Distribution of nucleotide differences between two randomly chosen cistrons in a population of variable size

The distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population is studied here when the population size changes from generation to generation. When genetic variability is measured by heterozygosity (i.e., the probability that two cistrons are different), by the probability that two cistrons differ at two or more nucleotide sites, or by mean number of site differences between cistrons, it is seen that in a population going through a small bottleneck all of these measures decline rapidly but, as soon as population size becomes large, they start to increase owing to new mutations. The amount of reduction in these measures depends not only on the size of bottleneck but also on the rate of population growth. The implications of this study explaining the observed variations in the rates of amino acid substitutions during the evolutionary process are also discussed.     

Prediction of identity by descent probabilities from marker-haplotypes

The prediction of identity by descent (IBD) probabilities is essential for all methods that map quantitative trait loci (QTL). The IBD probabilities may be predicted from marker genotypes and/or pedigree information. Here, a method is presented that predicts IBD probabilities at a given chromosomal location given data on a haplotype of markers spanning that position. The method is based on a simplification of the coalescence process, and assumes that the number of generations since the base population and effective population size is known, although effective size may be estimated from the data. The probability that two gametes are IBD at a particular locus increases as the number of markers surrounding the locus with identical alleles increases. This effect is more pronounced when effective population size is high. Hence as effective population size increases, the IBD probabilities become more sensitive to the marker data which should favour finer scale mapping of the QTL. The IBD probability prediction method was developed for the situation where the pedigree of the animals was unknown (i.e. all information came from the marker genotypes), and the situation where, say T, generations of unknown pedigree are followed by some generations where pedigree and marker genotypes are known.     

Exclusion probabilities for pedigree testing farm animals

Pedigree testing, using genetic markers, may be undertaken for a variety of situations, of which the classical paternity testing is only one. This has not always been made clear in the literature. Exclusion probabilities associated with various testing situations, including the use of autosomal or X-linked codominant marker systems with any number of alleles, are presented. These formulae can be used to determine the appropriate exclusion probability for the situation being investigated. One such situation is where sire groups of progeny are to be verified without knowledge of the dams' genotypes, in which case the classical paternity exclusion probability is too high, and if used may result in an optimistic declaration about the progeny that have not been excluded. On the other hand, if mating pairs are known then incorrect progeny can be excluded at a higher rate than suggested by paternity exclusion calculations. The formulae also assist in determining the usefulness of X-linked markers, particularly if the pedigree checks involve progeny of only one sex. A system of notation that is useful for the algebraic manipulation of genetic probabilities, including exclusion probabilities as presented here, is also given.     

Sample size determination

Scientists who use animals in research must justify the number of animals to be used, and committees that review proposals to use animals in research must review this justification to ensure the appropriateness of the number of animals to be used. This article discusses when the number of animals to be used can best be estimated from previous experience and when a simple power and sample size calculation should be performed. Even complicated experimental designs requiring sophisticated statistical models for analysis can usually be simplified to a single key or critical question so that simple formulae can be used to estimate the required sample size. Approaches to sample size estimation for various types of hypotheses are described, and equations are provided in the Appendix. Several web sites are cited for more information and for performing actual calculations     

An analytical formula to estimate confidence interval of QTL location with a saturated genetic map as a function of experimental design

Analytical formulae are derived for the confidence interval for location of a quantitative trait locus (QTL) using a saturated genetic map, as a function of the experimental design, the QTL allele substitution effect, and the number of individuals genotyped and phenotyped. The formulae are derived assuming evenly spaced recombination events, rather than the actual unevenly spaced distribution. The formulae are useful for determining desired sample size when designing a wide variety of QTL mapping experiments, and for evaluating a priori the potential of a given mapping population for defining the location of a QTL. The formulae do not take into account the finite number of recombination events in a given sample.     

Use of near-isogenic lines derived by backcrossing or selfing to map qualitative traits

Near-isogenic lines (NILs) are a valuable resource for detecting linkages between qualitative trait loci and molecular markers. Molecular marker studies are expensive and methods that require genotyping fewer individuals, such as the NIL-analysis method, are desirable. We present a theory for using sets of NILs to detect linkages between molecular markers and introgressed loci. The probability that a marker a specific distance from the introgressed gene will have a donor parent allele in a near-isogenic line is a function of the distance between the marker and the gene, and the number of back-crosses and/or selfs used in deriving the NIL. The binomial probability formula is used to calculate the probability of having a donor parent allele at a given marker when sets of NILs are used. The formulae given allow calculation of the probability that a marker is linked to the introgressed gene, as well as the probability that a gene will be successfully detected when using given numbers of NILs, backcrosses, and molecular markers.     

