Markov processes in single molecule fluorescence

This article examines the current status of Markov processes in single molecule fluorescence. For molecular dynamics to be described by a Markov process, the Markov process must include all states involved in the dynamics and the first-passage time (FPT) distributions out of those states must be describable by a simple exponential law. The observation of non-exponential FPT distributions or other evidence of non-Markovian dynamics is common in single molecule studies and offers an opportunity to expand the Markov model to include new dynamics or states that improve understanding of the system.     

Molecular dynamic theories in chromatography

The molecular dynamic model of chromatography is a microscopic model that consists of two fundamental processes: (i) the random migration of the molecules in the mobile phase, and (ii) the random adsorption-desorption of molecules on the stationary phase. The diffusion and drift of the molecules in the mobile phase is usually described with a simple one-dimensional random walk. The adsorption-desorption process is modeled most of the time by a Poisson process that assumes exponential sojourn times of the molecules in both the mobile and the stationary phases. The molecular dynamic model of chromatography can simply be used to characterize the chromatographic process on heterogeneous stationary phases. It has been applied to reversed phase, chiral, size-exclusion, and ion-exchange separations.     

Computing stationary distributions in equilibrium and nonequilibrium systems with forward flux sampling

We present a method for computing stationary distributions for activated processes in equilibrium and nonequilibrium systems using forward flux sampling. In this method, the stationary distributions are obtained directly from the rate constant calculations for the forward and backward reactions; there is no need to perform separate calculations for the stationary distribution and the rate constant. We apply the method to the nonequilibrium rare event problem proposed by Maier and Stein, to nucleation in a 2-dimensional Ising system, and to the flipping of a genetic switch.     

Benzyl esters of optically active malic acid stereocopolymers as obtained by ring-opening polymerization of (R)-(+) and (S)-(-)-benzyl malolactonates

(R)-(+) and (S)-(-)-benzyl malolactonates were copolymerized anionically by using triethylamine and aluminum porphyrin as initiators. Configurational composition of resulting optically active poly(benzyl β-malate) stereocopolymers depends on the enantiomeric purity in the monomer feed. Stereochemistry during the multistep synthesis starting from aspartic acid enantiomers used as precursors is discussed. It is shown that natural L(+)-aspartic acid yields (-)-poly(benzyl β-malate) which degrades to L(-)-malic acid in alkaline aqueous medium. Configurational structures were investigated by ¹³C NMR (75.47 MHz) and splittings of stereosensitive resonances are discussed in terms of stereosequence distributions. Results show that stereocopolymerization proceeds through random addition of chiral repeating units according to Bernoullian statistics. In order to exemplify the effect of enantiomeric composition on physical properties, thermal characteristics have been investigated by DSC for the whole series of stereocopolymers. Samples with more than 70% (S) or (R)-units are semicrystalline. Melting temperature and crystallinity increase with stereoregularity.     

由Mg(nBu)2制备的MgCl2载体催化剂的烯烃聚合动力学

讨论了催化剂ＣＤ，ＣＭ和ＣＤ－Ａ的乙烯和丙烯聚合动力学曲线，只有催化剂ＣＤ在丙烯聚合时加入外给电子体其动力学曲线为上升－衰减型，其它均为衰减型，催化剂ＣＤ丙烯聚合活性高于催化剂ＣＭ。丙烯聚合时加入外给电子体使总活性中心浓度降低，但等规中心浓度反应增加。同种载体制备的催化剂ＣＤ和ＣＤ－Ａ载钛过程中加入内给电子体，可使其聚合活性增加，在一定范围内，随Ａｌ／Ｔｉ增加，在一定范围内，随Ａｌ／Ｔｉ增加，催化     

Equivalence of the microscopic and macroscopic models of chromatography: stochastic-dispersive versus lumped kinetic model

The microscopic model of chromatography is a stochastic model that consists of two fundamental processes: (i) the random migration of the molecules in the mobile phase, and (ii) the random adsorption-desorption of molecules on the stationary phase contained in a chromatographic column. The diffusion and drift of the molecules in the mobile phase is described with a simple one-dimensional random walk. The adsorption-desorption process is modeled by a Poisson process that assumes exponential sojourn times of the molecules in both the mobile and the stationary phases. The microscopic, or molecular model of chromatography studied here turns out to be identical to the macroscopic lumped kinetic model of chromatography, whose solution is well known in chromatography. A complete equivalence of the two models is established via the identical expressions they provide for the band profiles.     

