Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence

Summary A Bayesian method was developed for identifying genetic markers linked to quantitative trait loci (QTL) by analyzing data from daughter or granddaughter designs and single markers or marker pairs. Traditional methods may yield unrealistic results because linkage tests depend on number of markers and QTL gene effects associated with selected markers are overestimated. The Bayesian or posterior probability of linkage combines information from a daughter or granddaughter design with the prior probability of linkage between a marker locus and a QTL. If the posterior probability exceeds a certain quantity, linkage is declared. Upon linkage acceptance, Bayesian estimates of marker-QTL recombination rate and QTL gene effects and frequencies are obtained. The Bayesian estimates of QTL gene effects account for different amounts of information by shrinking information from data toward the mean or mode of a prior exponential distribution of gene effects. Computation of the Bayesian analysis is feasible. Exact results are given for biallelic QTL, and extensions to multiallelic QTL are suggested.     

Efficient Bayesian inference for mechanistic modelling with high-throughput data

Bayesian methods are routinely used to combine experimental data with detailed mathematical models to obtain insights into physical phenomena. However, the computational cost of Bayesian computation with detailed models has been a notorious problem. Moreover, while high-throughput data presents opportunities to calibrate sophisticated models, comparing large amounts of data with model simulations quickly becomes computationally prohibitive. Inspired by the method of Stochastic Gradient Descent, we propose a minibatch approach to approximate Bayesian computation. Through a case study of a high-throughput imaging scratch assay experiment, we show that reliable inference can be performed at a fraction of the computational cost of a traditional Bayesian inference scheme. By applying a detailed mathematical model of single cell motility, proliferation and death to a data set of 118 gene knockdowns, we characterise functional subgroups of gene knockdowns, each displaying its own typical combination of local cell density-dependent and -independent motility and proliferation patterns. By comparing these patterns to experimental measurements of cell counts and wound closure, we find that density-dependent interactions play a crucial role in the process of wound healing.     

Admixture of genetic lineages of different glacial origin: a case study of <Emphasis Type="Italic">Abies alba</Emphasis> Mill. in the Carpathians

Genetic exchange between divergent lineages of silver fir (Abies alba Mill.) was studied in the Ukrainian Carpathians where two expanding populations originating from different glacial refugia meet. The study included 33 silver fir populations from Ukraine, Romania and Slovakia which were investigated using the maternally inherited mitochondrial nad5-4 marker and biparentally inherited nuclear microsatellites. The boundary between mitochondrial lineages is very sharp; only six populations containing a mixture of different haplotypes were found. Bayesian analysis of population structure based on seven nSSR loci revealed the existence of two clusters which coincided fairly well with mitochondrial lineages. Both haplotype frequencies and proportions of clusters identified by the Bayesian analysis exhibited a clinal transition over the contact zone, with cline widths of 17.6 km for mitochondrial haplotype frequencies (reflecting gene flow by seeds) and 119.6 km for Bayesian clusters based on nSSR (reflecting gene flow by pollen). Allelic richness and gene diversity differ significantly between mitochondrial lineages, the Balkan group being more variable, but an increase in gene diversity towards the boundary between lineages was observed only within the Balkan lineage. The observed patterns are suggested to reflect the postglacial colonization and historical gene flow. They demonstrated that in silver fir as a wind-dispersed species, the colonization front is quite continuous, and the survival of migrant seeds in already established populations is low. The relative contribution of pollen-mediated gene flow to genetic exchange between divergent lineages associated with glacial refugia is much higher than in the case of seeds, but pollen dispersal distance is lower than suggested by earlier studies.     

Bayesian estimates of unknown parameters of mathematical models of the dynamics of the mutation process and changes in the ratio of cells having passed different numbers of divisions in a culture

With the help of Bayesian methods, the conditions of solving experimental data samples and their divisions were established, equivalence of estimations of unknown linear dynamic models parameters proved, the estimations having been worked out by both accounting calculation errors and using their compensations with additional noise in the original model's discrete analog. The results are used in mathematical modeling of changing intensity of the process of hereditary pathology frequencies, and the process of changing the ratio of cells having passed different numbers of divisions in the culture.     