Sample size calculations based on ranking and selection in microarray experiments

Summary .   We develop formulae to calculate sample sizes for ranking and selection of differentially expressed genes among different clinical subtypes or prognostic classes of disease in genome-wide screening studies with microarrays. The formulae aim to control the probability that a selected subset of genes with fixed size contains enough truly top-ranking informative genes, which can be assessed on the basis of the distribution of ordered statistics from independent genes. We provide strategies for conservative designs to cope with issues of unknown number of informative genes and unknown correlation structure across genes. Application of the formulae to a clinical study for multiple myeloma is given.     

Contraception, reproduction and demography of free-ranging Etosha lions (Panthera leo)

The use of long-acting contraceptives as an effective population control mechanism for free-ranging lions was investigated during a three-year study in the Etosha National Park on five lion prides. Lions were immobilized with ketamine hydrochloride and xylazine hydrochloride. Following immobilization, lionesses were weighed, measured and aged, while serum steroid hormone levels were analysed and vaginal smears were obtained. Physical condition and sexual state were assessed. Individuals were used as control animals or given melengestrol acetate (contraceptive) implants, branded and released; the animals were observed for possible changes in behaviour, birth rate and mortality. Ten treated lionesses were recaptured at irregular intervals to reassess weight and steroid hormone levels, while three lionesses had their implants removed to determine if their fertility would return. The contraceptives prevented pregnancy, were reversible when removed and did not alter lion behaviour significantly, except that sexual behaviour was not recorded in treated lionesses.     

Genetic drift and estimation of effective population size

The statistical properties of the standardized variance of gene frequency changes (a quantity equivalent to Wright's inbreeding coefficient) in a random mating population are studied, and new formulae for estimating the effective population size are developed. The accuracy of the formulae depends on the ratio of sample size to effective size, the number of generations involved (t), and the number of loci or alleles used. It is shown that the standardized variance approximately follows the chi(2) distribution unless t is very large, and the confidence interval of the estimate of effective size can be obtained by using this property. Application of the formulae to data from an isolated population of Dacus oleae has shown that the effective size of this population is about one tenth of the minimum census size, though there was a possibility that the procedure of sampling genes was improper.     

THE EVOLUTION OF STRATEGY VARIATION: WILL AN ESS EVOLVE?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.     

Effect of Tooth Removal on Recaptures of Raccoons

Abstract: Researchers have extensively used mark—recapture techniques to obtain information on demographic parameters of wildlife populations. However, researchers have recognized that a number of factors can influence capture probabilities of wildlife species, which in turn can bias mark—recapture estimates of demographic parameters. Tooth extraction, which is a commonly used technique in studies of mesopredator species to obtain precise age estimates and to monitor the use of vaccine baits, is an aspect of animal handling that clearly might affect the recapture probability of individuals. However, the effect that tooth removal has on the individual recapture probabilities of wildlife species is unknown. During 2005, we trapped and marked 91 raccoons (Procyon lotor) in northern Indiana, USA, as part of a mark—recapture study designed specifically to determine if tooth extractions have an effect on recapture probabilities of individuals. We performed tooth extractions on 50% of the raccoons at the time of capture, and we attempted to balance tooth extractions with respect to sex and age of raccoons. We used logistic regression to model the effects of sex, age, and tooth removal on recapture probabilities, and we used Mann—Whitney U-tests to examine the effect of tooth removal on the number of times we recaptured individuals. The probability of recapture differed between sexes but did not differ as a function of tooth removal or among age classes. In addition, we failed to detect any difference in the mean number of times that we recaptured raccoons between the tooth removed and non—tooth-removed groups. Our results suggest that managers can use tooth extractions as an effective management tool without biasing population estimates or compromising other management objectives.     

Difficulties in parentage analysis: the probability that an offspring and parent have the same heterozygous genotype.

Parentage studies often estimate the number of parents contributing to half-sib progeny arrays by counting the number of alleles attributed to unshared parents. This approach is compromised when an offspring has the same heterozygous genotype as the shared parent, for then the contribution of the unshared parent cannot be unambiguously deduced. To determine how often such cases occur, formulae for co-dominant markers with n alleles are derived here for Ph, the probability that a given heterozygous parent has an offspring with the same heterozygous genotype, and Pa, the probability that a randomly chosen offspring has the same heterozygous genotype as the shared parent. These formulae have been derived assuming Mendelian segregation with either (1) an arbitrary mating system, (2) random mating or (3) mixed mating. The maximum value of Pa under random mating is 0.25 and occurs with any two alleles each at a frequency of 0.5. The behaviour with partial selfing (where reproduction is by selfing with probability s, and random mating otherwise) is more complex. For n < or = 3 alleles, the maximum value of Pa occurs with any two alleles each at a frequency of 0.5 if s < 0.25, and with three equally frequent alleles otherwise. Numerically, the maximum value of Pa for n > or = 4 alleles occurs with n* < or = n alleles at equal frequencies, where the maximizing number of alleles n* is an increasing function of the selfing rate. Analytically, the maximum occurs with all n alleles present and equally frequent if s > or = 2/3. In addition, the potential applicability of these formulae for evolutionary studies is briefly discussed.     

