Four Applications of a Bivariate Pareto Distribution

The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate PARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X, rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.     

A polydisperse linear random coil model for the quaternary structure of pig colonic mucin

The distribution of molecular weights for polymeric colonic mucus glycoprotein or ``mucin' isolated and solubilised in the presence of protease inhibitors from pig colons is shown to be considerably greater than its ``subunit' (thiol reduction product) and papain digested forms using the technique of size-exclusion chromatography coupled to multi-angle laser light scattering, and confirmed by sedimentation equilibrium measurements. The conformation of this mucin is probed by examining the molecular weight – intrinsic viscosity relationship in terms of the Mark-Houwink-Kuhn-Sakurada analysis for its polymeric (or ``whole'), reduced and papain-digested forms: an exponent ``a' of (1.1±0.1) is obtained indicating a linear random coil conformation consistent with other mucins. Size-exclusion chromatography coupled to multi-angle laser light scattering is shown to provide a relatively simple complementary technique to sedimentation equilibrium for the molecular weight distribution analysis of polydisperse materials. Received: 29 November 1996 / Accepted: 2 December 1996     

Random median sampling to enhance the precision of population estimates

We describe the characteristics of a sampling procedure called random median sampling that was proposed to enhance the precision of population estimates. In performing random median sampling, we first select a sampling item at random from the sampling area. We roughly compare the abundance of individuals in the selected item with that of the adjacent two items in order to identify the item that has median abundance, i.e., the item that has the second largest abundance among the three items. We count the number of individuals of the item having the median abundance. This procedure is repeated n times in the sampling area (i = 1, 2, ..., n). Let m _i be the ith median abundance. The estimates of the mean abundance per sampling item and the variance of estimates are given by Σm _i /n and Σ(m _i –Σm _i /n)²/n(n – 1), respectively. This method is a local application of the median ranked set sampling that was proposed by Muttlak (J Appl Stat Sci 6:245–255, 1997). Random median sampling is effective when the correlation coefficient between adjacent items is small. If the correlation coefficient is close to zero, random median sampling reduces the variance of estimates to 45 or 32% of that in simple random sampling when the distribution follows a normal distribution or a Laplace distribution, respectively. The sample size required to achieve a given precision of estimate decreases accordingly. The effectiveness of random median sampling, however, is small if the correlation coefficient is large. The condition in which random median sampling is superior to simple random sampling is also discussed.     

Equilibrium conditions in polysomic inheritance for a panmictic population

A general model of a large 2m-ploid breeding population, withr loci ands _h alleles at the _h ^th locus is considered. It is assumed that the population is bisexual, non-overlapping and breeds by random mating. The genotypic structure of the population is presented as a bilinear form in the gametic output vectors where the genotype distribution is in the matrix form. Using the concept of the segregation distribution, the genotype proportions in the (n+1)^st generation are given. An equilibrium condition for random chromosome segregation is obtained in terms of gene frequencies.     

Beyond the binary case in random nets

Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP ^N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP>2 nets is examined briefly.     

Factors controlling the distribution of planktonic foraminifera in the Red Sea and implications for the development of transfer functions

The Red Sea is an extreme marine environment, with conditions limiting the application of standard geochemical proxies for the reconstruction of paleoclimate. In order to develop paleoenvironmental reconstruction methods which are not dependent on chemical signals, we investigated the distribution of planktonic foraminifera in the surface sediments and assessed the viability of constructing foraminiferal transfer functions in this basin. We find a distinct gradient in the faunal assemblage along the basin's axis, which is reflected in a high correlation between faunal composition and all considered environmental parameters (temperature, salinity, chlorophyll a concentration, stratification, and oxycline depth). As a result, transfer functions constructed by different methods (ANN, MAT, IKM, WA-PLS) appear to be able to estimate all of these parameters with a high average accuracy (15% of the parameter's range in the Red Sea). However, redundancy analysis of the distribution of foraminiferal assemblages in surface sediments alone did not yield unambiguous results in terms of which of the considered factors exerts a primary control on the foraminifera distribution and which of the observed relationships are the result of the mutual correlation among the environmental factors. To disentangle the effect of individual environmental parameters, we applied the obtained transfer functions on a newly generated Holocene record from the central Red Sea. The integration of published paleoclimate reconstructions with our data allowed us to identify productivity as the most likely primary control of the planktonic foraminifera distribution in the Red Sea. The generated transfer functions can estimate paleoproductivity with acceptable accuracy (RMSEP chlorophyll a = 0.1 mg/m³; ~ 8% of recent range), but only under such conditions in the past when circulation patterns and salinity levels in the basin were fundamentally comparable to the present day. Since productivity in the central and southern Red Sea is closely linked with the Monsoon-driven water exchange across the Strait of Bab al Mandab, the resulting reconstructions can provide indirect information on the mode and intensity of the monsoonal system in the past.     

