whereby electron long-range interactions are obtained using an expansion in the metal/organic wave function overlap, while the electronic charge of both materials remains unchanged. This approach yields the pillow dipole and represents the first effect induced by the metal/organic interaction, resulting in a reduction of the metal work function. In a second step, we consider how charge is transferred between the metal and the organic material by means of the IDIS model: Charge transfer is determined by the relative position of the metal work function (corrected by the pillow effect) and the organic charge neutrality level, as well as by an interface parameter S, which measures how this potential difference is screened. In our approach, we show that the combined IDIS-pillow effects can be described in terms of the original IDIS alignment corrected by a screened pillow dipole. For the organic materials considered in this paper, we see that the IDIS dipole already represents most of the realignment induced at the metal/organic interface. We therefore conclude that the pillow effect yields minor corrections to the IDIS model.     

Activation processes with memory

We propose a mathematical treatment of the activated processes governed by stochastic Langevin dynamics with a colored random force, corresponding to a noise generated by an Ornstein-Uhlenbeck process. Such non-Markovian dynamics take place in a variety of chemical and biological systems. Using the path integral approach, we constructed the conditional probability for passing between two stationary states in configurational space. Our relations can be used for Monte Carlo sampling of evolution trajectories for systems with many degrees of freedom as well as for determining the reaction coordinate used in transition state theory. On the basis of our relation for a conditional probability, we generalize the method of determining the most probable path to the case of colored random force. Using the simple three-hole potential, we examine numerically the effect of nonzero correlation time (memory) on the evolution of the most probable path for a finite temperature.     

Kinetics of autocatalysis in small systems

Autocatalysis is a ubiquitous chemical process that drives a plethora of biological phenomena, including the self-propagation of prions etiological to the Creutzfeldt-Jakob disease and bovine spongiform encephalopathy. To explain the dynamics of these systems, we have solved the chemical master equation for the irreversible autocatalytic reaction A+B-->2A. This solution comprises the first closed form expression describing the probabilistic time evolution of the populations of autocatalytic and noncatalytic molecules from an arbitrary initial state. Grand probability distributions are likewise presented for autocatalysis in the equilibrium limit (A+B <==>2A), allowing for the first mechanistic comparison of this process with chemical isomerization (B<==>A) in small systems. Although the average population of autocatalytic (i.e., prion) molecules largely conforms to the predictions of the classical "rate law" approach in time and the law of mass action at equilibrium, thermodynamic differences between the entropies of isomerization and autocatalysis are revealed, suggesting a "mechanism dependence" of state variables for chemical reaction processes. These results demonstrate the importance of chemical mechanism and molecularity in the development of stochastic processes for chemical systems and the relationship between the stochastic approach to chemical kinetics and nonequilibrium thermodynamics.     

C6H6/Au(111): interface dipoles, band alignment, charging energy, and van der Waals interaction

We analyze the benzene/Au(111) interface taking into account charging energy effects to properly describe the electronic structure of the interface and van der Waals interactions to obtain the adsorption energy and geometry. We also analyze the interface dipoles and discuss the barrier formation as a function of the metal work-function. We interpret our DFT calculations within the induced density of interface states (IDIS) model. Our results compare well with experimental and other theoretical results, showing that the dipole formation of these interfaces is due to the charge transfer between the metal and benzene, as described in the IDIS model.     

Stochastic model of photodynamics in multichromophoric conjugated polymers

A stochastic model of triplet exciton dynamics in multichromophoric conjugated polymers is presented and analyzed in detail, with a focus on the single molecule spectroscopy observables. The model deals with the evolution of a discrete statistical distribution of triplets in isolated polymer molecules. This approach should provide more accurate quantitative information on the dynamic processes involved, as compared to the previously used two-state model which assumes that a conjugated polymer cannot contain more than one triplet. In particular, it allows for determination of the triplet-triplet annihilation rate.     

Derivation of quantum langevin equation from an explicit molecule-medium treatment in interaction picture

A quantum mechanical form of the Langevin equation is derived from an explicit consideration of the molecule-medium interaction, as advocated by Simons in 1978, and by using two identities in the interaction picture. This can be easily reduced to the classical regime, and further simplified to the macroscopic Langevin equation by considering the stochastic Langevin force autocorrelation function. One of the so-called Einstein relations appears as a byproduct. By following the methodology proposed by Simons, an exact expression for the momentum autocorrelation function is obtained. The latter can be used to calculate the zero-frequency macroscopic diffusion coefficient that is observed to satisfy the second Einstein relation. The formalism described above gives rise to the possibility of explicitly computing the transport characteristics such as friction constant and diffusion coefficient from the corresponding quantum statistical mechanical expressions. A discussion on the Langevin equation becomes complete only when the corresponding Fokker-Planck equation is obtained. Therefore, the probability of the evolution of states with a particular absolute magnitude of linear momentum from those of another momentum eigenvalue is quantum mechanically defined. This probability appears as a special average value of a projection operator and as a special projection operator correlation function. A classical identity is introduced that is shown to be valid also for the quantum mechanically defined probability function. By using this identity, the so-called Fokker-Planck equation for the evolution probability is easily established.     

A reduction method for multiple time scale stochastic reaction networks

In this paper we develop a reduction method for multiple time scale stochastic reaction networks. When the transition-rate matrix between different states of the species is available, we obtain systems of reduced equations, whose solutions can successively approximate, to any degree of accuracy, the exact probability that the reaction system be in any particular state. For the case when the transition-rate matrix is not available, one needs to rely on the chemical master equation. For this case, we obtain a corresponding reduced master equation with first-order accuracy. We illustrate the accuracy and efficiency of both approaches by simulating several motivating examples and comparing the results of our simulations with the results obtained by the exact method. Our examples include both linear and nonlinear reaction networks as well as a three time scale stochastic reaction-diffusion model arising from gene expression.     

受主方程控制的化学反应体系非平衡定态宏观稳定性判据之随机推广

借助扰动按分布参数分离法及主方程的Kramers-Moyal展开证明,随机熵相应于对最可几路径偏离的偏超量之时间导数与体系对外部扰动的响应性直接相关.该演变速率等价于偏超随机熵产生,并与根据随机位方法提出的随机超熵产生速率等效.对Poisson分布,该量表现为Gibbs超熵产生的等价量.局域平衡假定失效后,化学反应体系的宏观稳定性即决定于这个新的随机量.     

Simultaneous stochastic effects of a thermal reservoir and an electromagnetic field on the optical susceptibility of a two-level system

In this paper we derive analytical expressions for the optical susceptibility of a two-level system immersed in a thermal bath and interacting with an external electromagnetic field, where both of them are considered as noise sources. The dynamics of the system is described by a set of optical stochastic Bloch equations. The noise sources are modeled as Ornstein-Uhlenbeck processes. The optical stochastic Bloch equations are perturbatively solved up to second order in the external field. We found that each noise affects the dynamics in a different manner. Thus, at first order, the bath modifies the transverse relaxation time, whereas the effect of a random field can only be appreciated if the expansion is calculated up to second order, where correlations begin to be important.     

反应碰撞的动力学李代数描述

提出一种李代数方法描述分子反应碰撞问题．给出了含有主要动力学参量的S-矩阵元、分子碰撞跃迁几率以及反应体系能量统计平均值随时间演化的解析表达式．讨论了一个简单排斥势场中的原子-双原子分子共线反应体系，以阐明这种新方法的要点。