Identification of gametes and treatment of linear dependencies in the gametic QTL-relationship matrix and its inverse

The estimation of gametic effects via marker-assisted BLUP requires the inverse of the conditional gametic relationship matrix G. Both gametes of each animal can either be identified (distinguished) by markers or by parental origin. By example, it was shown that the conditional gametic relationship matrix is not unique but depends on the mode of gamete identification. The sum of both gametic effects of each animal – and therefore its estimated breeding value – remains however unaffected. A previously known algorithm for setting up the inverse of G was generalized in order to eliminate the dependencies between columns and rows of G. In the presence of dependencies the rank of G also depends on the mode of gamete identification. A unique transformation of estimates of QTL genotypic effects into QTL gametic effects was proven to be impossible. The properties of both modes of gamete identification in the fields of application are discussed.     

Estimation of the position and effect of a lethal factor locus on a molecular marker linkage map

In the mapping of DNA markers the distortion of segregation of marker genotypes is often observed, which may be caused by a lethal factor acting in filial generations derived from distant crosses. A method is presented for estimating the recombination values between a lethal factor locus and neighboring molecular markers, and the relative viability or fertilization ability of gametes or zygotes affected by the lethal factor in an F₂ population using the maximum likelihood method and the expectation conditional maximization (ECM) algorithm. Three selection models of gamete or zygote were considered, and the most likely one was determined by goodness of fit of the observed frequency of the phenotypes to the expected ones under the models. The method was applied to segregation data of molecular markers of an F₂ population consisting of 144 individuals derived from a cross between an Indica and a Japonica rice variety. The presence of a lethal factor locus (L) located on chromosome III that caused partial gametic selection in both the male and female sides was suggested. The locus L was tightly linked to RFLP marker number 23 of the RFLP linkage map of Saito et al. (1991a), and the fertilization chance of a male or female gamete possessing the lethal factor was, on average, 41.5% of that of the normal gamete.     

Derivation of single-locus relationship coefficients conditional on marker information

The coefficient of relationship is defined as the correlation between the additive genetic values of two individuals. This coefficient can be defined specifically for a single quantitative trait locus (QTL) and may deviate considerably from the overall expectation if it is taken conditional on information from linked marker loci. Conditional halfsib correlations are derived under a simple genetic model with a biallelic QTL linked to a biallelic marker locus. The conditional relationship coefficients are shown to depend on the recombination rate between the marker and the QTL and the population frequency of the marker alleles, but not on parameters of the QTL, i.e. number and frequency of QTL alleles, degree of dominance etc., nor on the (usually unknown) QTL genotype of the sire. Extensions to less simplified cases (multiple alleles at the marker locus and the QTL, two marker loci flanking the QTL) are given. For arbitrary pedigrees, conditional relationship coefficients can also be derived from the conditional gametic covariance matrix suggested by Fernando and Grossman (1989). The connection of these two approaches is discussed. The conditional relationship coefficient can be used for marker-assisted genetic evaluation as well as for the detection of QTL and the estimation of their effects.     

Estimation of Bayesian confidence intervals for the product of conditional survival probabilities

A model is derived to estimate the survival probability of a time interval when censorings occur. The time interval is divided into partial intervals in order to obtain the conditional survival probabilities, each of which is a parameter of a Binomial distributed random variable. To allow for the dependence between the events in the different intervals these parameters are transformed. Corresponding a priori density functions are formulated regarding both the Bayesian uniform distribution and the special model. The a posteriori density function is derived for the product of the conditional survival probabilities, and formulae for the BAYE sian confidence interval and the expectation are given. Lower and upper bounds for the confidence interval and the expectation are derived. Some examples are given to compare the results with other methods.     

The ecological genetics of conditional strategies

We develop a quantitative genetic model for conditional strategies that incorporates the ecological realism of previous strategic models. Similar to strategic models, the results show that environmental heterogeneity, cue reliability, and environment-dependent fitness trade-offs for the alternative tactics of the conditional strategy interact to determine when conditional strategies will be favored and that conditional strategies should be a common form of adaptive variation in nature. The results also show that conditional and unconditional development can be maintained in one of two ways: by frequency-dependent selection or by the maintenance of genetic variation that exceeds the threshold for induction. We then modified the model to take into account variance in exposures to the environmental cue as well as variance in response to the cue, which allows a derivation of a dose-response curve. Here the results showed that increasing the genetic variance for response both flattens and shifts the dose-response curve. Finally, we modify the model to derive the dose-response curve for a population polymorphic for a gene that blocks expression of the conditional strategy. We illustrate the utility of the model by application to predator-induced defense in an intertidal barnacle and compare the results with phenotypic models of selection.     