Steady State and Equilibrium in Reversible Copolymerization at Constant Comonomer Concentrations

Selected aspects of copolymerization processes carried on at constant comonomer concentrations are analyzed theoretically and modeled by Monte Carlo method. It is confirmed that some combinations of initial parameters lead to stationary conditions of copolymer formation for both irreversible and reversible systems which can be regarded as the first‐order Markov chain process. However, this study shows that for many copolymerization systems the stationary conditions are attained only at high number‐average degree of polymerization DP _n, and for some reversible copolymerizations, attaining equilibrium, stationary conditions are not observed at all. The analysis shows that the chain length distribution (CLD) for copolymerization carried out under steady state conditions at constant comonomer concentrations, equal to equilibrium concentrations for infinitely high DP _n, is approximately described by the modified Bessel and exponential functions. This type of CLD is analytically proved and confirmed by the Monte Carlo simulations for the analogous homopolymerization process.     

Polymerization of coordinated monomers. XVIII. Sequence distribution in methyl acrylate–styrene copolymers prepared in the presence of zinc chloride

Methyl acrylate and styrene have been copolymerized in the presence of zinc chloride either by photoinitiation or spontaneously. The copolymerization mechanism is investigated by analyses of copolymers composition and monomer sequence distribution. The resulting copolymers are not always alternating, their composition being dependent especially on the monomer feed ratio. Appreciable deviation to higher methyl acrylate unit content from an equimolar composition occurs at monomer feed fractions of methyl acrylate over 0.7. The larger deviation is induced by higher temperature, by photoirradiation, and by greater dilution of the reaction mixture with toluene. The ¹³C-NMR spectrum of the alternating copolymer shows a sharp singlet at the carbonyl region, whereas the spectra of random copolymers prepared by benzoyl peroxide initiation at 60°C show a triplet splitting at the carbonyl carbon region, irrespective of copolymer composition. The relative intensities of the triplet peaks for the random copolymers are in good correspondence to the contents of triad sequences calculated by means of conventional radical copolymerization theory. These results clearly indicate that the carbonyl splitting is caused predominantly by variation of the monomer sequence and not by variation of the stereosequence. The monomer sequence distribution in the copolymers is thus directly and quantitatively measured from the split carbonyl resonance. Although the same triplet splitting appears in the spectra of methyl acrylate–rich copolymers prepared in the presence of zinc chloride at high feed ratios (>0.7) of methyl acrylate, the relative intensities of the split peaks do not fit the sequence distributions of random copolymers calculated by means of the Lewis–Mayo equation. The copolymerization yielding these peculiar sequences and the alternating sequence in the presence of zinc chloride is fully comprehended by a copolymerization mechanism proceeding between two active coordinated monomers, i.e., the ternary molecular complex composed of zinc chloride, methyl methacrylate, and styrene, and the binary molecular complex composed of zinc chloride and methyl methacrylate.     

Effects of chemical exchange and self-diffusion on the spin-echo signals in two-spin systems

A theoretical study is reported for nuclear spin echo in a two-spin system. It is assumed that the decay of the echo signals is due to translational diffusion of the molecules and to chemical exchange between magnetically distinct nuclei. The behavior of the spin system is described by McConnell's equations, as modified to include diffusion. The method of stationary Markov processes is applied to solve the equations of motion. Theoretical explanations are given for spin-echo observations.     

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.     

Replica exchange and expanded ensemble simulations as Gibbs sampling: simple improvements for enhanced mixing

The widespread popularity of replica exchange and expanded ensemble algorithms for simulating complex molecular systems in chemistry and biophysics has generated much interest in discovering new ways to enhance the phase space mixing of these protocols in order to improve sampling of uncorrelated configurations. Here, we demonstrate how both of these classes of algorithms can be considered as special cases of Gibbs sampling within a Markov chain Monte Carlo framework. Gibbs sampling is a well-studied scheme in the field of statistical inference in which different random variables are alternately updated from conditional distributions. While the update of the conformational degrees of freedom by Metropolis Monte Carlo or molecular dynamics unavoidably generates correlated samples, we show how judicious updating of the thermodynamic state indices--corresponding to thermodynamic parameters such as temperature or alchemical coupling variables--can substantially increase mixing while still sampling from the desired distributions. We show how state update methods in common use can lead to suboptimal mixing, and present some simple, inexpensive alternatives that can increase mixing of the overall Markov chain, reducing simulation times necessary to obtain estimates of the desired precision. These improved schemes are demonstrated for several common applications, including an alchemical expanded ensemble simulation, parallel tempering, and multidimensional replica exchange umbrella sampling.     

