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Moment closure in a Moran model with recombination  [PDF]
Ellen Baake,Thiemo Hustedt
Quantitative Biology , 2011,
Abstract: We extend the Moran model with single-crossover recombination to include general recombination and mutation. We show that, in the case without resampling, the expectations of products of marginal processes defined via partitions of sites form a closed hierarchy, which is exhaustively described by a finite system of differential equations. One thus has the exceptional situation of moment closure in a nonlinear system. Surprisingly, this property is lost when resampling (i.e., genetic drift) is included.
An exactly solved model for mutation, recombination and selection  [PDF]
Michael Baake,Ellen Baake
Mathematics , 2002,
Abstract: It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of M\"obius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.
Moran sets and Moran classes
Zhiying Wen
Chinese Science Bulletin , 2001, DOI: 10.1007/BF02901155
Abstract: The purpose of this survey is to present Moran sets and Moran classes which generalize the classical selfsimilar sets from the following points: (i) The placements of the basic sets at each step of the constructions can be arbitrary; (ii) the contraction ratios may be different at each step; and (iii) the lower limit of the contraction ratios permits zero. In this discussion we will present geometrical properties and results of dimensions of these sets and classes, and discuss conformai Moran sets and random Moran sets as well.
Indica and Japonica Crosses Resulting in Linkage Block and Recombination Suppression on Rice Chromosome 12  [PDF]
Yulin Jia, Melissa H. Jia, Xueyan Wang, Guangjie Liu
PLOS ONE , 2012, DOI: 10.1371/journal.pone.0043066
Abstract: Understanding linkage block size and molecular mechanisms of recombination suppression is important for plant breeding. Previously large linkage blocks ranging from 14 megabases to 27 megabases were observed around the rice blast resistance gene Pi-ta in rice cultivars and backcross progeny involving an indica and japonica cross. In the present study, the same linkage block was further examined in 456 random recombinant individuals of rice involving 5 crosses ranging from F2 to F10 generation, with and without Pi-ta containing genomic indica regions with both indica and japonica germplasm. Simple sequence repeat markers spanning the entire chromosome 12 were used to detect recombination break points and to delimit physical size of linkage blocks. Large linkage blocks ranging from 4.1 megabases to 10 megabases were predicted from recombinant individuals involving genomic regions of indica and japonica. However, a significantly reduced block from less than 800 kb to 2.1megabases was identified from crosses of indica with indica rice regardless of the existence of Pi-ta. These findings suggest that crosses of indica and japonica rice have significant recombination suppression near the centromere on chromosome 12.
Single-crossover dynamics: finite versus infinite populations  [PDF]
Ellen Baake,Inke Herms
Mathematics , 2006,
Abstract: Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarise and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the {\em expected} type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.
Rate of decay in admixture linkage disequilibrium and its implication in gene mapping
Shiheng Tao,Xiaoming Liu,Jianhua Chu,Rongmei Zhang,Liping Du,Zewei Luo
Chinese Science Bulletin , 2001, DOI: 10.1007/BF03183263
Abstract: Modeling linkage disequilibria (LD) between genes usually observed in admixed natural populations has been shown an effective approach in high-resolution mapping of disease genes in humans. A prerequisite to obtain accurate estimation of recombination fraction between genes at a marker locus and the disease locus using the approach is a reliable prediction of the proportion of the admixture populations. The present study suggested the use of gene frequencies to predict the estimate of the admixture proportion based on the observation that the gene frequencies are much more stable quantities than the haplotype frequencies over evolution of the population. In this paper, we advanced the theory and methods by which the decay rate of nonlinear term of LD in admixed population may be used to estimate the recombination fraction between the genes. Theoretical analysis and simulation study indicate that, the larger the difference of gene frequencies between parental populations and the more closely the admixture proportion approaches 0.5, the more important the nonlinear term of the LD in the admixed population, and hence the more informative such admixed populations in the high-resolution gene mapping practice.
Rate of decay in admixture linkage disequilibrium and its implication in gene mapping

TAO Shiheng,LIU Xiaoming,CHU Jianhua,ZHANG Rongmei,DU Liping,LUO Zewei,

科学通报(英文版) , 2001,
Abstract: Modeling linkage disequilibria (LD) between genes usually observed in admixed natural populations has been shown an effective approach in high-resolution mapping of disease genes in humans. A prerequisite to obtain accurate estimation of recombination fraction between genes at a marker locus and the disease locus using the approach is a reliable prediction of the proportion of the admixture populations. The present study suggested the use of gene frequencies to predict the estimate of the admixture proportion based on the observation that the gene frequencies are much more stable quantities than the haplotype frequencies over evolution of the population. In this paper, we advanced the theory and methods by which the decay rate of nonlinear term of LD in admixed population may be used to estimate the recombination fraction between the genes. Theoretical analysis and simulation study indicate that, the larger the difference of gene frequencies between parental populations and the more closely the admixture proportion approaches 0.5, the more important the nonlinear term of the LD in the admixed population, and hence the more informative such admixed populations in the high-resolution gene mapping practice.
