Whole-genome duplication (WGD) is believed to be a significant source of major evolutionary innovation. Redundant genes resulting from WGD are thought to be lost or acquire new functions. However, the rates of gene loss and thus temporal process of genome reshaping after WGD remain unclear. The WGD shared by all teleost fish, one-half of all jawed vertebrates, was more recent than the two ancient WGDs that occurred before the origin of jawed vertebrates, and thus lends itself to analysis of gene loss and genome reshaping. Using a newly developed orthology identification pipeline, we inferred the post-teleost-specific WGD evolutionary histories of 6,892 protein-coding genes from nine phylogenetically representative teleost genomes on a time-calibrated tree. We found that rapid gene loss did occur in the first 60 My, with a loss of more than 70-80% of duplicated genes, and produced similar genomic gene arrangements within teleosts in that relatively short time. Mathematical modeling suggests that rapid gene loss occurred mainly by events involving simultaneous loss of multiple genes. We found that the subsequent 250 My were characterized by slow and steady loss of individual genes. Our pipeline also identified about 1,100 shared single-copy genes that are inferred to have become singletons before the divergence of clupeocephalan teleosts. Therefore, our comparative genome analysis suggests that rapid gene loss just after the WGD reshaped teleost genomes before the major divergence, and provides a useful set of marker genes for future phylogenetic analysis.orthologous gene | bony vertebrates | post-WGD genome evolution T he recent rapid growth of genome data has made it possible to clarify major evolutionary events that have shaped eukaryote genomes, such as gene duplication, chromosomal rearrangement, and whole-genome duplication (WGD) (1). In particular, WGD events, known to have occurred in several major lineages of flowering plants (2), budding yeasts (3), and vertebrates (4) (Fig. 1A), are considered to have had a major impact on genomic architecture and consequently organismal features.Duplicate genes generated by WGD are typically assumed to be redundant and therefore subsequently lost in a stochastic manner. Comparative genome studies have suggested that 90% of duplicate genes were rapidly lost (5) by a neutral process (6) after WGD in budding yeast, but 20-30% of them were retained in human (7) even after several hundred million years. However, few genome-wide studies have addressed the temporal pattern of gene loss or persistence after WGD with reference to a reliable timescale (but see refs. 6 and 8). Such examination is indispensable for understanding when duplicate genes were lost and, consequently, genome structures were reshaped, during vertebrate diversification after the WGD (Fig. 1).To examine the detailed process of duplicate gene loss after WGD, one needs to estimate the number (proportion) of remaining duplicates in extant and ancestral species. For this purpose, both (i) reliably time-calibrat...

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The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control. RésuméLe but de cet article est de présenter une condition suffisante permettant de garantir que le lieu de coupure d'une classe de métriques sur la 2-sphère de révolution est réduit à une branche simple. Ce travail est motivé par des problèmes de contrôle optimal en mécanique spatiale et mécanique quantique. Des résultats globaux d'optimalité sont obtenus en transfert orbital ainsi que dans le cas des équations de Lindblad en contrôle quantique.

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