TRAB: testing whether mutation frequencies are above an unknown background

To rigorously determine whether a gene or a set of genes have alterations that are involved in carcinogenesis requires a comparison of the prevalence of identified changes to a control mutation frequency present in tumor DNA. To facilitate this task, we develop a testing approach and the associated R library, called TRAB, that evaluates whether the frequency of somatic mutation in a given gene is higher than that observed in a control group of genes. Specifically, we test the null hypothesis that the frequency belongs to a control population of frequencies, against the alternative hypothesis that the frequency is higher. Mutation frequencies in the control group are themselves allowed to be variable. TRAB computes the a posteriori probability and the Bayes factor for the hypothesis using a hierarchical Bayesian approach.     

Statistical analysis of DNA sequences in the neighborhood of splice sites

Prediction of gene sequences and their exon-intron structure in large eukaryotic genomic sequences is one of the central problems of mathematical biology. Solving this problem involves, in particular, high-accuracy splice site recognition. Using statistical analysis of a splice site-containing human gene fragment database, some characteristic features were described for nucleotide sequences in the splicing site neighborhood, the frequencies of all nucleotides and dinucleotides were determined, and those with frequencies increased or decreased in comparison to a random sequence were identified. The results can be used in sequence annotation, splicing site prediction, and the recognition of the gene exon-intron structure.     

Empirical Bayes procedure for estimating genetic distance between populations and effective population size

We developed an empirical Bayes procedure to estimate genetic distances between populations using allele frequencies. This procedure makes it possible to describe the skewness of the genetic distance while taking full account of the uncertainty of the sample allele frequencies. Dirichlet priors of the allele frequencies are specified, and the posterior distributions of the various composite parameters are obtained by Monte Carlo simulation. To avoid overdependence on subjective priors, we adopt a hierarchical model and estimate hyperparameters by maximizing the joint marginal-likelihood function. Taking advantage of the empirical Bayesian procedure, we extend the method to estimate the effective population size using temporal changes in allele frequencies. The method is applied to data sets on red sea bream, herring, northern pike, and ayu broodstock. It is shown that overdispersion overestimates the genetic distance and underestimates the effective population size, if it is not taken into account during the analysis. The joint marginal-likelihood function also estimates the rate of gene flow into island populations.     

Reverse Engineering of Gene Regulatory Networks: A Comparative Study

Reverse engineering of gene regulatory networks has been an intensively studied topic in bioinformatics since it constitutes an intermediate step from explorative to causative gene expression analysis. Many methods have been proposed through recent years leading to a wide range of mathematical approaches. In practice, different mathematical approaches will generate different resulting network structures, thus, it is very important for users to assess the performance of these algorithms. We have conducted a comparative study with six different reverse engineering methods, including relevance networks, neural networks, and Bayesian networks. Our approach consists of the generation of defined benchmark data, the analysis of these data with the different methods, and the assessment of algorithmic performances by statistical analyses. Performance was judged by network size and noise levels. The results of the comparative study highlight the neural network approach as best performing method among those under study.     

DMLE+: Bayesian linkage disequilibrium gene mapping

SUMMARY: The program DMLE+ allows Bayesian inference of the location of a gene carrying a mutation influencing a discrete trait (such as a disease) and/or other parameters of interest (such as mutation age) based on the observed linkage disequilibrium at multiple genetic markers. DMLE+ uses either individual marker genotypes, or haplotypes, integrates over uncertain population allele frequencies, and can incorporate prior information about gene location from an annotated human genome sequence. AVAILABILITY: DMLE+ is available in both Windows GUI and portable UNIX command line versions at http://dmle.org.     

