Analysis of Kinetic Intermediates in Single-Particle Dwell-Time Distributions

Many biological and chemical processes proceed through one or more intermediate steps. Statistical analysis of dwell-time distributions from single molecule trajectories enables the study of intermediate steps that are not directly observable. Here, we discuss the application of the randomness parameter and model fitting in determining the number of steps in a stochastic process. Through simulated examples, we show some of the limitations of these techniques. We discuss how shot noise and heterogeneity among the transition rates of individual steps affect how accurately the number of steps can be determined. Finally, we explore dynamic disorder in multistep reactions and show that the phenomenon can obscure the presence of rate-limiting intermediate steps.     

Sample Sizes for Testing Equality of Location Parameters of Two Exponential Distributions

A table for determining minimal sample sizes n₁ = n₂ = n for testing the hypothesis of equality of location parameters a₄ and a₂ of two two-parametric exponential distributions for a first kind risk α=0,01 and 0,05 are given in such a way that the second kind risk β≦β₀ as long as |a₁–a>₂|>d.     

Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

In this article we study some properties of a new family of distributions, namely Exponentiated Exponential distribution, discussed in Gupta , Gupta , and Gupta (1998). The Exponentiated Exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed that many properties of this new family are quite similar to those of a Weibull or a gamma family, therefore this distribution can be used as a possible alternative to a Weibull or a gamma distribution. We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.     

Estimation of Exponential Parameters in the Presence of Outliers

In the present paper we discuss the estimation of the parameter and survival function of an exponential population. It is assumed that the sample based on which the estimation is done, contains k outliers that are also exponentially distributed with parameter, a constant multiple of that of the original distribution.     

Some Applications of the Family of Discrete Exponential Distributions in Biology and Toxicology

In this paper the relationship between the number of males and females in a community, between the number of boys and girls in a family, between the number of animals trapped and untrapped in an animal experiment, between the number of infected and uninfected in an epidemiological study, between the number of mutagens and non-mutagens in a given sample of toxicological experiments, and others, is studied. The general expression for the correlation coefficient, ?, is obtained when N, the community size N, has a discrete exponential family distribution recently studied by Janardan (1980). This is then specialized to some well known discrete distributions.     

The Relationship Between Two Statistical Tests for the Comparison of Two Exponential Distributions

To compare two exponential distributions with or without censoring, two different statistics are often used; one is the F test proposed by COX (1953) and the other is based on the efficient score procedure. In this paper, the relationship between these tests is investigated and it is shown that the efficient score test is a large-sample approximation of the F test.     

Estimation of Parameters in Mixtures of Distributions Arising in Cultivation Problems

In many processes one has to describe events in which given particles or elements divide themselves so that these elements are to observe only after the division. In biological problems one meets such a situation i.e. in the cultivation of bacteria. In this paper we investigate a mathematical model which describes such a phenomenon. We give estimators for the expectation μ and the variance σ² of the distribution of the undivided particles. Therefore we give a best linear unbiased estimator for μ and an estimator μ where a clustering of the observation is essentially. In an example for the judgment of a cultivation method we demonstrate the given estimation procedures.     

How Powerful is the Dwell-Time Analysis of Multichannel Records?

Exact algorithms for the kinetic analysis of multichannel patch-clamp records require hours to days for a single record. Thus, it may be reasonable to use a fast but less accurate method for the analysis of all data sets and to use the results for a reanalysis of some selected records with more sophisticated approaches. For the first run, the tools of single-channel analysis were used for the evaluation of the single-channel rate constants from multichannel dwell-time histograms. This could be achieved by presenting an ensemble of single channels by a ``macrochannel' comprising all possible states of the ensemble of channels. Equations for the calculations of the elements of the macrochannel transition matrix and for the steady-state concentrations for individual states are given. Simulations of multichannel records with 1 to 8 channels with two closed and one open states and with 2 channels with two open and two closed states were done in order to investigate under which conditions the one-dimensional dwell-time analysis itself already provides reliable results. Distributions of the evaluated single-channel rate constants show that a bias of the estimations of the single-channel rate constants of 10 to 20% has to be accepted. The comparison of simulations with signal-to-noise ratios of SNR = 1 or SNR = 25 demonstrates that the major problem is not the convergence of the fitting routine, but failures of the level detector algorithm which creates the dwell-times distributions from noisy time series. The macrochannel presentation allows the incorporation of constraints like channel interaction. The evaluation of simulated 4-channel records in which the rate-constant of opening increased by 20% per already open channel could reveal the interaction factor. Received: 9 June 1997/Revised: 28 April 1998     

A Maximum Likelihood Estimator for Shape Parameters of S‐Distributions

The S‐distribution is a four‐parameter distribution that is defined in terms of a differential equation, in which the cumulative is represented as the dependent variable: The article proposes a maximum likelihood estimator for the shape parameters of this distribution.     

