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Use of compound-probability distributions in the study of induced post-implantation dominant lethals.

The nature of the probability distribution of post-implantation dominant lethality was investigated in terms of the distribution of dead implants per female. It has been postulated that this distribution would be poisson in a control series of females but may follow a compound or a contagious distribution such as the beta binomial, negative binomial or Neyman type A in the treated series of females. The nature of these compound distributions for fitting mammalian mutagenicity has been examined. The implications of the results on the estimation of induced mutation rates are discussed.

Animals↗

Highlights of the sexual activity of the heterosexual population in the province of Quebec.

OBJECTIVES: To describe and quantify the level of sexual activity of the heterosexually active population of Quebec. METHODS: The data analysed included 2889 heterosexually active individuals aged 15-60 (agemed = 32) from a 1996-7 survey on the sexual lifestyles of the general population of Quebec. Various probability distributions were studied to assess their capacity to describe and quantify the lifetime and yearly numbers of sexual partners of the sampled population. To estimate the annual rates of new partner acquisition, a generalised linear model was fitted to the number of lifetime sexual partners as a function of age, years of sexual activity, and sex. RESULTS: The mean and variance of the number of lifetime sexual partners for men (mean = 11, s2 = 163) is higher than for women (mean = 6, s2 = 72). The negative binomial and lognormal probability distributions give the most adequate fit to the lifetime number of partners for both agglomerated and stratified (by sex and age) data. The estimated annual rates of new partner acquisition provide two important results for prevention: (1) the first year of sexual activity represents the highest annual rate of new partner acquisition independent of age, (2) annual rates of new partner acquisitions increase through mid-life (ages 40-50) combined with a decrease in condom use. CONCLUSION: Problems caused by the use of large categories in the estimation of mean and variance cannot totally be overcome by fitting probability distributions to the empirical data despite good fits. Furthermore, we believe that adequate estimates of the annual rate of new partner acquisition should be a better measure of the risk of HIV infection than the number of partners since the first is a measure of incidence while the second is a measure of prevalence.

Adolescent↗

Robustness of group testing in the estimation of proportions.

In binomial group testing, unlike one-at-a-time testing, the test unit consists of a group of individuals, and each group is declared to be defective or nondefective. A defective group is one that is presumed to include one or more defective (e.g., infected, positive) individuals and a nondefective group to contain only nondefective individuals. The usual binomial model considers the individuals being grouped as independent and identically distributed Bernoulli random variables. Under the binomial model and presuming that groups are tested and classified without error, it has been shown that, when the proportion of defective individuals is low, group testing is often preferable to individual testing for identifying infected individuals and for estimating proportions of defectives. We discuss the robustness of group testing for estimating proportions when the underlying assumptions of (i) no testing errors and (ii) independent individuals are violated. To evaluate the effect of these model violations, two dilution-effect models and a serial correlation model are considered. Group testing proved to be quite robust to serial correlation. In the presence of a dilution effect, smaller group sizes should be used, but most of the benefits of group testing can still be realized.

AIDS Serodiagnosis↗

A model for estimating abundance of cattle grub (Diptera: Oestridae) from the proportion of uninfested cattle as determined by serology.

A model is presented for determining the abundance of cattle grubs in the backs of calves from the proportion of uninfested calves in a herd. The distribution of grubs in calves' backs was compared with the negative binomial, but no relationships were found among the distribution parameters, suggesting that the negative binomial is an inappropriate choice for the basis of a sampling model. The relationship between the mean number of grubs per animal (mean), variance (delta 2), and proportion of uninfested calves (p0) in a herd was determined and used as the basis for the sampling model. The relationship between p0 and p0e [determined using serology (ELISA)] was evaluated. The variance of estimates of mean grubs per animal based on the regression model and uncertainty due to using p0e as an estimate of p0 was examined. A test of the model indicated that p0e could be used to obtain a reliable estimate of mean grubs per animal and that the method would be applicable for monitoring grub populations, assessing chemical control programmes, and determining release rates of sterile insects for control.

Animals↗

A survey of unequal crossover systems and their mathematical properties.