The "battle of the sexes": a genetic model with limit cycle behavior

A two-locus genetic model, based on Dawkins "sex war" game, with the fitness of the genotypes at each locus depending on the gene frequencies at the other, is shown to give rise to a stable limit cycle. The mathematical analysis involves averaging techniques and elliptic integrals.     

How does farmer connectivity influence livestock genetic structure? A case‐study in a Vietnamese goat population

Assessing how genes flow across populations is a key component of conservation genetics. Gene flow in a natural population depends on ecological traits and the local environment, whereas for a livestock population, gene flow is driven by human activities. Spatial organization, relationships between farmers and their husbandry practices will define the farmer's network and so determine farmer connectivity. It is thus assumed that farmer connectivity will affect the genetic structure of their livestock. To test this hypothesis, goats reared by four different ethnic groups in a Vietnamese province were genotyped using 16 microsatellites. A Bayesian approach and spatial multivariate analysis (spatial principal component analysis, sPCA) were used to identify subpopulations and spatial organization. Ethnic group frequencies, husbandry practices and altitude were used to create cost maps that were implemented in a least-cost path approach. Genetic diversity in the Vietnamese goat population was low (0.508) compared to other local Asian breeds. Using a Bayesian approach, three clusters were identified. sPCA confirmed these three clusters and also that the genetic structure showed a significant spatial pattern. The least-cost path analysis showed that genetic differentiation was significantly correlated (0.131–0.207) to ethnic frequencies and husbandry practices. In brief, the spatial pattern observed in the goat population was the result of complex gene flow governed by the spatial distribution of ethnic groups, ethnicity and husbandry practices. In this study, we clearly linked the livestock genetic pattern to farmer connectivity and showed the importance of taking into account spatial information in genetic studies.     

Genetic Structure of PILEOLARIA PSEUDOMILITARIS (Polychaeta: Spirorbidae)

The genetic structure of Pileolaria pseudomilitaris was studied by means of gene-diversity analysis of allozyme frequencies. At an esterase locus, most of the gene diversity was due to subdivision of the population into colonies and subpopulations separated by less than 100 meters. Gene frequencies at a phosphoglucose isomerase locus were similar over many kilometers, but differed between two habitat types. Differences between colonies are attributed to drift and founder effect; similarities over greater distances are attributed to similar selection pressures. A mathematical appendix details the method of gene diversity analysis for a multi-leveled, hierarchically subdivided population.     

Simultaneous learning of instantaneous and time-delayed genetic interactions using novel information theoretic scoring technique

ABSTRACT: BACKGROUND: Understanding gene interactions is a fundamental question in systems biology. Currently, modeling of gene regulations using the Bayesian Network (BN) formalism assumes that genes interact either instantaneously or with a certain amount of time delay. However in reality, biological regulations, both instantaneous and time-delayed, occur simultaneously. A framework that can detect and model both these two types of interactions simultaneously would represent gene regulatory networks more accurately. RESULTS: In this paper, we introduce a framework based on the Bayesian Network (BN) formalism that can represent both instantaneous and time-delayed interactions between genes simultaneously. A novel scoring metric having rm mathematical underpinnings is also proposed that, unlike other recent methods, can score both interactions concurrently and takes into account the reality that multiple regulators can regulate a gene jointly, rather than in an isolated pair-wise manner. Further, a gene regulatory network (GRN) inference method employing an evolutionary search that makes use of the framework and the scoring metric is also presented. CONCLUSION: By taking into consideration the biological fact that both instantaneous and time-delayed regulations can occur among genes, our approach models gene interactions with greater accuracy. The proposed framework is efcient and can be used to infer gene networkshaving multiple orders of instantaneous and time-delayed regulations simultaneously. Experiments are carried out using three different synthetic networks (with three different mechanisms for generating synthetic data) as well as real life networks of Saccharomyces cerevisiae, E. coli and cyanobacteria gene expression data. The results show the effectiveness of our approach.     