种群增长的分段指数模型及其参数估计

本文给出了种群增长的分段指数模型其中N（t）是在时刻t种群的密度,No=N（t0）,r0和rl是群群的内禀增长率,t0是转变点,H（t-t0）=1,t≥t0,H（t-t0）=0,t＜t0．利用非线性模型的正割法（DUD,Doesn’tusederivatives）,可同时确定模型的所有参数（包括交点t0在内）．并用于描述长爪沙鼠种群动态．     

Equivariant Estimation for the Parameters of an Exponential Model Based on Censored Sampling

Lehmann (1983) provides a detailed discussion on equivariant estimation of the parameters of location, scale and location-scale models. Edwin Prabakaran and Chandrasekar (1994) developed simultaneous equivariant estimation approach and illustrated the method with examples. In this paper, we consider exponential models and obtain minimum risk equivariant estimators of the parameters based on Type II censored samples. The equivariant method of estimation is illustrated with biological and reliability applications.     

Bayesian Inference of Genetic Parameters Based on Conditional Decompositions of Multivariate Normal Distributions

It is widely recognized that the mixed linear model is an important tool for parameter estimation in the analysis of complex pedigrees, which includes both pedigree and genomic information, and where mutually dependent genetic factors are often assumed to follow multivariate normal distributions of high dimension. We have developed a Bayesian statistical method based on the decomposition of the multivariate normal prior distribution into products of conditional univariate distributions. This procedure permits computationally demanding genetic evaluations of complex pedigrees, within the user-friendly computer package WinBUGS. To demonstrate and evaluate the flexibility of the method, we analyzed two example pedigrees: a large noninbred pedigree of Scots pine (Pinus sylvestris L.) that includes additive and dominance polygenic relationships and a simulated pedigree where genomic relationships have been calculated on the basis of a dense marker map. The analysis showed that our method was fast and provided accurate estimates and that it should therefore be a helpful tool for estimating genetic parameters of complex pedigrees quickly and reliably.MUCH effort in genetics has been devoted to revealing the underlying genetic architecture of quantitative or complex traits. Traditionally, the polygenic model has been used extensively to estimate genetic variances and breeding values of natural and breeding populations, where an infinite number of genes is assumed to code for the trait of interest (Bulmer 1971; Falconer and Mackay 1996). The genetic variance of a quantitative trait can be decomposed into an additive part that corresponds to the effects of individual alleles and a part that is nonadditive because of interactions between alleles. Attention has generally been focused on the estimation of additive genetic variance (and heritability), since additive variation is directly proportional to the response of selection via the breeder''s equation (Falconer and Mackay 1996, Chap. 11). However, to estimate additive genetic variation and heritability accurately, it can be important to identify potential nonadditive sources in genetic evaluations (Misztal 1997; Ovaskainen et al. 2008; Waldmann et al. 2008), especially if the pedigree being analyzed contains a large proportion of full-sibs and clones, as these in particular give rise to nonadditive genetic relationships (Lynch and Walsh 1998, pp. 145). The polygenic model using pedigree and phenotypic information, i.e., the animal model (Henderson 1984), has been the model of choice for estimating genetic parameters in breeding and natural populations (Abney et al. 2000; Sorensen and Gianola 2002; O′Hara et al. 2008).Recent breakthroughs in molecular techniques have made it possible to create genome-wide, single nucleotide polymorphism (SNP) maps. These maps have helped to uncover a vast amount of new loci responsible for trait expression and have provided general insights into the genetic architecture of quantitative traits (e.g., Valdar et al. 2006; Visscher 2008; Flint and Mackay 2009). These insights can help when calculating disease risks in humans, when attempting to increase the yield from breeding programs, and when estimating relatedness in conservation programs. High-density SNPs of many species of importance to science and agriculture can now be scored quickly and relatively cheaply, for example, in mice (Valdar et al. 2006), chickens (Muir et al. 2008), and dairy cattle (VanRaden et al. 2009).In the analysis of populations of breeding stock, the inclusion of dense marker data has improved the predictive ability (i.e., reliability) of genetic evaluations compared to the traditional phenotype model, both in simulations (Meuwissen et al. 2001; Calus et al. 2008; Hayes et al. 2009) and when using real data (Legarra et al. 2008; VanRaden et al. 2009; González-Recio et al. 2009). Meuwissen et al. (2001) suggested that the effect of all markers should first be estimated, and then summed, to obtain genomic estimated breeding values (GEBVs). An alternative procedure, where all markers are used to compute the genomic relationship matrix (in place of the additive polygenic relationship matrix) has also been suggested (e.g., Villanueva et al. 2005; VanRaden 2008; Hayes et al. 2009); this matrix is then incorporated into the statistical analysis to estimate GEBVs. A comparison of both procedures (VanRaden 2008) yielded similar estimates of GEBVs in cases where the effect of an individual allele was small. In addition, if not all pedigree members have marker information, a combined relationship matrix derived from both genotyped and ungenotyped individuals could be computed; this has been shown to increase the accuracy of GEBVs (Legarra et al. 2009; Misztal et al. 2009). Another plausible option to incorporate marker information is to use low-density SNP panels within families and to trace the effect of SNPs from high-density genotyped ancestors, as suggested by Habier et al. (2009) and Weigel et al. (2009). However, fast and powerful computer algorithms, which can use the marker information as efficiently as possible in the analysis of quantitative traits, are needed to obtain accurate GEBVs from genome-wide marker data.This study describes the development of an efficient Bayesian method for incorporating general relationships into the genetic evaluation procedure. The method is based on expressing the multivariate normal prior distribution as a product of one-dimensional normal distributions, each conditioned on the descending variables. When evaluating the genetic parameters of natural and breeding populations, high-dimensional distributions are often used as prior distributions of various genetic effects, such as the additive polygenic effect (Wang et al. 1993), multivariate additive polygenic effects (Van Tassell and Van Vleck 1996), and quantitative trait loci (QTL) effects via the identical-by-decent matrix (Yi and Xu 2000). A Bayesian framework is adopted to obtain posterior distributions of all unknown parameters, estimated by using Markov chain Monte Carlo (MCMC) sampling algorithms in the software package WinBUGS (Lunn et al. 2000, 2009). By performing prior calculations in the form of the factorized product of simple univariate conditional distributions, the computational time of the MCMC estimation procedure is reduced considerably. This feature permits rapid inference for both the polygenic model and the genomic relationship model. Moreover, the decomposition allows for inbreeding of varying degree, since the correct genetic covariance structure can be inferred into the analysis. In this article, we test the method on two previously published pedigree data sets: phenotype data from a large pedigree of Scots pine, incorporation of information on both additive and dominance genetic relationships (Waldmann et al. 2008); and genomic information obtained from a genome-wide scan of a simulated animal population (Lund et al. 2009).     

