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.

Host genetics and HIV-1: the final phase?

This is a crucial transition time for human genetics in general, and for HIV host genetics in particular. After years of equivocal results from candidate gene analyses, several genome-wide association studies have been published that looked at plasma viral load or disease progression. Results from other studies that used various large-scale approaches (siRNA screens, transcriptome or proteome analysis, comparative genomics) have also shed new light on retroviral pathogenesis. However, most of the inter-individual variability in response to HIV-1 infection remains to be explained: genome resequencing and systems biology approaches are now required to progress toward a better understanding of the complex interactions between HIV-1 and its human host.

Working at the interface of phylogenetics and population genetics: a biogeographical analysis of Triaenops spp. (Chiroptera: Hipposideridae).

New applications of genetic data to questions of historical biogeography have revolutionized our understanding of how organisms have come to occupy their present distributions. Phylogenetic methods in combination with divergence time estimation can reveal biogeographical centres of origin, differentiate between hypotheses of vicariance and dispersal, and reveal the directionality of dispersal events. Despite their power, however, phylogenetic methods can sometimes yield patterns that are compatible with multiple, equally well-supported biogeographical hypotheses. In such cases, additional approaches must be integrated to differentiate among conflicting dispersal hypotheses. Here, we use a synthetic approach that draws upon the analytical strengths of coalescent and population genetic methods to augment phylogenetic analyses in order to assess the biogeographical history of Madagascar's Triaenops bats (Chiroptera: Hipposideridae). Phylogenetic analyses of mitochondrial DNA sequence data for Malagasy and east African Triaenops reveal a pattern that equally supports two competing hypotheses. While the phylogeny cannot determine whether Africa or Madagascar was the centre of origin for the species investigated, it serves as the essential backbone for the application of coalescent and population genetic methods. From the application of these methods, we conclude that a hypothesis of two independent but unidirectional dispersal events from Africa to Madagascar is best supported by the data.

Can Genetics Predict Response to Complex Behavioral Interventions? Evidence from a Genetic Analysis of the Fast Track Randomized Control Trial.

Early interventions are a preferred method for addressing behavioral problems in high-risk children, but often have only modest effects. Identifying sources of variation in intervention effects can suggest means to improve efficiency. One potential source of such variation is the genome. We conducted a genetic analysis of the Fast Track randomized control trial, a 10-year-long intervention to prevent high-risk kindergarteners from developing adult externalizing problems including substance abuse and antisocial behavior. We tested whether variants of the glucocorticoid receptor gene NR3C1 were associated with differences in response to the Fast Track intervention. We found that in European-American children, a variant of NR3C1 identified by the single-nucleotide polymorphism rs10482672 was associated with increased risk for externalizing psychopathology in control group children and decreased risk for externalizing psychopathology in intervention group children. Variation in NR3C1 measured in this study was not associated with differential intervention response in African-American children. We discuss implications for efforts to prevent externalizing problems in high-risk children and for public policy in the genomic era.

Application of a rank-based genetic association test to age-at-onset data from the Collaborative Study on the Genetics of Alcoholism study.

Association studies of quantitative traits have often relied on methods in which a normal distribution of the trait is assumed. However, quantitative phenotypes from complex human diseases are often censored, highly skewed, or contaminated with outlying values. We recently developed a rank-based association method that takes into account censoring and makes no distributional assumptions about the trait. In this study, we applied our new method to age-at-onset data on ALDX1 and ALDX2. Both traits are highly skewed (skewness > 1.9) and often censored. We performed a whole genome association study of age at onset of the ALDX1 trait using Illumina single-nucleotide polymorphisms. Only slightly more than 5% of markers were significant. However, we identified two regions on chromosomes 14 and 15, which each have at least four significant markers clustering together. These two regions may harbor genes that regulate age at onset of ALDX1 and ALDX2. Future fine mapping of these two regions with densely spaced markers is warranted.

Quantitative genetics of CTCF binding reveal local sequence effects and different modes of X-chromosome association.

Associating genetic variation with quantitative measures of gene regulation offers a way to bridge the gap between genotype and complex phenotypes. In order to identify quantitative trait loci (QTLs) that influence the binding of a transcription factor in humans, we measured binding of the multifunctional transcription and chromatin factor CTCF in 51 HapMap cell lines. We identified thousands of QTLs in which genotype differences were associated with differences in CTCF binding strength, hundreds of them confirmed by directly observable allele-specific binding bias. The majority of QTLs were either within 1 kb of the CTCF binding motif, or in linkage disequilibrium with a variant within 1 kb of the motif. On the X chromosome we observed three classes of binding sites: a minority class bound only to the active copy of the X chromosome, the majority class bound to both the active and inactive X, and a small set of female-specific CTCF sites associated with two non-coding RNA genes. In sum, our data reveal extensive genetic effects on CTCF binding, both direct and indirect, and identify a diversity of patterns of CTCF binding on the X chromosome.