Dynamics in a discrete two-species competition model: coexistence and over-compensation

The dynamic features of an over-compensating discrete two-species competition system with stable coexistence are recaptured, and it is shown how the probabilities of the different possible ecological scenarios, e.g. coexistence, may be calculated when the assumption of no over-compensation is loosened. A Bayesian methodology for calculating the probability that stable oscillations or chaos may occur in plant populations or communities is outlined. The methodology is exemplified using an experimental population of Arabidopsis thaliana. It is concluded that, when making ecological predictions it is preferable and possibly important to test for the possibility of chaotic population dynamics due to over-compensation rather than assuming a priori that over-compensation does not occur.     

Designing occupancy surveys and interpreting non‐detection when observations are imperfect

Aim Conservation practitioners use biological surveys to ascertain whether or not a site is occupied by a particular species. Widely used statistical methods estimate the probability that a species will be detected in a survey of an occupied site. However, these estimates of detection probability are alone not sufficient to calculate the probability that a species is present given that it was not detected. The aim of this paper is to demonstrate methods for correctly calculating (1) the probability a species occupies a site given one or more non‐detections, and (2) the number of sequential non‐detections necessary to assert, with a pre‐specified confidence, that a species is absent from a site. Location Occupancy data for a tree frog in eastern Australia serve to illustrate methods that may be applied anywhere species’ occupancy data are used and detection probabilities are < 1. Methods Building on Bayesian expressions for the probability that a site is occupied by a species when it is not detected, and the number of non‐detections necessary to assert absence with a pre‐specified confidence, we estimate occupancy probabilities across tree frog survey locations, drawing on information about where and when the species was detected during surveys. Results We show that the number of sequential non‐detections necessary to assert that a species is absent increases nonlinearly with the prior probability of occupancy, the probability of detection if present, and the desired level of confidence about absence. Main conclusions If used more widely, the Bayesian analytical approaches illustrated here would improve collection and interpretation of biological survey data, providing a coherent way to incorporate detection probability estimates in the design of minimum survey requirements for monitoring, impact assessment and distribution modelling.     

Reproductive success, mortality and sexual size dimorphism in the adder, Vipera berus

From 1981 to 1986 an isolated adder population was studied in the extreme south of Sweden. During this period 48 adult males and 44 adult females were marked. Male adders did not grow as large as the females. Large males had a significantly higher annual mating success and were engaged in more combats than smaller males. The mean length of recaptured males was significantly lower than that of those not recaptured, indicating a higher motality of larger males. Females brood size was positively correlated with body size. In females there was no difference in mean length of recaptured vs not recaptured individuals. The adder is one of the few snake species with male combat where males are smaller than females. I suggest that this is due to stronger selective advantages for large body size in females than in males.     

Sample size considerations in genetic polymorphism studies.

OBJECTIVES: Molecular studies for genetic polymorphisms are being carried out for a number of different applications, such as genetic disorders in different populations, pharmacogenomics, genetic identification of ethnic groups for forensic and legal applications, genetic identification of breed/stock in animals and plants for commercial applications and conservation of germ plasm. In this paper, for a random sampling scheme, we address two questions: (A) What should be the minimum size of the sample so that, with a prespecified probability, all alleles at a given locus (or haplotypes at a given set of loci) are detected? (B) What should be the sample size so that the allele frequency distribution at a given locus (or haplotype frequency distribution at a given set of loci) is estimated reliably within permissible error limits? METHODS: We have used combinatorial probabilistic arguments and Monte Carlo simulations to answer these questions. RESULTS: We found that the minimum sample size required in case A depends mainly on the prespecified probability of detecting all alleles, while in case B, it varies greatly depending on the permissible error in estimation (which will vary with the application). We have obtained the minimum sample sizes for different degrees of polymorphism at a locus under high stringency, as well as a relaxed level of permissible error. We present a detailed sampling procedure for estimating allele frequencies at a given locus, which will be of use in practical applications. CONCLUSION: Since the sample size required for reliable estimation of allele frequency distribution increases with the number of alleles at the locus, there is a strong case for using biallelic markers (like single nucleotide polymorphisms) when the available sample size is about 800 or less.     

