A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Structure, function, and phylogeny of the mating locus in the Rhizopus oryzae complex.

The Rhizopus oryzae species complex is a group of zygomycete fungi that are common, cosmopolitan saprotrophs. Some strains are used beneficially for production of Asian fermented foods but they can also act as opportunistic human pathogens. Although R. oryzae reportedly has a heterothallic (+/-) mating system, most strains have not been observed to undergo sexual reproduction and the genetic structure of its mating locus has not been characterized. Here we report on the mating behavior and genetic structure of the mating locus for 54 isolates of the R. oryzae complex. All 54 strains have a mating locus similar in overall organization to Phycomyces blakesleeanus and Mucor circinelloides (Mucoromycotina, Zygomycota). In all of these fungi, the minus (-) allele features the SexM high mobility group (HMG) gene flanked by an RNA helicase gene and a TP transporter gene (TPT). Within the R. oryzae complex, the plus (+) mating allele includes an inserted region that codes for a BTB/POZ domain gene and the SexP HMG gene. Phylogenetic analyses of multiple genes, including the mating loci (HMG, TPT, RNA helicase), ITS1-5.8S-ITS2 rDNA, RPB2, and LDH genes, identified two distinct groups of strains. These correspond to previously described sibling species R. oryzae sensu stricto and R. delemar. Within each species, discordant gene phylogenies among multiple loci suggest an outcrossing population structure. The hypothesis of random-mating is also supported by a 50:50 ratio of plus and minus mating types in both cryptic species. When crossed with tester strains of the opposite mating type, most isolates of R. delemar failed to produce zygospores, while isolates of R. oryzae produced sterile zygospores. In spite of the reluctance of most strains to mate in vitro, the conserved sex locus structure and evidence for outcrossing suggest that a normal sexual cycle occurs in both species.

Nucleolar organization, ribosomal DNA array stability, and acrocentric chromosome integrity are linked to telomere function.

The short arms of the ten acrocentric human chromosomes share several repetitive DNAs, including ribosomal RNA genes (rDNA). The rDNA arrays correspond to nucleolar organizing regions that coalesce each cell cycle to form the nucleolus. Telomere disruption by expressing a mutant version of telomere binding protein TRF2 (dnTRF2) causes non-random acrocentric fusions, as well as large-scale nucleolar defects. The mechanisms responsible for acrocentric chromosome sensitivity to dysfunctional telomeres are unclear. In this study, we show that TRF2 normally associates with the nucleolus and rDNA. However, when telomeres are crippled by dnTRF2 or RNAi knockdown of TRF2, gross nucleolar and chromosomal changes occur. We used the controllable dnTRF2 system to precisely dissect the timing and progression of nucleolar and chromosomal instability induced by telomere dysfunction, demonstrating that nucleolar changes precede the DNA damage and morphological changes that occur at acrocentric short arms. The rDNA repeat arrays on the short arms decondense, and are coated by RNA polymerase I transcription binding factor UBF, physically linking acrocentrics to one another as they become fusogenic. These results highlight the importance of telomere function in nucleolar stability and structural integrity of acrocentric chromosomes, particularly the rDNA arrays. Telomeric stress is widely accepted to cause DNA damage at chromosome ends, but our findings suggest that it also disrupts chromosome structure beyond the telomere region, specifically within the rDNA arrays located on acrocentric chromosomes. These results have relevance for Robertsonian translocation formation in humans and mechanisms by which acrocentric-acrocentric fusions are promoted by DNA damage and repair.