The evolution of proteins from random amino acid sequences: II. Evidence from the statistical distributions of the lengths of modern protein sequences

This paper continues an examination of the hypothesis that modern proteins evolved from random heteropeptide sequences. In support of the hypothesis, White and Jacobs (1993, J Mol Evol 36:79–95) have shown that any sequence chosen randomly from a large collection of nonhomologous proteins has a 90% or better chance of having a lengthwise distribution of amino acids that is indistinguishable from the random expectation regardless of amino acid type. The goal of the present study was to investigate the possibility that the random-origin hypothesis could explain the lengths of modern protein sequences without invoking specific mechanisms such as gene duplication or exon splicing. The sets of sequences examined were taken from the 1989 PIR database and consisted of 1,792 super-family proteins selected to have little sequence identity, 623 E. coli sequences, and 398 human sequences. The length distributions of the proteins could be described with high significance by either of two closely related probability density functions: The gamma distribution with parameter 2 or the distribution for the sum of two exponential random independent variables. A simple theory for the distributions was developed which assumes that (1) protoprotein sequences had exponentially distributed random independent lengths, (2) the length dependence of protein stability determined which of these protoproteins could fold into compact primitive proteins and thereby attain the potential for biochemical activity, (3) the useful protein sequences were preserved by the primitive genome, and (4) the resulting distribution of sequence lengths is reflected by modern proteins. The theory successfully predicts the two observed distributions which can be distinguished by the functional form of the dependence of protein stability on length.The theory leads to three interesting conclusions. First, it predicts that a tetra-nucleotide was the signal for primitive translation termination. This prediction is entirely consistent with the observations of Brown et al. (1990a,b, Nucleic Acids Res 18:2079–2086 and 18: 6339-6345) which show that tetra-nucleotides (stop codon plus following nucleotide) are the actual signals for termination of translation in both prokaryotes and eukaryotes. Second, the strong dependence of statistical length distributions on sequence-termination signaling codes implies that the evolution of stop codons and translation-termination processes was as important as gene splicing in early evolution. Third, because the theory is based upon a simple no-exon stochastic model, it provides a plausible alternative to a limited universe of exons from which all proteins evolved by gene duplication and exon splicing (Dorit et al. 1990, Science 250:1377–1382).     

Bayesian Estimation in a Generalized Negative Binomial Distribution

A generalized negative binomial (GNB) distribution was introduced by JAIN and CONSUL (1971) and was modified by NELSON (1975). The probability function of the distribution is defined by the function p(x; m, β, θ)= θ^x (1 - θ)^m+βx—x for x=0, 1, …, and zero otherwise, where m>0, 0<θ<1 and β=0 or 1≦β<θ^?1. The Bayes estimators for a number of parametric functions of θ when m and β are known are derived. The prior information on θ may be given by a beta distribution, B(a, b), to which no subjective significance is attached. It has been illustrated that the parameters in the prior distribution can be assigned by a computer. Comparisons are made of the Bayes estimate of P(X=k) to the corresponding ML estimate and the MVU estimate for any given sample to the order n^?1 for different values of k..     

Salinity-yield response functions of barley genotypes assessed with a triple line source sprinkler system

Evaluation of the salt tolerance of crop cultivars under field conditions is greatly complicated by the typical temporal and spatial variability of soil salinity. We obtained the grain yield – salinity response functions of 124 barley genotypes by growing them in ten salinity treatments imposed by a Triple Line Source Sprinkler (TLS) system during five consecutive years. Additional objectives were to ascertain the consistency and reproducibility over years of these functions, to quantify the deleterious effects of saline sprinkling irrigations, and to assess correlations between salinity tolerance and leaf sap salt concentration. The consistency and reproducibility of the response functions within and between years were adequate (only 8% of the response functions were discarded for statistical reasons). The Y _m (grain yield without salinity) and the EC₅₀ (the EC _e that reduces yield by 50%) estimates were not correlated (P > 0.05) suggesting that the most productive genotypes were not necessarily less salinity tolerant. Y _m was positively and significantly (P < 0.01) correlated with Y₆ and Y₁₂ (fitted grain yields at EC _e values of 6 dS m^-1, and 12 dS m^-1, respectively), indicating that it is a useful statistic in the selection of barley genotypes most productive under medium and high salinities. Foliar salt uptake due to saline sprinkling irrigations decreased the EC₅₀ by around 50% as compared with the salinity tolerance obtained with surface irrigation systems. No consistent relationships were found between either Y _m or EC₅₀ and the leaf sap osmotic potential, Cl, Ca, Na and K concentrations. They could not therefore be used in screening for salinity tolerance of barley. On the basis of the evidence from the present study, Y _m is the best statistic for predicting the most productive barley genotypes in salt-affected soils. This revised version was published online in June 2006 with corrections to the Cover Date.     

