On Countable Mixtures of Bivariate Binomial Distributions

A unified treatment is given for mixtures of bivariate binomial distributions with respect to their index parameter(s). The use of probability generating functions is employed and a number of interesting properties including probabilities, factorial moments, factorial cumulants and conditional distributions are derived. Five classes of such mixtures are examined and several well known bivariate discrete distributions are used as illustrative examples. Biological applications are indicated including the fit of three bivariate distributions to an actual set of human family data.     

The Structure of Compounded Trivariate Binomial Distributions

A family of trivariate binomial mixtures with respect to their exponent parameter is introduced and its structure is studied by the use of probability generating functions. Expressions for probabilities, factorial moments and factorial cumulants are given. Conditional distributions are also examined. Illustrative examples include the trivariate Poisson, binomial, negative binomial and modified logarithmic series distributions. In addition, properties of the compounded trivariate Poisson distribution are discussed. Finally biological, medical and ecological applications are indicated.     

Estimating sexual dimorphism by method-of-moments

Estimating the degree of sexual dimorphism is difficult in fossil species because most specimens lack indicators of sex. We present a procedure that estimates sexual dimorphism in samples of unknown sex using method-of-moments. We assume that the distribution of a metric trait is composed of two underlying normal distributions, one for males and one for females. We use three moments around the mean of the combined-sex distribution to estimate the means and the common standard deviation of the two underlying distributions. This procedure has advantages over previous methods: it is relatively simple to use, specimens need not be assigned to sex a priori, no reference to living species analogs is required, and the method provides conservative estimates of dimorphism under a variety of conditions. The method performs best when the male and female distributions overlap minimally but also works well when overlap is substantial. Simulations indicate that this relatively simple method is more accurate and reliable than previous methods for estimating dimorphism. © 1996 Wiley-Liss, Inc.     

Analysis of the probability distribution of small random samples by nonlinear fitting of integrated probabilities

Small random samples of biochemical and biological data are often representative of complex distribution functions and are difficult to analyze in detail by conventional means. The common approaches reduce the data to a few representative parameters (such as their moments) or combine the data into a histogram plot. Both approaches reduce the information content of the data. By fitting the empirical cumulative distribution function itself with models of integrated probability distributions, the information content of the raw data can be fully utilized. This approach, distribution analysis by nonlinear fitting of integrated probabilities, allows analysis of normally distributed samples, truncated data sets, and multimodal distributions with a single, powerful data processing procedure.     

Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns?1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.     

On the stochastic theory of compartments: Solution forn-compartment systems with irreversible,time-dependent transition probabilities

The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.     

A Report on a Monte Carlo Simulation of the Sampling Distribution of the Incidence Test and its Two Related Tests

A large-scale (5000 throws) Monte Carlo simulation experiment was carried out to study the nature of the sampling distributions of the incidence test and its two related tests, the FRIEDMAN test and the dual test, and to evaluate the goodness of the proposed gamma approximation relative to the conventional chi-square approximation. For all three tests, on the basis of the experimental results obtained for k = 3, 4, 5, and N = k(1) (120/k), the gamma approximation should be preferred over the chi-square. Exact companion tabulations show that use of the chi-square approximation entails an appreciable conservative bias. Extensive listings are provided of the moments of the experimental distributions of the three test statistics, and of the observed distributions of the corresponding gamma probabilities. The simulation experiment was run on a UNIVAC 1108 large-scale computer system.     

Analysis of distributions of amino acids and amino acid pairs in human tumour necrosis factor precursor and its eight mutations according to a random mechanism

In this study, we use the random principle to analyse the distributions of amino acids and amino acid pairs in human tumour necrosis factor precursor (TNF-!) and its eight mutations, to compare the measured distribution probability with the theoretical distribution probability and to rank the measured distribution probability against the theoretical distribution probability. In this way, we can suggest that distributions with a high random rank should not be deliberately evolved and conserved and those with a low random rank should be deliberately evolved and conserved in human TNF-!. An increased distribution probability in a mutation means probabilistically that the mutation is more likely to occur spontaneously, whereas a decreased distribution probability in a mutation means probabilistically that the mutation is less likely to occur spontaneously and perhaps is more related to a certain cause. The results, for example, show that the distributions of 30% of the amino acids are identical with their probabilistic simplest distributions, and the distributions of some of the remaining amino acids are very close to their probabilistic simplest distributions. With respect to probabilities of distributions of amino acids in mutations, the results show that mutations lead to an increase in eight probabilities, which are thus more likely to occur. Eight probabilities decrease and are thus less likely to occur. With respect to the random ranks against the theoretical probabilities of distributions of amino acids, the results show that mutations lead to an increase in seven and a decrease in seven probabilities, with two probabilities unchanged.     