Population size influences amphibian detection probability: implications for biodiversity monitoring programs

Monitoring is an integral part of species conservation. Monitoring programs must take imperfect detection of species into account in order to be reliable. Theory suggests that detection probability may be determined by population size but this relationship has not yet been assessed empirically. Population size is particularly important because it may induce heterogeneity in detection probability and thereby cause bias in estimates of biodiversity. We used a site occupancy model to analyse data from a volunteer-based amphibian monitoring program to assess how well different variables explain variation in detection probability. An index to population size best explained detection probabilities for four out of six species (to avoid circular reasoning, we used the count of individuals at a previous site visit as an index to current population size). The relationship between the population index and detection probability was positive. Commonly used weather variables best explained detection probabilities for two out of six species. Estimates of site occupancy probabilities differed depending on whether the population index was or was not used to model detection probability. The relationship between the population index and detectability has implications for the design of monitoring and species conservation. Most importantly, because many small populations are likely to be overlooked, monitoring programs should be designed in such a way that small populations are not overlooked. The results also imply that methods cannot be standardized in such a way that detection probabilities are constant. As we have shown here, one can easily account for variation in population size in the analysis of data from long-term monitoring programs by using counts of individuals from surveys at the same site in previous years. Accounting for variation in population size is important because it can affect the results of long-term monitoring programs and ultimately the conservation of imperiled species.     

The effects of strain heterology on the epidemiology of equine influenza in a vaccinated population

We assess the effects of strain heterology (strains that are immunologically similar but not identical) on equine influenza in a vaccinated population. Using data relating to individual animals, for both homologous and heterologous vaccinees, we estimate distributions for the latent and infectious periods, quantify the risk of becoming infected in terms of the quantity of cross-reactive antibodies to a key surface protein of the virus (haemagglutinin) and estimate the probability of excreting virus (i.e. becoming infectious) given that infection has occurred. The data suggest that the infectious period, the risk of becoming infected (for a given vaccine-induced level of cross-reactive antibodies) and the probability of excreting virus are increased for heterologously vaccinated animals when compared with homologously vaccinated animals. The data are used to parameterize a modified susceptible, exposed, infectious and recovered/resistant (SEIR) model, which shows that these relatively small differences combine to have a large effect at the population level, where populations of heterologous vaccinees face a significantly increased risk of an epidemic occurring.     

A Two-Locus Forensic Match Probability for Subdivided Populations

A two-locus match probability is presented that incorporates the effects of within-subpopulation inbreeding (consanguinity) in addition to population subdivision. The usual practice of calculating multi-locus match probabilities as the product of single-locus probabilities assumes independence between loci. There are a number of population genetics phenomena that can violate this assumption: in addition to consanguinity, which increases homozygosity at all loci simultaneously, gametic disequilibrium will introduce dependence into DNA profiles. However, in forensics the latter problem is usually addressed in part by the careful choice of unlinked loci. Hence, as is conventional, we assume gametic equilibrium here, and focus instead on between-locus dependence due to consanguinity. The resulting match probability formulae are an extension of existing methods in the literature, and are shown to be more conservative than these methods in the case of double homozygote matches. For two-locus profiles involving one or more heterozygous genotypes, results are similar to, or smaller than, the existing approaches. This revised version was published online in August 2006 with corrections to the Cover Date.     

Bayesian analysis of genetic differentiation between populations

We introduce a Bayesian method for estimating hidden population substructure using multilocus molecular markers and geographical information provided by the sampling design. The joint posterior distribution of the substructure and allele frequencies of the respective populations is available in an analytical form when the number of populations is small, whereas an approximation based on a Markov chain Monte Carlo simulation approach can be obtained for a moderate or large number of populations. Using the joint posterior distribution, posteriors can also be derived for any evolutionary population parameters, such as the traditional fixation indices. A major advantage compared to most earlier methods is that the number of populations is treated here as an unknown parameter. What is traditionally considered as two genetically distinct populations, either recently founded or connected by considerable gene flow, is here considered as one panmictic population with a certain probability based on marker data and prior information. Analyses of previously published data on the Moroccan argan tree (Argania spinosa) and of simulated data sets suggest that our method is capable of estimating a population substructure, while not artificially enforcing a substructure when it does not exist. The software (BAPS) used for the computations is freely available from http://www.rni.helsinki.fi/~mjs.