In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson-Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ , respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ + 1) distribution, while the reverse is true for Coag α,θ . This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ). Repeated application of the Frag α operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation-fragmentation duality.
Exonic deletions disrupting the autism susceptibility candidate 2 (AUTS2) gene have been demonstrated as causal variants leading to neurodevelopmental disorders (NDDs) such as autism spectrum disorder (ASD) and developmental delay (DD). Here, we report on 830 kb de novo deletion at chromosome 7q11.22 in a 4-year-old male patient with ASD and DD. This deletion disrupts the promoter region and exon 1 of AUTS2, potentially leading to complete haploinsuffiency of the gene. In addition, we discuss the clinical presentation of the de novo deletion in the light of the previous studies describing deletions of AUTS2 in NDDs.
Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In [7] authors exploit an analogy with the theory of regenerative composition and partition structures, and provide various characterizations of the associated exchangeable random partitions. This paper gives parallel results for the further generalized model with lines of descent following a coalescent with simultaneous multiple collisions.
Classification of DNA sequences is an important issue in the bioinformatics study, yet most existing methods for phylogenetic analysis including Multiple Sequence Alignment (MSA) are time-consuming and computationally expensive. The alignment-free methods are popular nowadays, whereas the manual intervention in those methods usually decreases the accuracy. Also, the interactions among nucleotides are neglected in most methods. Here we propose a new Accumulated Natural Vector (ANV) method which represents each DNA sequence by a point in ℝ
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. By calculating the Accumulated Indicator Functions of nucleotides, we can further find an Accumulated Natural Vector for each sequence. This new Accumulated Natural Vector not only can capture the distribution of each nucleotide, but also provide the covariance among nucleotides. Thus global comparison of DNA sequences or genomes can be done easily in ℝ
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. The tests of ANV of datasets of different sizes and types have proved the accuracy and time-efficiency of the new proposed ANV method.
GRM7, the gene encoding metabotropic glutamate receptor 7 (mGluR7), have been implicated in multiple neuropsychiatric disorders and shown to mediate excitatory synaptic neurotransmitter signaling and plasticity in the mammalian brain. Here we report a 303 kb de novo deletion at band 3p26.1, disrupting five coding exons of GRM7 in a proband with autism spectrum disorder, and hyperactivity. Our exon transcriptome-mutation contingency index method shows that three of the exons within the breakpoint boundaries are under purifying selection and highly expressed in prenatal brain regions. Based on our results and a thorough review of the literature, we propose that haploinsufficiency of the GRM7 product (mGluR7) contributes to autism spectrum disorders and hyperactivity phenotype as seen in the patient described here.
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