The S-Distribution A Tool for Approximation and Classification of Univariate,Unimodal Probability Distributions

In many statistical applications a data set needs to be evaluated but there is no solid information about which probability distribution might be most appropriate. Typical solutions to this problems are: to make assumptions that facilitate mathematical treatment; to use a family of distribution functions that contains all relevant distributions as special cases; or, to employ nonparametric methods. All three solutions have disadvantages since assumptions are usually difficult to justify, families of distributions contain too many parameters to be of practical use, and nonparametric methods make it difficult to characterize data in a succinct quantitative form. The S-distribution introduced here is a compromise between the conflicting goals of simplicity in analysis and generality in scope. It is characterized by four parameters, one of which reflects its location, the second one its spread, and the remaining two its shape; transformation to a standard form reduces the number of free parameters to two. Cumulatives and densities are computed numerically in fractions of seconds, key features like quantiles and moments are easily obtained, and results can be presented in terms of parameter values. The S-distribution rather accurately models different distribution functions, including central and noncentral distributions, and thus competes in flexibility with some distribution families. As an approximation, the S-distribution provides a graphical method for demonstrating relationships between distributions, such as the relationships between central F, χ² and χ^?2 or central and noncentral t, χ^-1, and normal.     

Scalability Properties of the S-Distribution

Previous work has shown that the S-distribution is a valuable tool for data analysis and for the classification of continuous and discrete distribution functions. The distribution has four parameters: one determines its location, one is related to the variance, and two control its shape. The distributional structure allows symmetry as well as skewness to the right or the left. This offers great flexibility and, among other analyses, facilitates the simultaneous investigation of random variables whose distributions differ in shape. The present paper demonstrates that mean, variance, and quantiles of any S-distribution can be computed with simple algebraic operations from corresponding properties of a standard S-distribution, sSd , which is characterized by only the two shape parameters. This scalability is comparable with the use of z-scores when dealing with normal distributions.     

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.     

Discrete Distributions of Adenosine Receptors in Mammalian Retina

Binding sites for both the adenosine A1 receptor agonists [3H]phenylisopropyladenosine and [3H]cyclohexyladenosine and the mixed A1-A2 agonist N-[3H]ethylcarboxamidoadenosine [( 3H]NECA) were localized in rabbit and mouse retinas using autoradiographic techniques. These two classes of agonists bound to very different regions of mammalian retinas. A1 agonist binding was localized to the inner retina, particularly over the inner plexiform layer. The binding of [3H]NECA was observed primarily over the retinal pigmented epithelium and the outer and inner segments of photoreceptors. [3H]NECA labeling was not affected either by including a low concentration of unlabeled A1 agonist or by pretreating tissue with N-ethylmaleimide to inhibit ligand binding at A1 sites. While virtually all of the [3H]NECA binding was displaced by an excess of unlabeled NECA, displacement with antagonist or a large excess of cyclohexyladenosine revealed that approximately 30% of the [3H]NECA binding was at non-A1,A2 sites. The majority of the binding in the outer retina thus labeled A2 receptor sites. The unique localizations of the two classes of adenosine receptors suggest different functions in visual processing.     

A Suprasystem of Probability Distributions

A set of differential equations is described whose solutions represent a general system of probability distribution functions. Previously reported systems of such distribution functions are special cases of this general system. The differential equations can be used to represent distribution functions and generate their related statistics in cases for which no simple formula for the distribution is known. The generality of this suprasystem of distribution functions and its potential utility are discussed.     

General Mathematical Frame for Open or Closed Biomembranes (Part I): Equilibrium Theory and Geometrically Constraint Equation

This paper aims at constructing a general mathematical frame for the equilibrium theory of open or closed biomembranes. Based on the generalized potential functional, the equilibrium differential equation for open biomembrane (with free edge) or closed one (without boundary) is derived. The boundary conditions for open biomembranes are obtained. Besides, the geometrically constraint equation for the existence, formation and disintegration of open or closed biomembranes is revealed. The physical and biological meanings of the equilibrium differential equation and the geometrically constraint equation are discussed. Numerical simulation results for axisymmetric open biomembranes show the effectiveness and convenience of the present theory.     

A Deterministic Approximation to the Number of Mutants in Growing Clones

The use of an approximation to the median of the Poisson distribution to represent each occurrence of mutations in a growing clone permits the prediction of the number of mutants per clone without the limitations imposed by more heuristic expressions. Its application to the evaluation of mutation rates yields results comparable to those obtained by fluctuation analysis.     

Genome-wide and molecular evolution analysis of the subtilase gene family in Vitis vinifera

Background

Vitis vinifera (grape) is one of the most economically significant fruit crops in the world. The availability of the recently released grape genome sequence offers an opportunity to identify and analyze some important gene families in this species. Subtilases are a group of subtilisin-like serine proteases that are involved in many biological processes in plants. However, no comprehensive study incorporating phylogeny, chromosomal location and gene duplication, gene organization, functional divergence, selective pressure and expression profiling has been reported so far for the grape.

Results

In the present study, a comprehensive analysis of the subtilase gene family in V. vinifera was performed. Eighty subtilase genes were identified. Phylogenetic analyses indicated that these subtilase genes comprised eight groups. The gene organization is considerably conserved among the groups. Distribution of the subtilase genes is non-random across the chromosomes. A high proportion of these genes are preferentially clustered, indicating that tandem duplications may have contributed significantly to the expansion of the subtilase gene family. Analyses of divergence and adaptive evolution show that while purifying selection may have been the main force driving the evolution of grape subtilases, some of the critical sites responsible for the divergence may have been under positive selection. Further analyses of real-time PCR data suggested that many subtilase genes might be important in the stress response and functional development of plants.