We present a model of gene duplication by means of unequal crossover (UCO) where the probability of any given pairing between homologous sequences scales as a penalty factor pz < or = 1, with z the number of mismatches due to asymmetric sequence alignment. From this general representation, we derive several limiting case models of UCO, some of which have been treated elsewhere in the literature. One limiting case is random unequal crossover (RUCO), obtained by setting p = 1 (corresponding to equiprobable pairings at each site). Another limiting case scenario (the 'Krueger-Vogel' model) proposes an optimal 'endpoint' alignment which strongly penalizes both overhang and deviations from endpoint matching positions. For both of these scenarios, we make use of the symmetry properties of the transition operator (together with the more general UCO properties of copy number conservation and equal parent-offspring mean copy number) to derive the stationary distribution of gene copy number generated by UCO. For RUCO, the stationary distribution of genotypes is shown to be a negative binomial, or alternatively, a convolution of geometric distributions on 'haplotype' frequencies. A different type of model derived from the general representation only allows recombination without overhang (internal UCO or IntUCO). This process has the special property of converging to a single copy length or a distribution on a pair of copy lengths in the absence of any other evolutionary forces. For UCO systems in general, we also show that selection can readily act on gene copy number in all of the UCO systems we investigate due to the perfect heritability (h 2 = 1) imposed by conservation of copy number. Finally, some preliminary work is presented which suggests that the more general models based on misalignment probabilities seem to also converge to stationary distributions, which are most likely functions of parameter value p.

Animals↗

Mass isotopomer distribution analysis: a technique for measuring biosynthesis and turnover of polymers.

Mass isotopomer distribution analysis (MIDA) is a technique for measuring biosynthesis and turnover of polymers in vivo. A stable isotopically enriched precursor is administered, and the relative abundances of different mass isotopomers in the polymer of interest are measured by mass spectrometry (MS). By comparison of statistical distributions predicted from the binomial or multinomial expansion to the pattern of excess isotopomer frequencies observed in the polymer, the enrichment of the biosynthetic precursor subunits (p) for newly synthesized polymers is calculated. MIDA thereby provides a solution to the problem of determining the isotope content in the actual precursor molecules that entered a particular polymeric product (the "true" precursor). The fraction of polymer molecules in a mixture that were newly synthesized during an isotopic experiment (fractional synthesis) can then be calculated. We describe some mathematical characteristics of MIDA and point out certain advantageous features. For example, mathematical estimates of p remain valid even if there does not exist a single anatomic or functional precursor pool. The interpretation of decay curves of endogenously labeled polymers may be improved by the use of higher mass isotopomers, which better fulfill the assumption of flash labeling. By combining fractional synthesis values with rate constants of decay, absolute endogenous synthesis rates can be calculated. Thus, by using probability logic combined with MS analysis, MIDA allows dynamic measurements to be made through analyses on a polymer alone during both isotopic incorporation and decay phases. The method has been applied to fatty acids, cholesterol, and glucose and is potentially applicable to nucleic acids, porphyrins, perhaps proteins, and many other classes of polymers.

Indicator Dilution Techniques↗

Mass isotopomer distribution analysis at eight years: theoretical, analytic, and experimental considerations.