A mixed polyploid model for linkage analysis in outcrossing tetraploids using a pseudo-test backcross design.

Based on how chromosomes pair at meiosis, the nature of polyploids can be described by bivalent polyploids, multivalent polyploids, and mixed polyploids. In bivalent polyploids, only two chromosomes pair, during which two more similar chromosomes have a higher pairing probability (preferential pairing) than two less similar chromosomes, whereas in multivalent polyploids more than two chromosomes pair at a time, which results in double reduction. Preferential chromosome pairings and double reduction affect the frequencies of gamete formation and, therefore, linkage analysis of polymorphic markers in bivalent and multivalent polyploids, respectively. For mixed polyploids, in which both bivalent and multivalent formations occur simultaneously, linkage analysis is affected by both preferential pairings and double reduction. In this study, we develop a hierarchical maximum likelihood model for discerning gamete genotypes derived from different pairing mechanisms and different formation modes. The first-stage model in the hierarchy is formulated to characterize the relative frequencies of bivalent and multivalent pairing configurations in terms of the preferential pairing factor. The second-stage model is derived to rule out identical gamete genotypes into their different formation modes with relative probabilities determined by the recombination fraction. The first-stage pairing mechanism and second-stage formation mode are integrated to provide the simultaneous maximum likelihood estimates of the preferential pairing factor, the frequency of double reduction, and the recombination fraction, by implementing the EM algorithm. We performed extensive simulation studies to demonstrate the statistical properties of our hierarchical model for linkage analysis in tetraploids. The implications of our model for polyploid linkage mapping are discussed.     

Mixture model for inferring susceptibility to mastitis in dairy cattle: a procedure for likelihood-based inference

A Gaussian mixture model with a finite number of components and correlated random effects is described. The ultimate objective is to model somatic cell count information in dairy cattle and to develop criteria for genetic selection against mastitis, an important udder disease. Parameter estimation is by maximum likelihood or by an extension of restricted maximum likelihood. A Monte Carlo expectation-maximization algorithm is used for this purpose. The expectation step is carried out using Gibbs sampling, whereas the maximization step is deterministic. Ranking rules based on the conditional probability of membership in a putative group of uninfected animals, given the somatic cell information, are discussed. Several extensions of the model are suggested.     

First and second moment of counts of words in random texts generated by Markov chains

An exact expression for the variance of random frequency thata given word has in text generated by a Markov chain is presented.The result is applied to periodic Markov chains, which describethe protein-coding DNA sequences better than simple Markov chains.A new solution to the problem of word overlap is proposed. Itwas found that the expected frequency and overlapping propertiesdetermine most of the variance. The expectation and varianceof counts for triplets are compared with experimental countsin Escherichia coli coding sequences.     

Genotype dynamics in populations with breeding pairs

A new model of pair fecundity is proposed for populations with breeding pairs. The model is based upon the assumption of strong predominance of male gamete production over female. Corresponding equations for genotype dynamics are derived. It is established that the usual equation for allele frequency dynamics in panmictic populations cannot be applied to the considered system. In spite of accidental pair formation and accidental gametic fusion within each pair, gamete panmixia of the population as a whole is violated and gene dynamics should be described by means of genotype frequencies. The condition of reduction to the “classical” dynamics is achieved.     

A mixed model method to predict QTL-cluster effects using trait and marker information in a multi-group population.

In this study, a mixed model method using trait phenotype and marker information was developed for genetic evaluation of animals in a crossbred population originated from several founder genetic groups. The situation in which a cluster of QTLs is located in a particular chromosome region and is marked by two flanking markers is considered. With this method, the conditional expectation of the identity-by-descent proportion for the QTL-cluster marked and the genetic variances and covariances, given genetic group and marker information, are properly taken into account. The structure of segregation variance used in this method is different from that in the case of a single QTL marked. The current method provides best linear unbiased estimation of the relevant fixed effects and best linear unbiased prediction of the additive effects for the QTL-cluster marked and of the additive effects of the remaining polygenes. A small numerical example is given to illustrate the current prediction procedure.     

Persistence of Repeated Sequences That Evolve by Replication Slippage

The evolution of short repeated sequences by replication slippage under the assumption of selective neutrality is modeled using a linear birth and death process. The equilibrium distribution, the distribution of the life expectancy of a repeated sequence when the process starts from two repeats, the age distribution of repeats, the probability of obtaining two genes with i and j copies which diverged t generations ago and the conditional variance of copy number given the repeat number is more than one are computed. The distributions of life expectancy and age are shown to have long tails. Also the statistic which estimates the conditional variance is shown to have a large coefficient of variation. Using these theoretical results, we develop an approximate test of our model and analyze persistent repeated sequences found in the primate beta-globin gene region and Oenothera chloroplast DNA which are polymorphic within species. We found one sequence in Oenothera chloroplast DNA which does not fit to our neutral model.     

