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Linear mixed models (LMMs) have been widely used in genome-wide association studies to control for population stratification and cryptic relatedness. However, estimating LMM parameters is computationally expensive, necessitating large-scale matrix operations to build the genetic relationship matrix (GRM). Over the past 25 years, Randomized Linear Algebra has provided alternative approaches to such matrix operations by leveragingmatrix sketching, which often results in provably accurate fast and efficient approximations. We leverage matrix sketching to develop a fast and efficient LMM method calledMatrix-Sketching LMM (MaSk-LMM) by sketching the genotype matrix to reduce its dimensions and speed up computations. Our framework comes with both theoretical guarantees and a strong empirical performance compared to the current state-of-the-art for simulated traits and complex diseases. 

Abstract India represents an intricate tapestry of population substructure shaped by geography, language, culture, and social stratification. Although geography closely correlates with genetic structure in other parts of the world, the strict endogamy imposed by the Indian caste system and the large number of spoken languages add further levels of complexity to understand Indian population structure. To date, no study has attempted to model and evaluate how these factors have interacted to shape the patterns of genetic diversity within India. We merged all publicly available data from the Indian subcontinent into a data set of 891 individuals from 90 well-defined groups. Bringing together geography, genetics, and demographic factors, we developed Correlation Optimization of Genetics and Geodemographics to build a model that explains the observed population genetic substructure. We show that shared language along with social structure have been the most powerful forces in creating paths of gene flow in the subcontinent. Furthermore, we discover the ethnic groups that best capture the diverse genetic substructure using a ridge leverage score statistic. Integrating data from India with a data set of additional 1,323 individuals from 50 Eurasian populations, we find that Indo-European and Dravidian speakers of India show shared genetic drift with Europeans, whereas the Tibeto-Burman speaking tribal groups have maximum shared genetic drift with East Asians. 

Despite advances in sequencing, lack of standardization makes comparisons across studies challenging and hampers insights into the structure and function of microbial communities across multiple habitats on a planetary scale. Here we present a multi-omics analysis of a diverse set of 880 microbial community samples collected for the Earth Microbiome Project. We include amplicon (16S, 18S, ITS) and shotgun metagenomic sequence data, and untargeted metabolomics data (liquid chromatography-tandem mass spectrometry and gas chromatography mass spectrometry). We used standardized protocols and analytical methods to characterize microbial communities, focusing on relationships and co-occurrences of microbially related metabolites and microbial taxa across environments, thus allowing us to explore diversity at extraordinary scale. In addition to a reference database for metagenomic and metabolomic data, we provide a framework for incorporating additional studies, enabling the expansion of existing knowledge in the form of an evolving community resource. We demonstrate the utility of this database by testing the hypothesis that every microbe and metabolite is everywhere but the environment selects. Our results show that metabolite diversity exhibits turnover and nestedness related to both microbial communities and the environment, whereas the relative abundances of microbially related metabolites vary and co-occur with specific microbial consortia in a habitat-specific manner. We additionally show the power of certain chemistry, in particular terpenoids, in distinguishing Earth’s environments (for example, terrestrial plant surfaces and soils, freshwater and marine animal stool), as well as that of certain microbes including Conexibacter woesei (terrestrial soils), Haloquadratum walsbyi (marine deposits) and Pantoea dispersa (terrestrial plant detritus). This Resource provides insight into the taxa and metabolites within microbial communities from diverse habitats across Earth, informing both microbial and chemical ecology, and provides a foundation and methods for multi-omics microbiome studies of hosts and the environment.