Linear Rank Tests Derived from a Superpopulation Model

A superpopulation model generates the probabilities of a Bernouilli random variable. The ranks of the involved variables are considered as survey weights. The distribution f each linear rank statistic is derived under the null hypothesis for the two sample problem and for the case k2 when a simple random sampling or stratified sampling is used. The growth of a population of insects and the behavior of patients with imsomnia are studied using these procedures.     

Modelling the distribution of a potential invasive tropical fern,Cyclosorus afer in Nigeria

In this study, we predicted the distribution of Cyclosorus afer in Nigeria using the Maximum Entropy (Maxent) technique. We used 95 occurrence points in one State to extrapolate its spread in other States in Nigeria. Three sites of sizes 500 × 500 m² separated by minimum distance of 1,000 m² were sampled in the study area. Seven bioclimatic and elevation variables were selected for the model. Maxent model was run using standard settings with 70% of the occurrence data as training and remaining 30% as test data. The result showed that Maxent performed better than random prediction due to the area under curve for receiver operating characteristics of training (0.990) and test data (0.987) closer to 1. The sensitivity of the Maxent model for occurrence data of C. afer was found to be 100%. The model predicted higher probability of occurrence covering an area of 26019.11 km² in 4 States of Nigeria. Jackknife evaluation of the model revealed that the environmental predictors of C. afer in Nigeria are annual mean temperature, mean temperature of driest quarter, precipitation seasonality, Precipitation of driest quarter and precipitation of coldest quarter. These variables all showed higher gain and contributions to the model.     

A theoretical study of concentration of profiles of primary cytochemical-enzyme reaction products in membrane-bound cell organelles and its application to lysosomal acid phosphatase

Synopsis A numerical method was developed for computing the steady-state concentration gradient of a diffusible enzyme reaction product in a membrane-limited compartment of a simplified theoretical cell model. In cytochemical enzyme reactions proceeding according to the metal-capture principle, the local concentration of the primary reaction product is an important factor in the onset of the precipitation process and in the distribution of the final reaction product. The following variables were incorporated into the model: enzyme activity, substrate concentration,K _m, diffusion coefficient of substrate and product, particle radius and cell radius.The method was applied to lysosomal acid phosphatase. Numerical values for the variables were estimated from experimental data in the literature. The results show that the calculated phosphate concentrations inside lysosomes are several orders of magnitude lower than the critical concentrations for efficient phosphate capture found in a previous experimental model study. Reasons for this apparent discrepancy are discussed.     

On the origin of coding sequences from random open reading frames

Summary The size distribution of 411 randomly selected mammalian exons was investigated. This distribution was found to be unimodal with a frequency maximum of 120 bp. Detailed analysis of the distribution demonstrated that larger exons (>150 bp) have a high goodness of fit to the size distribution of open reading frames (ORFs) in a random sequence, i.e., (61/64)^t in which t is the number of triplets. Based on this observation, the general character of the total exon size distribution suggested that this could be defined by a theoretical distribution by superimposing a sigmoid function on the ORF generating function, i.e., (61/64)^t×f_s(t)×E in which f_s(t) is a sigmoid function and E is a constant. We tested this distribution for fitness to the exon distribution using two sigmoid functions. f_s(t)=(t) and f_s(t)=Be^kt/1+Be^kt. In both cases a very high goodness of fit was attained. It is concluded that exons have been generated from ORFs in random sequences, that ORFs larger than 150 bp have been selected, irrespective of size, as exons, and that a lower size limit exists below which the probability of an ORF being selected as an exon is very low. These results provide evidence at the molecular level to support the ideas that (1) larger exons have been selected from random ORFs without primary correlation to structural or functional properties at the protein level, (2) there exists a restriction on smaller ORFs to be selected as exons, and (3) the interrupted coding sequences found in eukaryotes represent the ancient form of gene organization that existed prior to the divergence of prokaryotes and eukaryotes.     

Reference conditions and historical changes in an unharvested ponderosa pine stand on sedimentary soil

Much of the previous research on spatial reference conditions in dry frequent fire pine forests have come from stand‐level patterns under regionally average ecosystem conditions (e.g. soil type and precipitation). We evaluated the 1883 reference conditions of an uncut ponderosa pine stand representing a far end of the range of variability in terms of regionally unusual environmental conditions. Using a forest reconstruction model, univariate and bivariate Ripley's K functions, and regression analysis, we determined 1883 structural and spatial reference conditions, and compared those to the contemporary (2010) stand. Historical stand density was 77 trees/ha with a basal area of 8.0 m²/ha. Reference spatial patterns were significantly aggregated from 1 to 2 m and randomly distributed at distances greater than 2 m. Nearly 40% of the reconstructed trees were individuals, the average patch size was 2.9 trees, and the largest patch had 7 members. The contemporary stand had considerably greater densities and basal area than historical conditions and showed aggregation at all distances. Bivariate spatial analysis indicated attraction of post‐settlement recruitment to live pre‐settlement trees from 1 to 6 m and no association at distances greater than 6 m. We speculate that the historically random tree pattern is the product of a variety of factors including soil parent material, climate, and more homogeneous resource partitioning.     