Genetic linkage map of a wild genome: genomic structure, recombination and sexual dimorphism in bighorn sheep
Jocelyn Poissant, John T Hogg, Corey S Davis, Joshua M Miller, Jillian F Maddox, David W Coltman
BMC Genomics , 2010, DOI: 10.1186/1471-2164-11-524
Abstract: Bighorn sheep population-specific maps differed slightly in contiguity but were otherwise very similar in terms of genomic structure and recombination rates. The joint analysis of the two pedigrees resulted in a highly contiguous map composed of 247 microsatellite markers distributed along all 26 autosomes and the X chromosome. The map is estimated to cover about 84% of the bighorn sheep genome and contains 240 unique positions spanning a sex-averaged distance of 3051 cM with an average inter-marker distance of 14.3 cM. Marker synteny, order, sex-averaged interval lengths and sex-averaged total map lengths were all very similar between sheep species. However, in contrast to domestic sheep, but consistent with the usual pattern for a placental mammal, recombination rates in bighorn sheep were significantly greater in females than in males (~12% difference), resulting in an autosomal female map of 3166 cM and an autosomal male map of 2831 cM. Despite differing genome-wide patterns of heterochiasmy between the sheep species, sexual dimorphism in recombination rates was correlated between orthologous intervals.We have developed a first-generation bighorn sheep linkage map that will facilitate future studies of the genetic architecture of trait variation in this species. While domestication has been hypothesized to be responsible for the elevated mean recombination rate observed in domestic sheep, our results suggest that it is a characteristic of Ovis species. However, domestication may have played a role in altering patterns of heterochiasmy. Finally, we found that interval-specific patterns of sexual dimorphism were preserved among closely related Ovis species, possibly due to the conserved position of these intervals relative to the centromeres and telomeres. This study exemplifies how transferring genomic resources from domesticated species to close wild relative can benefit evolutionary ecologists while providing insights into the evolution of genomic structure and
Global haplotype partitioning for maximal associated SNP pairs
Ali Katanforoush, Mehdi Sadeghi, Hamid Pezeshk, Elahe Elahi
BMC Bioinformatics , 2009, DOI: 10.1186/1471-2105-10-269
Abstract: In comparison with other methods, our algorithm reports blocks of larger average size. Nevertheless, the haplotype diversity within the blocks is captured by a small number of tagSNPs. Resampling HapMap haplotypes under a block-based model of recombination showed that our algorithm is robust in reproducing the same partitioning for recombinant samples. Our algorithm performed better than previously reported models in a case-control association study aimed at mapping a single locus trait, based on simulation results that were evaluated by a block-based statistical test. Compared to methods of haplotype block partitioning, we performed best on detection of recombination hotspots.Our proposed method divides chromosomes into the regions within which allelic associations of SNP pairs are maximized. This approach presents a native design for dimension reduction in genome-wide association studies. Our results show that the pairwise allelic association of SNPs can describe various features of genomic variation, in particular recombination hotspots.Analysis of Single Nucleotide Polymorphisms (SNPs) in the DNA of unrelated individuals revealed a block-like structure of haplotype variation along the human genome. Using the first available genome-wide data of SNPs on chromosome 21, Patil et al. [1] showed that in particular regions on the chromosome, the observed diversity of SNP haplotypes is less than the expected. Almost at the same time, a similar structure in haplotypes within a region of 103 SNPs on chromosome region 5q31 was reported by Daly et al. [2]. In the latter study, a block structure of haplotypes was revealed using a Hidden Markov Model for estimating recombination rates. This approach, unlike models based on haplotype diversity, incorporated a quantity measuring Linkage Disequilibrium (LD) between pairs of SNPs.It is well known that effects such as population bottlenecks, geographic isolation, and natural selection can increase the extent of linkage disequilibr
Local dimensions in Moran constructions  [PDF]
Antti K?enm?ki,Bing Li,Ville Suomala
Mathematics , 2015,
Abstract: We study the dimensional properties of Moran sets and Moran measures in doubling metric spaces. In particular, we consider local dimensions and $L^q$-dimensions. We generalize and extend several existing results in this area.
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