Mass isotopomer distribution analysis (MIDA) is a technique for measuring the synthesis of biological polymers. First developed approximately eight years ago, MIDA has been used for measuring the synthesis of lipids, carbohydrates, and proteins. The technique involves quantifying by mass spectrometry the relative abundances of molecular species of a polymer differing only in mass (mass isotopomers), after introduction of a stable isotope-labeled precursor. The mass isotopomer pattern, or distribution, is analyzed according to a combinatorial probability model by comparing measured abundances to theoretical distributions predicted from the binomial or multinomial expansion. For combinatorial probabilities to be applicable, a labeled precursor must therefore combine with itself in the form of two or more repeating subunits. MIDA allows dilution in the monomeric (precursor) and polymeric (product) pools to be determined. Kinetic parameters can then be calculated (e.g., replacement rate of the polymer, fractional contribution from the endogenous biosynthetic pathway, absolute rate of biosynthesis). Several issues remain unresolved, however. We consider here the impact of various deviations from the simple combinatorial probability model of biosynthesis and describe the analytic requirements for successful use of MIDA. A formal mathematical algorithm is presented for generating tables and equations (APPENDIX), on the basis of which effects of various confounding factors are simulated. These include variations in natural isotope abundances, isotopic disequilibrium in the precursor pool, more than one biosynthetic precursor pool, incorrect values for number of subunits present, and concurrent measurement of turnover from exogenously labeled polymers. We describe a strategy for testing whether isotopic inhomogeneity (e.g., an isotopic gradient or separate biosynthetic sites) is present in the precursor pool by comparing higher-mass (multiply labeled) to lower-mass (single- and double-labeled) isotopomer patterns. Also, an algebraic correction is presented for calculating fractional synthesis when an incomplete ion spectrum is monitored, and an approach for assessing the sensitivity of biosynthetic parameters to measurement error is described. The different calculation algorithms published for MIDA are compared; all share a common model, use overlapping solutions to computational problems, and generate identical results. Finally, we discuss the major practical issue for using MIDA at present: quantitative inaccuracy of instruments. The nature and causes of analytic inaccuracy, strategies for evaluating instrument performance, and guidelines for optimizing accuracy and reducing impact on biosynthetic parameters are suggested. Adherence to certain analytic guidelines, particularly attention to concentration effects on mass isotopomer ratios and maximizing enrichments in the isotopomers of interest, reduces error. Improving instrument accuracy for quantification of isotopomer ratios is perhaps the highest priority for this field. In conclusion, MIDA remains the "equation for biosynthesis," but attention to potentially confounding factors and analytic performance is required for optimal application.

Isotopes↗

Ocular malingering: a surprising visual acuity test.

OBJECTIVE: To describe a visual acuity test for use in identifying psychogenic visual impairment and malingering. METHODS: The test contained 32 white plates with a black Landolt C printed in the center. The sequence of the 4 alternative directions of the C was not predictable. After plate 21, 4 circles were interspersed among the remaining plates. The test is administered at a distance from which the subject is presumed to be able to recognize the optotypes. He or she is requested to identify the direction of the C within 2 seconds, and the responses are noted. Two elements were evaluated: (1) The number of correct answers was compared with the corresponding value of the distribution function of the binomial formula that represents the probability of reaching this rate of correct answers by pure chance. (2) The response to the first circle (appearing after plate 21) was compared with the responses to the previous 19 Cs. The test was administered to 20 volunteer pseudomalingerers and 15 patients believed to be true malingerers. RESULTS: Malingering was detected in 14 (74%) of 19 volunteers included in the evaluation and in 12 (80%) of 15 patients by too many or too few correct answers to the stimuli or by their response to one of the circles. CONCLUSIONS: This test is useful in proving malingering. It may also provide evidence of a minimum visual recognition acuity within the gross dimension of the actual acuity.

Adult↗

Information from false statements concerning visual acuity and visual field in cases of psychogenic visual impairment.

BACKGROUND: If visual acuity (VA) and/or visual field (VF) is claimed to be worse than in reality, it can be difficult to estimate the actual VA/VF. This paper describes a method of proving malingering and of estimating the actual VA by statistical evaluation of the subjective responses to discrete visual stimuli. METHODS: VA is measured by using a Landolt or Snellen optotype with four possible directions. There are n (> or = 16) questions at each acuity level. Each direction of the optotype occurs with the same frequency. The sequence is not predictable. In testing the VF, the stimulus is presented in an unpredictable sequence but with the same frequency at each of four distinct locations outside the claimed field at the horizontal and the vertical meridian. The field is divided into four quadrants defined by the 45 degrees-225 degrees and 135 degrees-315 degrees meridians. The candidate is requested to search for the light. The target quadrant of the first eye movement is recorded. At each VA level, the distribution function of the binomial formula allows estimation of the probability Pc of < or = kc and > or = kc correct answers to n questions by mere accident. Regarding the VF, the direction of the first saccade at each stimulus presentation can be processed accordingly. RESULTS: If n = 32 and kc = 0,1,2, the chance P that this pattern is merely accidental is about 0.0001, 0.001, 0.007. Values of P < or = 0.01 strongly suggest that the answers were voluntarily wrong. Preference for the opposite direction can also point to psychogenic influence: If ko = 15,16,17,18, P is about 0.006, 0.002, 0.0006, 0.0002. CONCLUSION: Systematically false statements can yield valuable information about the actual visual functions.

Adult↗

Foveal cone detection statistics in color-normals and dichromats.