Evaluations of Interventions Using Mathematical Models with Exponential and Non-exponential Distributions for Disease Stages: The Case of Ebola

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511–1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.     

R-Factor Mutant Capable of Specifying Hypersynthesis of Penicillinase

The physical characteristics of a mutant, R(M201-2), capable of conferring high and stable ampicillion resistance was analyzed. The R(M201-2) and its parent R-factor deoxyribonucleic acid (DNA) could be isolated as an extrachromosomal and covalently closed circular form. Their buoyant densities were both 1.712 g/cm(3), and their molecular weights were about 82 x 10(6) and 64 x 10(6), respectively, when measured by CsCl and sucrose density gradient analyses. The contour lengths by electron microscopy were 35.9 +/- 0.6 and 31.0 +/- 0.6 mum, respectively. By using the extracted R-factor DNA, the mutant and parent characters were transformable to another Escherichia coli strain. The mutant R factor showed an increased amount of DNA even after conjugal transfer to Proteus. An increase in the size of R-factor DNA was thus considered to be the cause of the high level of ampicillin resistance.     

Coordination of Sex Pili with their Specifying R Factors

A single bacterial cell can simultaneously carry both F-like (fi⁺) and I-like (fi^?) R factors and, when the R factors are de-repressed, most cells produce both F-like and I-like sex pili. These pili can be distinguished immunologically and by their capacity to adsorb different phages¹. The F pilus is the receptor for RNA phages such as MS2 and filamentous DNA phages such as M13. The I pilus is the receptor for other filamentous DNA phages such as If1 and If2. Electron microscopy suggests that these filamentous DNA phages, both F-specific and I-specific, adsorb to the tip of the pilus^2,3.     

The Power of Two-Dimensional Dwell-Time Analysis for Model Discrimination, Temporal Resolution, Multichannel Analysis and Level Detection

Two-dimensional (2D) dwell-time analysis of time series of single-channel patch-clamp current was improved by employing a Hinkley detector for jump detection, introducing a genetic fit algorithm, replacing maximum likelihood by a least square criterion, averaging over a field of 9 or 25 bins in the 2D plane and normalizing per measuring time, not per events. Using simulated time series for the generation of the “theoretical” 2D histograms from assumed Markov models enabled the incorporation of the measured filter response and noise. The effects of these improvements were tested with respect to the temporal resolution, accuracy of the determination of the rate constants of the Markov model, sensitivity to noise and requirement of open time and length of the time series. The 2D fit was better than the classical hidden Markov model (HMM) fit in all tested fields. The temporal resolution of the two most efficient algorithms, the 2D fit and the subsequent HMM/beta fit, enabled the determination of rate constants 10 times faster than the corner frequency of the low-pass filter. The 2D fit was much less sensitive to noise. The requirement of computing time is a problem of the 2D fit (100 times that of the HMM fit) but can now be handled by personal computers. The studies revealed a fringe benefit of 2D analysis: it can reveal the “true” single-channel current when the filter has reduced the apparent current level by averaging over undetected fast gating.