GAMETE INTERACTIONS AND GENETIC DIFFERENTIATION AMONG THREE SYMPATRIC POLYCHAETES

The evolution of gamete incompatibility between free-spawning marine invertebrate species has been explained by three hypotheses: (1) independent divergence at gamete recognition loci; (2) selection against hybrids; and (3) a process of sexual selection involving polymorphic gamete recognition loci (Metz and Palumbi 1996). The first two hypotheses predict that gamete incompatibility appears only after gene flow has been halted for other reasons and the third that gamete incompatibility appears simultanously with blocks to gene flow. Here I show that gametes of three sympatric polychaetes in the genus Arctonoe are compatible in all crosses, over a broad range of gamete concentrations and contact times. Although at least some hybrid crosses produce fertile adults, allozyme and mitochondrial DNA sequence data indicate that the three species do not regularly exchange genes. These data are consistent with predictions of the first two hypotheses for the evolution of gamete incompatibility, but allow rejection of the third hypothesis. Gametes of the three species are compatible despite estimated divergence times of 1–3 M.Y.B.P.; in several other marine invertebrates, divergence times of the same magnitude are associated with asymmetric or complete gamete incompatibility. It appears likely that segregation of symbiotic adults on their respective host species restricts mating opportunities, and thus gene flow, among Arctonoe species.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Variance as a signature of neural computations during decision making

Traditionally, insights into neural computation have been furnished by averaged firing rates from many stimulus repetitions or trials. We pursue an analysis of neural response variance to unveil neural computations that cannot be discerned from measures of average firing rate. We analyzed single-neuron recordings from the lateral intraparietal area (LIP), during a perceptual decision-making task. Spike count variance was divided into two components using the law of total variance for doubly stochastic processes: (1) variance of counts that would be produced by a stochastic point process with a given rate, and loosely (2) the variance of the rates that would produce those counts (i.e., "conditional expectation"). The variance and correlation of the conditional expectation exposed several neural mechanisms: mixtures of firing rate states preceding the decision, accumulation of stochastic "evidence" during decision formation, and a stereotyped response at decision end. These analyses help to differentiate among several alternative decision-making models.     

Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations

Summary The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.     

Estimation of the marker gene frequency and linkage disequilibrium from conditional marker data.

A method is proposed to calculate the maximum likelihood estimate of gene frequency and linkage disequilibrium from disease-codominant marker conditional data. The method is illustrated using data on sickle-cell anemia and Duchenne muscular dystrophy and linked polymorphic restriction endonuclease cleavage sites.     

Markov Model of Haploid Random Mating with Given Distribution of Population Size

An exact Markov chain model is formulated and computed for random mating in a haploid gamete pool. There are two versions of the gamete, and there is a finite number of diploid monoecious organisms. The founder population is given, and the subsequent generations allow a prescribed statistical distribution over different population sizes. The non-homogeneous Markov chain works on the haploid gamete level provided the probability of self-fertilization is 1/n, where n is the number of diploid individuals. Standard deviations of gamete frequencies and fixation probabilities are calculated. Effective population sizes for different population size distributions are estimated, including periodic bottlenecks.     

The effect of recurrent mutation on the frequency spectrum of a segregating site and the age of an allele

The sample frequency spectrum of a segregating site is the probability distribution of a sample of alleles from a genetic locus, conditional on observing the sample to be polymorphic. This distribution is widely used in population genetic inferences, including statistical tests of neutrality in which a skew in the observed frequency spectrum across independent sites is taken as a signature of departure from neutral evolution. Theoretical aspects of the frequency spectrum have been well studied and several interesting results are available, but they are usually under the assumption that a site has undergone at most one mutation event in the history of the sample. Here, we extend previous theoretical results by allowing for at most two mutation events per site, under a general finite allele model in which the mutation rate is independent of current allelic state but the transition matrix is otherwise completely arbitrary. Our results apply to both nested and nonnested mutations. Only the former has been addressed previously, whereas here we show it is the latter that is more likely to be observed except for very small sample sizes. Further, for any mutation transition matrix, we obtain the joint sample frequency spectrum of the two mutant alleles at a triallelic site, and derive a closed-form formula for the expected age of the younger of the two mutations given their frequencies in the population. Several large-scale resequencing projects for various species are presently under way and the resulting data will include some triallelic polymorphisms. The theoretical results described in this paper should prove useful in population genomic analyses of such data.