Demographic and population structure inference is one of the most important problems in genomics. Population parameters such as effective population sizes, population split times and migration rates are of high interest both themselves and for many applications, e.g. for genome-wide association studies. Hidden Markov Model (HMM) based methods, such as PSMC, MSMC, coalHMM etc., proved to be powerful and useful for estimation of these parameters in many population genetics studies. At the same time, machine and deep learning have began to be used in natural science widely. In particular, deep learning based approaches have already substituted hidden Markov models in many areas, such as speech recognition or user input prediction. We develop a deep learning (DL) approach for local coalescent time estimation from one whole diploid genome. Our DL models are trained on simulated datasets. Importantly, demographic and population parameters can be inferred based on the distribution of coalescent times. We expect that our approach will be useful under complex population scenarios, which cannot be studied with existing HMM based methods. Our work is also a crucial step in developing a deep learning framework which would allow to create population genomics methods for different genomic data representations.
Adaptive introgression—the flow of adaptive genetic variation between species or populations—has attracted significant interest in recent years and it has been implicated in a number of cases of adaptation, from pesticide resistance and immunity, to local adaptation. Despite this, methods for identification of adaptive introgression from population genomic data are lacking. Here, we present Ancestry_HMM-S, a hidden Markov model-based method for identifying genes undergoing adaptive introgression and quantifying the strength of selection acting on them. Through extensive validation, we show that this method performs well on moderately sized data sets for realistic population and selection parameters. We apply Ancestry_HMM-S to a data set of an admixed Drosophila melanogaster population from South Africa and we identify 17 loci which show signatures of adaptive introgression, four of which have previously been shown to confer resistance to insecticides. Ancestry_HMM-S provides a powerful method for inferring adaptive introgression in data sets that are typically collected when studying admixed populations. This method will enable powerful insights into the genetic consequences of admixture across diverse populations. Ancestry_HMM-S can be downloaded from https://github.com/jesvedberg/Ancestry_HMM-S/.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.