We measured for six male observers, the psychometric functions for the detection of two simultaneously presented points of light. The test stimuli were two 1 min point sources separated by 17 min arc, and pulsed for 0.5 msec. The stimuli varied in wavelength from 500 to 620 nm. The psychometric functions were fit with a model that assumes ideal detection and the following properties: (1) Poisson-distributed quantal absorptions; (2) binomial sampling of foveal long-wavelength-sensitive (LWS) and middle-wavelength-sensitive (MWS) cones; (3) independent responses of the LWS and MWS cones; and (4) the Smith-Pokorny fundamentals for cone spectral sensitivities. Based on chi 2 fits to the psychometric functions for detecting neither, one or both of the two-point stimuli presented, estimates were derived for the minimum quantal catch by a single cone for detection (C), the number of effective cones illuminated by a point stimulus at threshold (N) and the proportion of central foveal cones of the LWS type (PL). Three observers were color-normal, two were protanopes and one was a deuteranope. A second deuteranope was included in the design but his data were too unreliable for an unambiguous solution. The estimated quantal requirement C was consistently near 5(4-6), and the effective number of illuminated cones always was 1 or 2. The plausible range of PL (98% confidence interval) for the color-normal observers was 0.48-0.68 (observer YY), 0.76-0.90 (observer MW) and 0.77-0.95 (observer DF). The best fitting PL solution for these observers were 0.61, 0.82 and 0.88, respectively. These were comparable to the values obtained from flicker photometric data. The best PL value for each of the protanopes was 0.00 and for the deuteranope the best PL value was 0.98.

Color Perception↗

Analysis of aggregation, a worked example: numbers of ticks on red grouse chicks.

The statistical aggregation of parasites among hosts is often described empirically by the negative binomial (Poisson-gamma) distribution. Alternatively, the Poisson-lognormal model can be used. This has the advantage that it can be fitted as a generalized linear mixed model, thereby quantifying the sources of aggregation in terms of both fixed and random effects. We give a worked example, assigning aggregation in the distribution of sheep ticks Ixodes ricinus on red grouse Lagopus lagopus scoticus chicks to temporal (year), spatial (altitude and location), brood and individual effects. Apparent aggregation among random individuals in random broods fell 8-fold when spatial and temporal effects had been accounted for.

Animals↗

1,2,4,5-Tetra([5]trovacenyl)benzene: an organometallic tetraradical displaying pronounced electro- and magnetocommunication.

The organometallic tetraradical 1,2,4,5-[(eta)7-C7H7)V(all eta]5-C5H4)]4C6H2 has been prepared and structurally characterized. The isotropic EPR spectrum displays 29 a(51V) hyperfine lines, the intensity distribution slightly deviating from binomial. Exchange coupling therefore approaches the strong exchange limit, J(ortho) not equal to J(meta) not equal to J(para) greater or approximately 50 a(51V) with a(51V)= 0.0067 cm(-1). According to magnetic susceptometry, the interaction is antiferromagnetic. While redox splittings deltaE(1/2) are resolved for the four reduction steps this is not the case for oxidation.

Electrochemistry↗

Sampling errors and the precision associated with counting very low numbers of white cells in blood components.

Attention to the accurate and precise measurement of the white cell (WBC) content of transfused products has risen in response to awareness of the potential benefits of WBC-depleted components and the development of technical capabilities to produce these components. The techniques thus far reported have focused on the reliability of detecting a WBC, provided it is present in the test system. The likelihood of selecting a WBC from the product of interest for counting in the analytical system--that is, the sampling error--must also be considered. The occurrence of a WBC in a WBC-depleted component is a rare event and may be modeled with the binomial or the Poisson distribution. Several assay techniques were analyzed by using these distribution models to determine the confidence intervals of the WBC content. The 95-percent confidence intervals spanned more than 2 logs10 for some methods at 3 x 10(5) WBCs per product. It is concluded that the reporting of WBC content for research provides not only the estimate of the mean but also a confidence interval for this estimate. Quality control procedures should be designed to verify that the WBC content is less than the targeted amount and should provide an associated statement of confidence.

Blood Platelets↗

Bayesian analysis of mutational spectra.