Markov processes follow from the principle of maximum caliber

Markov models are widely used to describe stochastic dynamics. Here, we show that Markov models follow directly from the dynamical principle of maximum caliber (Max Cal). Max Cal is a method of deriving dynamical models based on maximizing the path entropy subject to dynamical constraints. We give three different cases. First, we show that if constraints (or data) are given in the form of singlet statistics (average occupation probabilities), then maximizing the caliber predicts a time-independent process that is modeled by identical, independently distributed random variables. Second, we show that if constraints are given in the form of sequential pairwise statistics, then maximizing the caliber dictates that the kinetic process will be Markovian with a uniform initial distribution. Third, if the initial distribution is known and is not uniform we show that the only process that maximizes the path entropy is still the Markov process. We give an example of how Max Cal can be used to discriminate between different dynamical models given data.     

A Bayesian method for construction of Markov models to describe dynamics on various time-scales

The dynamics of many biological processes of interest, such as the folding of a protein, are slow and complicated enough that a single molecular dynamics simulation trajectory of the entire process is difficult to obtain in any reasonable amount of time. Moreover, one such simulation may not be sufficient to develop an understanding of the mechanism of the process, and multiple simulations may be necessary. One approach to circumvent this computational barrier is the use of Markov state models. These models are useful because they can be constructed using data from a large number of shorter simulations instead of a single long simulation. This paper presents a new Bayesian method for the construction of Markov models from simulation data. A Markov model is specified by (τ,P,T), where τ is the mesoscopic time step, P is a partition of configuration space into mesostates, and T is an N(P)×N(P) transition rate matrix for transitions between the mesostates in one mesoscopic time step, where N(P) is the number of mesostates in P. The method presented here is different from previous Bayesian methods in several ways. (1) The method uses Bayesian analysis to determine the partition as well as the transition probabilities. (2) The method allows the construction of a Markov model for any chosen mesoscopic time-scale τ. (3) It constructs Markov models for which the diagonal elements of T are all equal to or greater than 0.5. Such a model will be called a "consistent mesoscopic Markov model" (CMMM). Such models have important advantages for providing an understanding of the dynamics on a mesoscopic time-scale. The Bayesian method uses simulation data to find a posterior probability distribution for (P,T) for any chosen τ. This distribution can be regarded as the Bayesian probability that the kinetics observed in the atomistic simulation data on the mesoscopic time-scale τ was generated by the CMMM specified by (P,T). An optimization algorithm is used to find the most probable CMMM for the chosen mesoscopic time step. We applied this method of Markov model construction to several toy systems (random walks in one and two dimensions) as well as the dynamics of alanine dipeptide in water. The resulting Markov state models were indeed successful in capturing the dynamics of our test systems on a variety of mesoscopic time-scales.     

Theory for a wavelet analysis of electrochemical noise in the laplace space with use of two operational frequencies

The image of a random process in the Laplace space may be viewed as resulting from use of a oneway continuous wavelet transform with an exponential as the basic function, i.e. as resulting from the application of the Laplace wavelet. If the Laplace-wavelet variance of an electrochemical noise allows one to determine the Laplace transform of a time-correlation function, i.e. a factual operational spectral density of the noise, then the covariance of two Laplace waveletes of an electrochemical noise, each of which corresponds to its own operational frequency, allows one to verify a local consistency of the initial experimental noise data. The Laplace waveletes are applied rather widely. In fact, any stationary random process and stationary random time sequence can be described with operational noise spectra. Dedicated to the ninetieth anniversary of Ya.M. Kolotyrkin’s birth.     

Random frog: An efficient reversible jump Markov Chain Monte Carlo-like approach for variable selection with applications to gene selection and disease classification

The identification of disease-relevant genes represents a challenge in microarray-based disease diagnosis where the sample size is often limited. Among established methods, reversible jump Markov Chain Monte Carlo (RJMCMC) methods have proven to be quite promising for variable selection. However, the design and application of an RJMCMC algorithm requires, for example, special criteria for prior distributions. Also, the simulation from joint posterior distributions of models is computationally extensive, and may even be mathematically intractable. These disadvantages may limit the applications of RJMCMC algorithms. Therefore, the development of algorithms that possess the advantages of RJMCMC methods and are also efficient and easy to follow for selecting disease-associated genes is required. Here we report a RJMCMC-like method, called random frog that possesses the advantages of RJMCMC methods and is much easier to implement. Using the colon and the estrogen gene expression datasets, we show that random frog is effective in identifying discriminating genes. The top 2 ranked genes for colon and estrogen are Z50753, U00968, and Y10871_at, Z22536_at, respectively. (The source codes with GNU General Public License Version 2.0 are freely available to non-commercial users at: http://code.google.com/p/randomfrog/.)     

