From non-Brownian Functionals to a Fractional Schr\"odinger Equation

We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.Comment: 5 page

The variance of identity-by-descent sharing in the Wright-Fisher model

Widespread sharing of long, identical-by-descent (IBD) genetic segments is a hallmark of populations that have experienced recent genetic drift. Detection of these IBD segments has recently become feasible, enabling a wide range of applications from phasing and imputation to demographic inference. Here, we study the distribution of IBD sharing in the Wright-Fisher model. Specifically, using coalescent theory, we calculate the variance of the total sharing between random pairs of individuals. We then investigate the cohort-averaged sharing: the average total sharing between one individual and the rest of the cohort. We find that for large cohorts, the cohort-averaged sharing is distributed approximately normally. Surprisingly, the variance of this distribution does not vanish even for large cohorts, implying the existence of "hyper-sharing" individuals. The presence of such individuals has consequences for the design of sequencing studies, since, if they are selected for whole-genome sequencing, a larger fraction of the cohort can be subsequently imputed. We calculate the expected gain in power of imputation by IBD, and subsequently, in power to detect an association, when individuals are either randomly selected or specifically chosen to be the hyper-sharing individuals. Using our framework, we also compute the variance of an estimator of the population size that is based on the mean IBD sharing and the variance in the sharing between inbred siblings. Finally, we study IBD sharing in an admixture pulse model, and show that in the Ashkenazi Jewish population the admixture fraction is correlated with the cohort-averaged sharing.Comment: Includes Supplementary Materia

Genome-wide data from medieval German Jews show that the Ashkenazi founder event pre-dated the 14th century

We report genome-wide data from 33 Ashkenazi Jews (AJ), dated to the 14th century, obtained following a salvage excavation at the medieval Jewish cemetery of Erfurt, Germany. The Erfurt individuals are genetically similar to modern AJ, but they show more variability in Eastern European-related ancestry than modern AJ. A third of the Erfurt individuals carried a mitochondrial lineage common in modern AJ and eight carried pathogenic variants known to affect AJ today. These observations, together with high levels of runs of homozygosity, suggest that the Erfurt community had already experienced the major reduction in size that affected modern AJ. The Erfurt bottleneck was more severe, implying substructure in medieval AJ. Overall, our results suggest that the AJ founder event and the acquisition of the main sources of ancestry pre-dated the 14th century and highlight late medieval genetic heterogeneity no longer present in modern AJ

The Transcriptome of the Human Pathogen Trypanosoma brucei at Single-Nucleotide Resolution

The genome of Trypanosoma brucei, the causative agent of African trypanosomiasis, was published five years ago, yet identification of all genes and their transcripts remains to be accomplished. Annotation is challenged by the organization of genes transcribed by RNA polymerase II (Pol II) into long unidirectional gene clusters with no knowledge of how transcription is initiated. Here we report a single-nucleotide resolution genomic map of the T. brucei transcriptome, adding 1,114 new transcripts, including 103 non-coding RNAs, confirming and correcting many of the annotated features and revealing an extensive heterogeneity of 5′ and 3′ ends. Some of the new transcripts encode polypeptides that are either conserved in T. cruzi and Leishmania major or were previously detected in mass spectrometry analyses. High-throughput RNA sequencing (RNA-Seq) was sensitive enough to detect transcripts at putative Pol II transcription initiation sites. Our results, as well as recent data from the literature, indicate that transcription initiation is not solely restricted to regions at the beginning of gene clusters, but may occur at internal sites. We also provide evidence that transcription at all putative initiation sites in T. brucei is bidirectional, a recently recognized fundamental property of eukaryotic promoters. Our results have implications for gene expression patterns in other important human pathogens with similar genome organization (Trypanosoma cruzi, Leishmania sp.) and revealed heterogeneity in pre-mRNA processing that could potentially contribute to the survival and success of the parasite population in the insect vector and the mammalian host