Studies that examine both the frequency of gene mutation and the pattern or spectrum of mutational changes can be used to identify chemical mutagens and to explore the molecular mechanisms of mutagenesis. In this article, we propose a Bayesian hierarchical modeling approach for the analysis of mutational spectra. We assume that the total number of independent mutations and the numbers of mutations falling into different response categories, defined by location within a gene and/or type of alteration, follow binomial and multinomial sampling distributions, respectively. We use prior distributions to summarize past information about the overall mutation frequency and the probabilities corresponding to the different mutational categories. These priors can be chosen on the basis of data from previous studies using an approach that accounts for heterogeneity among studies. Inferences about the overall mutation frequency, the proportions of mutations in each response category, and the category-specific mutation frequencies can be based on posterior distributions, which incorporate past and current data on the mutant frequency and on DNA sequence alterations. Methods are described for comparing groups and for assessing dose-related trends. We illustrate our approach using data from the literature.

Animals↗

Non-Gaussian equilibrium distributions arising from the Langevin equation.

We study the Langevin equation of a point particle driven by random noise, modeled as a two-state Markov process. The corresponding master equation differs from the Fokker-Planck equation. In equilibrium, the velocity of the particle is distributed according to a binomial power law. We discuss transient (i.e., nonequilibrium) behavior, and the consequences of non-Markovian noise statistics.

Journal Article↗

Properties of single potassium channels in vesicles formed from the sarcolemma of frog skeletal muscle.

The patch-clamp method was used to study unitary delayed rectifier K+ channels in large vesicles formed from the membrane of frog skeletal muscle. Channels were activated by depolarizing pulses. Single-channel conductance was about 15 pS in physiological [K+]o and was doubled by raising [K+]o to 120 mM. TEA+ caused an apparent reduction in single-channel current, which we attribute to a rapid block. When depolarizations were repeated at brief intervals, records with and without channel openings were ordered non-randomly, providing evidence for a slow process which was probably inactivation. In multichannel patches the relation between variance and mean current, binomial analysis, and the distribution of times for single and double openings were all consistent with channels behaving independently. Open times were distributed exponentially. Mean open time, tau o, increased with depolarization so that 1/tau o was an exponential function of voltage. First latency histograms peaked at times later than zero and could not be fitted by a scheme having only two closed states. Channel openings occurred in bursts and closed time histograms could be fitted by the sum of three exponentials. Our results imply a scheme with at least three closed states, an open and an inactivated state.

Action Potentials↗

Ag-NORs variation and banding homologies in two species of Bovidae: Bubalus bubalis and Bos taurus.

G,Q and R-binding pattern comparisons between the Murrah type of Bubalus bubalis L. (2n - 50) and the Holstein Friesian breed of Bos taurus L. (2n = 60) chromosomes revealed that the autosomes are similar in both species; in the former, the five pairs of submetacentrics correspond to centric fusions of chromosomes 1-29, 2-22, 8-19, 5-28 and 16-25 in the latter. Silver staining on somatic cells of buffalo revealed telomeric AG-NORs located on six pairs of autosomes, identified as 3p, 4p, 8, 21, 23 and 24. Only one pair of nucleolus organizer chromosomes is common to both species, namely the 4p of buffalo which corresponds to the 28 of cattle. The remaining NORs are located on different pairs. Out of 18 individuals, the number of Ag-NORs per cell varied between 3 and 10, with a mean value of 6.3 +/- 1.7; the mean number of nucleolar associations per cell was between 0.1 and 1.8. In the total population of 317 cells the two variables followed the binomial and the Poisson distributions, respectively, and their correlation coefficient was not statistically significant (r = +0.36). In addition to the mean number of Ag-NORs, the mean number of nucleolar associations per cell should be used for a better understanding of the differences among individuals.

Animals↗

Heart wall motion: improved method of spatial modulation of magnetization for MR imaging.

A previously reported method of using magnetic resonance (MR) imaging to study heart wall motion involves a pair of nonselective radio-frequency (RF) pulses, separated by a magnetic field gradient pulse, prior to imaging; this produces images with a regular pattern of stripes that move with the heart wall and that have a sinusoidal intensity profile. It is demonstrated in this study that the substitution of more RF pulses, with their relative amplitudes distributed according to the binomial sequence, results in sharper stripes. This permits the use of a two-dimensional grid of stripes for more detailed studies of heart wall motion and provides a unique method of analyzing regional ventricular myocardial strain.

Heart↗