Protein denaturant binding polynomials

We show how moments of the denaturant binding distribution function can be extracted from experimental data on the denaturation of a protein as a function of the concentration of denaturant and how in turn these moments can be used to construct the denaturant binding distribution function. This approach is similar to our recent work on using the maximum-entropy method to construct ligand-binding distributions from moments obtained from titration curves for nucleic acids and proteins. As an example we take literature data on the denaturation of ferro- and ferricytochrome c by guanidine hydrochloride and from it construct the denaturant binding polynomial and binding distribution function for the unfolded protein.     

Maximum-entropy calculation of free energy distributions for two forms of myoglobin

The temperature dependence of the heat capacity of myoglobin depends dramatically on pH. At low pH (near 4.5), there are two weak maxima in the heat capacity at low and intermediate temperatures, respectively, whereas at high pH (near 10.7), there is one strong maximum at high temperature. Using literature data for the low-pH form (Hallerbach and Hinz, 1999) and for the high-pH form (Makhatadze and Privalov, 1995), we applied a recently developed technique (Poland, 2001d) to calculate the free energy distributions for the two forms of the protein. In this method, the temperature dependence of the heat capacity is used to calculate moments of the protein enthalpy distribution function, which in turn, using the maximum-entropy method, are used to construct the actual distribution function. The enthalpy distribution function for a protein gives the fraction of protein molecules in solution having a given value of the enthalpy, which can be interpreted as the probability that a molecule picked at random has a given enthalpy value. Given the enthalpy distribution functions at several temperatures, one can then construct a master free energy function from which the probability distributions at all temperatures can be calculated. For the high-pH form of myoglobin, the enthalpy distribution function that is obtained exhibits bimodal behavior at the temperature corresponding to the maximum in the heat capacity (Poland, 2001a), reflecting the presence of two populations of molecules (native and unfolded). For this form of myoglobin, the temperature evolution of the relative probabilities of the two populations can be obtained in detail from the master free energy function. In contrast, the enthalpy distribution function for the low-pH form of myoglobin does not show any special structure at any temperature. In this form of myoglobin the enthalpy distribution function simply exhibits a single maximum at all temperatures, with the position of the maximum increasing to higher enthalpy values as the temperature is increased, indicating that in this case there is a continuous evolution of species rather than a shift between two distinct population of molecules.     

Fluorescence polarization in a planar array of pigment molecules: theoretical treatment and application to flavins incorporated into artificial membranes.

A quantitative fluorescence polarization theory of molecules bound to two-dimensional plane layers has been developed when the electronic transition moments of absorption and emission are parallel within the fluorescent molecules. The transition moments are assumed to be in preferred orientation with respect to the normal to the plane and to be randomly oriented within the plane (rotational symmetry with the normal as axis of symmetry). Three basic model distributions of transition moments are investigated quantitatively. These model distributions represent a simplification but in most cases may be expected to describe reality with sufficient accuracy. For all distributions, two cases of different mobility of molecules are treated: (a) the lifetime of fluorescence is small compared with the characteristic relaxation time of the distribution, and (b) the lifetime of fluorescence is long, so that a complete reorientation of transition moments during the excited state can take place. From the quantitative calculations four characteristic quantities are derived, which are appropriate for the analysis of experimental data. Experiments are carried out with phosphatidylcholine bilayer membranes which contain three differently substituted amphiphilic flavins. All three flavins yield similar data. Their analyses predict free and fast mobility of the flavin chromophore.     

A Class of Distributions in the First Emptiness of a Semi-Infinite Reservoir

This paper considers the probability distribution of the volume of a certain substance (e.g. river discharge, rainfall, deposites of clay, organism, etc.) that flows into a semi-infinite reservoir before its first emptiness for continuous and homogeneous input process when the substance is released at unit rate per unit of time. A few moments of the distribution have been computed. A generalized gamma, and a generalized exponential distributions as particular cases are also discussed. Some possible applications of the generalized negative exponential distribution have been mentioned. These distributions are in fact the continuous analogues of the discrete LAGRANGE distributions recently considered by JAIN and others.     

Linear programming analysis of VA/Q distributions: limits on central moments

Linear programming examines the boundaries of infinite sets. We used this method with the multiple-inert gas-elimination technique to examine the central moments and arterial blood gases of the infinite family of ventilation perfusion (VA/Q) distributions that are compatible with a measured inert gas-retention set. A linear program was applied with Monte-Carlo error simulation to theoretical retention data, and 95% confidence intervals were constructed for the first three moments (mean, dispersion, and skew) and the arterial PO2 and PCO2 of all compatible blood flow distributions. Six typical cases were studied. Results demonstrate narrow confidence intervals for both the lower moments and predicted arterial blood gases of all test cases, which widen as moment number or error increase. We conclude that the blood gas composition and basic structure of all compatible VA/Q distributions are tightly constrained and that even subtle changes in this structure, as may occur experimentally, can be identified.     