Time series analysis of collective motions in proteins

The dynamics of alpha-amylase inhibitor tendamistat around its native state is investigated using time series analysis of the principal components of the C(alpha) atomic displacements obtained from molecular dynamics trajectories. Collective motion along a principal component is modeled as a homogeneous nonstationary process, which is the result of the damped oscillations in local minima superimposed on a random walk. The motion in local minima is described by a stationary autoregressive moving average model, consisting of the frequency, damping factor, moving average parameters and random shock terms. Frequencies for the first 50 principal components are found to be in the 3-25 cm(-1) range, which are well correlated with the principal component indices and also with atomistic normal mode analysis results. Damping factors, though their correlation is less pronounced, decrease as principal component indices increase, indicating that low frequency motions are less affected by friction. The existence of a positive moving average parameter indicates that the stochastic force term is likely to disturb the mode in opposite directions for two successive sampling times, showing the modes tendency to stay close to minimum. All these four parameters affect the mean square fluctuations of a principal mode within a single minimum. The inter-minima transitions are described by a random walk model, which is driven by a random shock term considerably smaller than that for the intra-minimum motion. The principal modes are classified into three subspaces based on their dynamics: essential, semiconstrained, and constrained, at least in partial consistency with previous studies. The Gaussian-type distributions of the intermediate modes, called "semiconstrained" modes, are explained by asserting that this random walk behavior is not completely free but between energy barriers.     

Discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise

A discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise is developed. Based on the discrete version of Wiener-Khinchin theorem, the theoretical discrete Chebyshev spectrum for the Markov random process is calculated. It is characterized by two parameters: the dispersion and the relaxation frequency (or relaxation time). The noise of corrosion process and the noise of recording equipment are measured. Using the theoretical Chebyshev spectrum, the Markov parameters were found both for the noise of the corrosion process and for the noise of the measuring equipment.     

新型非对称二醚给电子体丙烯聚合催化剂研究

1,3－丙二醚类化合物为给电子体合成的新一类复相Ziegler－Natta催化剂（Z-N催化剂）,其用于丙烯聚合时,在无需外加给电子体的情况下,可得到高活性的催化剂和高等规度的聚丙烯,催化剂的活性是以邻苯二甲酸二丁酯为内给电子体合成的Z-N催化剂的2～3倍,且得到聚丙烯的等规度大于95％[1-4]．由于1,3－丙二醚类化合物与载体的配位作用较强,不易与AlR3反应,因此在丙烯聚合时无需外给电子体,并能降低反应体系的复杂性,有利于研究活性中心结构和聚合机理［5-7]．以往研究均采用对称结构的 1,3－丙二醚类化合物作内给电子体［2-7],其结构如 Scheme 1 所示．本文采用一种新的     

Separation of Poly(propylene) Samples According to Tacticity Using a Hypercarb Column

Summary: Samples of polypropylene having different stereoregularities, i.e., differing in the isotactic or syndiotactic stereosequence distribution, were separated by means of high-temperature gradient adsorption liquid chromatography. The porous graphite was used as stationary phase in the column packing (Hypercarb®). Predominantly isotactic samples eluted in 1-decanol, while predominantly syndiotactic samples eluted in a binary gradient composed of 1-decanol and 1,2,4-trichlorobenzene. Their elution volumes increased with the average content of the syndiotactic units (racemo dyads mole fraction as determined with the NMR spectroscopy) in the samples. Thus these chromatographic separations represent a new method for the analysis and characterization of stereoregular polyolefins. It requires substantially less time and solvents than the commonly used methods.     

Classical theory of collisional entropy production

A classical dynamical theory of elementary collision processes is formulated in analogy to the quantum theory of the dynamical scattering matrix, which can be defined for a pure quantum stationary scattering state. The elements of this matrix are probability amplitudes for transitions between internal states defined for given values of a reaction coordinate. The squared magnitudes of these amplitudes, modeled in the proposed classical theory, define normalized internal state population distributions suitable for information theoretical analysis. Statistical entropy and surprisal are defined as dynamical functions of a reaction coordinate. This formalism differs fundamentally from concepts based on the classical Liouville equation.