Conclusions

Tandem duplications as well as purifying and positive selections have contributed to the functional divergence of subtilase genes in V. vinifera. The data may contribute to a better understanding of the grape subtilase gene family.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2164-15-1116) contains supplementary material, which is available to authorized users.     

Approximation of continuous growth of Cephalotaxus harringtonia plant cell cultures using fed-batch operation

In Cephalotaxus harringtonia plant cell cultures, periods of batch growth that are limited by hexose uptake are too short to make an accurate estimate of the Monod saturation constant. Continuous cultures are infeasible on a laboratory scale, and semicontinuous cultures require too frequent sampling. Fed-batch operation, consisting of intermittent removal from a culture that is fed continuously, was investigated as a possible solution to these problems. For a constant feed rate, computer simulations showed that a steady state can be achieved which is useful for studying growth at different specific growth rates. In terms of the dilution rate it was confirmed that the operation is essentially equivalent to continuous culture when the samples represent a small fraction of the total culture volume. Experiments with glucose or fructose as the carbon source were carried out in shake flasks fed by a multichannel syringe pump. Results indicate that Monod kinetics based on medium glucose levels cannot adequately describe growth under these conditions. Monod's expression for specific growth rate using internal glucose concentration gives an improved correlation.     

Use of the Score Statistic for Testing Goodness-of-Fit of Some Generalised Distributions

The use of the score statistic to test whether a generalised distribution gives an improved fit over a non-generalised distribution is recommended. The score statistic for a generalised exponential family is derived. Several specific examples are given.     

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous ‘singularity models’ is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a ‘posterior tilt,’ and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 μm/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system.     

A Comment on Maximum Likelihood Estimation for Finite Mixtures of Distributions

Iterative numerical methods are necessary to find the maximum likelihood estimates for finite mixture distributions. This paper shows that it will often be possible to analytically reduce the number of equations that must ultimately be solved numerically. Such a reduction in dimensionality has not generally been used, or sought after, for mixture distributions. Yet such results are easily derived when each mixture component is assumed to be from the same parametric model within the exponential family.     

Numerical Simulation of Extinction Spectra of Plasmonically Coupled Nanospheres Using Discrete Dipole Approximation: Influence of Compositional Asymmetry

We demonstrate plasmon coupling phenomenon between equivalent (homodimer) and non-equivalent (heterodimer) spherical shape noble metal nanoparticle (Ag, Au and Al). A systematic comparison of surface plasmon resonance (SPR) and extinction properties of various configurations (monomer, homodimer and heterodimer) has been investigated to observe the effect of compositional asymmetry. Numerical simulation has been done by using discrete dipole approximation method to study the optical properties of plasmonically coupled metal nanoparticles (MNPs). Plasmon coupling between similar nanoparticles allows only higher wavelength bonding plasmon mode while both the plasmon modes lower wavelength antibonding mode as well as higher wavelength bonding mode in the case of heterodimer. Au monomer of radius 50 nm shows resonance peak at 518 nm while plasmon coupling between Au-Au homodimer results in a spectral red shift around 609 nm. Au-Ag plasmonic heterodimer (radius 50 nm) reveals two resonant modes corresponding to higher energy antibonding mode (422 nm) as well as lower energy bonding mode (533 nm). Further, we have shown that interparticle edge-to-edge separation is the most significant parameter affecting the surface plasmon resonances of MNPs. As the inter particle separation decreases, resonance wavelength shows red spectral shift which is maximum for the touching condition. It is shown that plasmon coupling is a reliable strategy to tune the SPR.

     

Estimation of Sheet Thickness Distributions from Linear and Plane Sections

A biological “membrane” or a “barrier” can often be modelled by a sheet, namely a portion of space comprised between two smooth, quasi-parallel faces, such that sheet thickness is in general variable but very small in relation to face extent. Sheet thickness measurements are often of interest to the morphometrist, the physiologist and the pathologist. In the present paper, some stereological techniques are developed for estimating true sheet thickness distributions, and their moments, from the apparent thickness or intercept length measurements obtained, respectively, via random plane or linear sections. An important premise for the models to work is that of isotropic random (‘non-preferential’) sheet orientation relative to the sectioning probe.     

Computation of Some Exact Properties of Wald's Sprt when Sampling from a Class of Discrete Distributions

An algorithm is developed for determining exact values of the Operating Characteristic and Average Sample Number functions for Wald's Sequential Probability Ratio Test (SPRT) when observations are drawn from a discrete, Koopman-Darmois density with positive probability on the nonnegative integers. The sample size distribution and percentiles of sample size are also considered. An example is given for the negative binomial distribution.     

Modeling antibiotic resistance in hospitals: the impact of minimizing treatment duration

Infections caused by antibiotic-resistant pathogens are a global public health problem. Numerous individual- and population-level factors contribute to the emergence and spread of these pathogens. An individual-based model (IBM), formulated as a system of stochastically determined events, was developed to describe the complexities of the transmission dynamics of antibiotic-resistant bacteria. To simplify the interpretation and application of the model's conclusions, a corresponding deterministic model was created, which describes the average behavior of the IBM over a large number of simulations. The integration of these two model systems provides a quantitative analysis of the emergence and spread of antibiotic-resistant bacteria, and demonstrates that early initiation of treatment and minimization of its duration mitigates antibiotic resistance epidemics in hospitals.