Polyploids are organisms whose genomes consist of more than two complete sets of chromosomes. Both autopolyploids and allopolyploids may display polysomic inheritance. A peculiarity of polysomic inheritance is multivalent formation during meiosis resulting in double-reduction, which occurs when sister chromatid fragments segregate into the same gamete. Double-reduction can result in gametes carrying identical-by-descent alleles and slightly increasing homozygosity. This will cause the genotypic frequencies to deviate from expected values and will thus bias the results of standard population genetic analytical methods used in molecular ecology and selective breeding. In this study, we extend existing double-reduction models to account for any even level of ploidy, and derive the symbolic expressions for genotypic frequencies via two methods. Inbreeding coefficients and heterozygosity under double-reduction and inbreeding are also calculated. Numerical solutions obtained by computer simulations are compared with analytical solutions predicted by the model to validate the model.

5Polyploids are organisms whose genomes consist of more than two complete sets 6 of chromosomes. Both autopolyploids and allopolyploids may display polysomic 7 inheritance. A peculiarity of polysomic inheritance is multivalent formation during 8 meiosis resulting in double-reduction, which occurs when sister chromatid fragments 9 are segregated into the same gamete. Double-reduction can result in gametes carry-10 ing identical-by-descent alleles and slightly increasing homozygosity. This will cause 11 the genotypic frequencies to deviate from expected values and will thus bias the re-12 sults of standard population genetic analytical methods used in molecular ecology 13 and selective breeding. In this study, we extend existing double-reduction models to 14 account for any even level of ploidy, and derive the symbolic expressions for geno-15 typic frequencies via two methods. Inbreeding coefficients and heterozygosity under 16 double-reduction and inbreeding are also calculated. Numerical solutions obtained 17 by computer simulations are compared with analytical solutions predicted by the 18 model to validate the model.19 20 ing coefficient, heterozygosity.21 3Polyploids are organisms whose genomes consist of more than two complete sets of chromosomes 23 (Madlung, 2013). They represent a significant proportion of plant species, with 30-80% of angiosperm 24 species showing polyploidy (Burow et al., 2001) and most lineages showing evidence of paleoploidy (Otto, 25 2007). Polyploid plants can arise spontaneously in nature by several mechanisms, including meiotic or 26 mitotic failures, and fusion of unreduced gametes (Comai, 2005). 27 There are two distinct mechanisms of genome duplication that result in polyploidy: allopolyploidy 28 and autopolyploidy. Autopolyploids are usually thought to arise within a species by the doubling of 29 structurally similar homologous genomes, whereas allopolyploids arise via interspecific hybridization and 30 subsequent doubling of non-homologous genomes (Parisod et al., 2010). Both autopolyploids and al-31 lopolyploids can be found among both wild and domesticated plant species. Although rare, polyploidy is 32 also found in a few species of vertebrates such as some salmonid fish (Limborg et al., 2017), the weather 33 loach (Misgurnus anguillicaudatus) (Zhou et al., 2016), the common carp (Cyprinus carpio) (David et al., 34 2003), and the African clawed frog (Xenopus laevis) (Session et al., 2016). 35In autopolyploids, more than two homologous chromosomes can pair at meiosis, resulting in the 36 formation of multivalents and polysomic inheritance (Rieger et al., 1968). A peculiarity of polysomic 37 inheritance is the possibility that a gamete inherits a single gene copy twice, termed double-reduction 38 (Butruille and Boiteux, 2000). For example, an autotetraploid individual produces a gamete . 39 In prophase I, crossovers can happen between the locus and the centromere, resulting in an exchange 40 of chromatid fragments between pairing chromosomes. In a multivalent configuration, the...

Duplicated loci, for example those associated with major histocompatibility complex (MHC) genes, often have similar DNA sequences that can be coamplified with a pair of primers. This results in genotyping difficulties and inaccurate analyses. Here, we present a method to assign alleles to different loci in amplifications of duplicated loci. This method simultaneously considers several factors that may each affect correct allele assignment. These are the sharing of identical alleles among loci, null alleles, copy number variation, negative amplification, heterozygote excess or heterozygote deficiency, and linkage disequilibrium. The possible multilocus genotypes are extracted from the alleles for each individual and weighted to estimate the allele frequencies. The likelihood of an allele configuration is calculated and is optimized with a heuristic algorithm. Monte‐Carlo simulations and three empirical MHC data sets are used as examples to evaluate the efficacy of our method under different conditions. Our new software, mhc‐typer V1.1, is freely available at https://github.com/huangkang1987/mhc-typer.

Parentage analysis is an important method that is used widely in zoological and ecological studies. Current mathematical models of parentage analyses usually assume that a population has a uniform genetic structure and that mating is panmictic. In a natural population, the geographic or social structure of a population, and/or nonrandom mating, usually leads to a genetic structure and results in genotypic frequencies deviating from those expected under the Hardy-Weinberg equilibrium (HWE). In addition, in the presence of null alleles, an observed genotype represents one of several possible true genotypes. The true father of a given offspring may thus be erroneously excluded in parentage analyses, or may have a low or negative LOD score. Here, we present a new mathematical model to estimate parentage that includes simultaneously the effects of inbreeding, null alleles, and negative amplification. The influences of these three factors on previous model are evaluated by Monte-Carlo simulations and empirical data, and the performance of our new model is compared under controlled conditions. We found that, for both simulated and empirical data, our new model outperformed other methods in many situations. We make available our methods in a new, free software package entitled parentage. This can be downloaded via http://github.com/huangkang1987/parentage.

Given the design difficulty and poor accuracy of tooth profile of noncircular gear used in noncircular gear hydraulic motor, it is proposed to reduce the design difficulty and improve the design accuracy by using arc-shaped pitch curve instead of noncircular pitch curve with continuously varying curvature. Based on the geometric relationship and transmission relationship of the noncircular planetary gear mechanism, a nonlinear programming model is constructed for the circular arc-shaped pitch curve. By solving the nonlinear programming model, a noncircular planetary gear mechanism with a modulus of m = 1.5 is designed. The noncircular gear mechanism with the arc-shaped pitch curve was machined and installed in a hydraulic motor, and an efficiency comparison experiment was conducted with a high-order elliptical noncircular gear mechanism with a continuously varying curvature. The experiment shows that the efficiency of the two noncircular gear mechanisms is basically the same, and the best speed range is 100–400 rpm. The noncircular planetary gear mechanism with an arc-shaped pitch curve designed in this paper has reasonable structure, correct transmission relationship, and simple design method, which shows that the design method proposed in this paper has a good engineering application value.