Replication of minichromosomes in a host in which chromosome replication is random

Minichromosomes are plasmids with the origin of chromosome replication, oriC, as their only origin of replication. In Escherichia coli, minichromosomes are compatible with the chromosome and replicate in a cell-cycle-specific manner at the same time as oriC located on the chromosome initiates replication. In int strains, oriC has been inactivated and replaced by a plasmid origin. Because plasmids control their own replication, chromosome replication is uncoupled from the normal cell-cycle control and is random with respect to the cell cycle in the int strains. We have used an intP1 strain to address the question of whether minicromosome replication is coupled to the replication of the chromosome or is governed by cell-cycle-specific signals. Minichromosome replication was analysed by density-shift experiments and found not to be random in the randomly replicating intP1 host. This suggests that the cell-cycle-specific control functions of oriC replication are operating also in the intP1 strain.     

Engineering of a bacterial tyrosinase for improved catalytic efficiency towards D‐tyrosine using random and site directed mutagenesis approaches

The tyrosinase gene from Ralstonia solanacearum (GenBank NP518458) was subjected to random mutagenesis resulting in tyrosinase variants (RVC10 and RV145) with up to 3.2‐fold improvement in k_cat, 5.2‐fold lower K_m and 16‐fold improvement in catalytic efficiency for D ‐tyrosine. Based on RVC10 and RV145 mutated sequences, single mutation variants were generated with all variants showing increased k_cat for D ‐tyrosine compared to the wild type (WT). All single mutation variants based on RV145 had a higher k_cat and K_m value compared to the RV145 and thus the combination of four mutations in RV145 was antagonistic for turnover, but synergistic for affinity of the enzyme for D ‐tyrosine. Single mutation variant 145_V153A exhibited the highest (6.9‐fold) improvement in k_cat and a 2.4‐fold increase in K_m compared to the WT. Two single mutation variants, C10_N322S and C10_T183I reduced the K_m up to 2.6‐fold for D ‐tyrosine but one variant 145_V153A increased the K_m 2.4‐fold compared to the WT. Homology based modeling of R. solanacearum tyrosinase showed that mutation V153A disrupts the van der Waals interactions with an α‐helix providing one of the conserved histidine residues of the active site. The k_cat and K_m values for L ‐tyrosine decreased for RV145 and RVC10 compared to the WT. RV145 exhibited a 2.1‐fold high catalytic efficiency compared to the WT which is a 7.6‐fold lower improvement compared to D ‐tyrosine. RV145 exhibited a threefold higher monophenolase:diphenolase activity ratio for D ‐tyrosine:D ‐DOPA and a 1.4‐fold higher L ‐tyrosine:L ‐DOPA activity ratio compared to the WT. Biotechnol. Bioeng. 2013; 110: 1849–1857. © 2013 Wiley Periodicals, Inc.     

A modeled time-varying density function for the incubation period of AIDS

Building on the Weibull distribution, we develop a modeled time-varying density function of the incubation time between exposure to HIV infection and full-blown AIDS. This approach leads to a series of cohort-specific density functions that take into account the increasing impact of new therapies such as zidovudine (AZT). The resulting modeled density functions are studied in detail, particularly with regard to their modes and medians. The mode is sensitive to changes in the period incubation time distribution, with even a possibility of a bimodal distribution for certain combinations of the parameters that determine the rate at which the period median incubation time changes. An important substantive result is that when a period median incubation period slowly increases to some leveling off value, say m(x _c ), then it is surprisingly early on that cohorts of infected individuals have a median incubation period very close to that ultimate value m(x _c ).     

Assessment of the Quality of Energy Functions for Protein Folding by Using a Criterion Derived With the Help of the Noisy Go Model

We propose a simple criterion based on the Z-scoreto assess the quality of energy functions for protein folding: one should obtain Z<–10 for the equilibrium ensembleat about native conditions. We derive this criterionby studying a Go model with random errors added to the native interactions. The dependence of the Z-score on the thermodynamic parameters,including the noise, can be precisely obtained in this case,as the ground state of the model is known exactly.We apply this criterion to rapidly rule out two otherwise promisingpairwise energy approximations.The advantage of adopting the present criterionis that it is not necessary to know the ground state of an energy function to assess its quality. It is sufficient to compute the Z-scorefrom a single equilibrium simulation at around the folding temperature.