Localization of a quantitative trait locus via a Bayesian approach

A Bayesian approach to the direct mapping of a quantitative trait locus (QTL), fully utilizing information from multiple linked gene markers, is presented in this paper. The joint posterior distribution (a mixture distribution modeling the linkage between a biallelic QTL and N gene markers) is computationally challenging and invites exploration via Markov chain Monte Carlo methods. The parameter's complete marginal posterior densities are obtained, allowing a diverse range of inferences. Parameters estimated include the QTL genotype probabilities for the sires and the offspring, the allele frequencies for the QTL, and the position and additive and dominance effects of the QTL. The methodology is applied through simulation to a half-sib design to form an outbred pedigree structure where there is an entire class of missing information. The capacity of the technique to accurately estimate parameters is examined for a range of scenarios.     

The effect of a fusion of subpopulations on the total fixation index

Summary A general mathematical expression is found for the decrease in the fixation index of a population where subpopulations with different gene frequencies fuse. It is shown that the use of Wright's formulas for a hierarchic structure will not necessarily give the correct result in this situation, since the conditions for their application are usually not satisfied. Two examples are given, one with fusions among subpopulations with a continuously distributed gene frequency, and one with data from real observations producing a discrete distribution.     

Evolution of Primate Gene Expression: Drift and Corrective Sweeps?

Changes in gene expression play an important role in species' evolution. Earlier studies uncovered evidence that the effect of mutations on expression levels within the primate order is skewed, with many small downregulations balanced by fewer but larger upregulations. In addition, brain-expressed genes appeared to show an increased rate of evolution on the branch leading to human. However, the lack of a mathematical model adequately describing the evolution of gene expression precluded the rigorous establishment of these observations. Here, we develop mathematical tools that allow us to revisit these earlier observations in a model-testing and inference framework. We introduce a model for skewed gene-expression evolution within a phylogenetic tree and use a separate model to account for biological or experimental outliers. A Bayesian Markov chain Monte Carlo inference procedure allows us to infer the phylogeny and other evolutionary parameters, while quantifying the confidence in these inferences. Our results support previous observations; in particular, we find strong evidence for a sustained positive skew in the distribution of gene-expression changes in primate evolution. We propose a “corrective sweep” scenario to explain this phenomenon.     

Fundamental differences between the methods of maximum likelihood and maximum posterior probability in phylogenetics

Using a four-taxon example under a simple model of evolution, we show that the methods of maximum likelihood and maximum posterior probability (which is a Bayesian method of inference) may not arrive at the same optimal tree topology. Some patterns that are separately uninformative under the maximum likelihood method are separately informative under the Bayesian method. We also show that this difference has impact on the bootstrap frequencies and the posterior probabilities of topologies, which therefore are not necessarily approximately equal. Efron et al. (Proc. Natl. Acad. Sci. USA 93:13429-13434, 1996) stated that bootstrap frequencies can, under certain circumstances, be interpreted as posterior probabilities. This is true only if one includes a non-informative prior distribution of the possible data patterns, and most often the prior distributions are instead specified in terms of topology and branch lengths. [Bayesian inference; maximum likelihood method; Phylogeny; support.].     

A multiorganism based method for Bayesian gene network estimation

The primary goal of this article is to infer genetic interactions based on gene expression data. A new method for multiorganism Bayesian gene network estimation is presented based on multitask learning. When the input datasets are sparse, as is the case in microarray gene expression data, it becomes difficult to separate random correlations from true correlations that would lead to actual edges when modeling the gene interactions as a Bayesian network. Multitask learning takes advantage of the similarity between related tasks, in order to construct a more accurate model of the underlying relationships represented by the Bayesian networks. The proposed method is tested on synthetic data to illustrate its validity. Then it is iteratively applied on real gene expression data to learn the genetic regulatory networks of two organisms with homologous genes.