Relation of transition probabilities in the Leslie model to those from experimental cumulative distributions

We explore the relationship between transition probabilities in the Leslie model and those derived from experimental cumulative distributions. The nature of the two kinds of probabilities are discussed, and a formula derived for converting from one to the other. A numerical example is given to illustrate the differences.     

Ambiguity of the solution of the inverse problem of electroencephalography

It has been shown for homogeneous conducting medium that if different stationary distributions of current sources form identical potential fields on the plane boundary, the difference between these two distributions is the distribution of current sources with all multiple moments equaling zero. Examples of current source distributions are presented which satisfy this condition.     

Spatial scaling of species abundance distributions

Species abundance distributions are an essential tool in describing the biodiversity of ecological communities. We now know that their shape changes as a function of the size of area sampled. Here we analyze the scaling properties of species abundance distributions by using the moments of the logarithmically transformed number of individuals. We find that the moments as a function of area size are well fitted by power laws and we use this pattern to estimate the species abundance distribution for areas larger than those sampled. To reconstruct the species abundance distribution from its moments, we use discrete Tchebichef polynomials. We exemplify the method with data on tree and shrub species from a 50 ha plot of tropical rain forest on Barro Colorado Island, Panama. We test the method within the 50 ha plot, and then we extrapolate the species abundance distribution for areas up to 5 km². Our results project that for areas above 50 ha the species abundance distributions have a bimodal shape with a local maximum occurring for the singleton classes and that this maximum increases with sampled area size.     

Stochastic inequality probabilities for adaptively randomized clinical trials

We examine stochastic inequality probabilities of the form P (X > Y) and P (X > max (Y, Z)) where X, Y, and Z are random variables with beta, gamma, or inverse gamma distributions. We discuss the applications of such inequality probabilities to adaptively randomized clinical trials as well as methods for calculating their values.     

The probabilities of one- and multi-track events for modeling radiation-induced cell kill

In view of the clinical importance of hypofractionated radiotherapy, track models which are based on multi-hit events are currently reinvestigated. These models are often criticized, because it is believed that the probability of multi-track hits is negligible. In this work, the probabilities for one- and multi-track events are determined for different biological targets. The obtained probabilities can be used with nano-dosimetric cluster size distributions to obtain the parameters of track models. We quantitatively determined the probabilities for one- and multi-track events for 100, 500 and 1000 keV electrons, respectively. It is assumed that the single tracks are statistically independent and follow a Poisson distribution. Three different biological targets were investigated: (1) a DNA strand (2 nm scale); (2) two adjacent chromatin fibers (60 nm); and (3) fiber loops (300 nm). It was shown that the probabilities for one- and multi-track events are increasing with energy, size of the sensitive target structure, and dose. For a 2 × 2 × 2 nm³ target, one-track events are around 10,000 times more frequent than multi-track events. If the size of the sensitive structure is increased to 100–300 nm, the probabilities for one- and multi-track events are of the same order of magnitude. It was shown that target theories can play a role for describing radiation-induced cell death if the targets are of the size of two adjacent chromatin fibers or fiber loops. The obtained probabilities can be used together with the nano-dosimetric cluster size distributions to determine model parameters for target theories.     

Distributions of positive signals in pyrosequencing

Pyrosequencing is one of the important next-generation sequencing technologies. We derive the distribution of the number of positive signals in pyrograms of this sequencing technology as a function of flow cycle numbers and nucleotide probabilities of the target sequences. As for the distribution of sequence length, we also derive the distribution of positive signals for the fixed flow cycle model. Explicit formulas are derived for the mean and variance of the distributions. A simple result for the mean of the distribution is that the mean number of positive signals in a pyrogram is approximately twice the number of flow cycles, regardless of nucleotide probabilities. The statistical distributions will be useful for instrument and software development for pyrosequencing and other related platforms.     

Coding DNA sequences: statistical distributions

The base distributions in coding DNA sequences (CDS) are investigated. We explore the scaling properties of the 4-dimensional directed random walk and compare them with that for the DNA sequences. Inference from these observation are, however, contradicted by alternate analysis using factorial moments. To resolve this conflict we look directly at the nucleotide base distributions. In all the cases the base distributions change from gaussian to non-gaussian as the scale size is increased. The CDS, therefore, have nucleotide distributions different from the random.