Motivation: Next-generation sequencing technology is increasingly being used for clinical diagnostic tests. Unlike research cell lines, clinical samples are often genomically heterogeneous due to low sample purity or the presence of genetic subpopulations. Therefore, a variant calling algorithm for calling low-frequency polymorphisms in heterogeneous samples is needed. Result: We present a novel variant calling algorithm that uses a hierarchical Bayesian model to estimate allele frequency and call variants in heterogeneous samples. We show that our algorithm improves upon current classifiers and has higher sensitivity and specificity over a wide range of median read depth and minor allele frequency. We apply our model and identify twelve mutations in the PAXP1 gene in a matched clinical breast ductal carcinoma tumor sample; two of which are loss-of-heterozygosity events.

"Recently, sequencing technologies have generated massive and heterogeneous data sets. However, interpretation of these data sets is a major barrier to understand genomic heterogeneity in complex diseases. In this dissertation, we develop a Bayesian statistical method for single nucleotide level analysis and a global optimization method for gene expression level analysis to characterize genomic heterogeneity in mixed samples. The detection of rare single nucleotide variants (SNVs) is important for understanding genetic heterogeneity using next-generation sequencing (NGS) data. Various computational algorithms have been proposed to detect variants at the single nucleotide level in mixed samples. Yet, the noise inherent in the biological processes involved in NGS technology necessitates the development of statistically accurate methods to identify true rare variants. At the single nucleotide level, we propose a Bayesian probabilistic model and a variational expectation maximization (EM) algorithm to estimate non-reference allele frequency (NRAF) and identify SNVs in heterogeneous cell populations. We demonstrate that our variational EM algorithm has comparable sensitivity and specificity compared with a Markov Chain Monte Carlo (MCMC) sampling inference algorithm, and is more computationally efficient on tests of relatively low coverage (27x and 298x) data. Furthermore, we show that our model with a variational EM inference algorithm has higher specificity than many state-of-the-art algorithms. In an analysis of a directed evolution longitudinal yeast data set, we are able to identify a time-series trend in non-reference allele frequency and detect novel variants that have not yet been reported. Our model also detects the emergence of a beneficial variant earlier than was previously shown, and a pair of concomitant variants. Characterization of heterogeneity in gene expression data is a critical challenge for personalized treatment and drug resistance due to intra-tumor heterogeneity. Mixed membership factorization has become popular for analyzing data sets that have within-sample heterogeneity. In recent years, several algorithms have been developed for mixed membership matrix factorization, but they only guarantee estimates from a local optimum. At the gene expression level, we derive a global optimization (GOP) algorithm that provides a guaranteed epsilon-global optimum for a sparse mixed membership matrix factorization problem for molecular subtype classification. We test the algorithm on simulated data and find the algorithm always bounds the global optimum across random initializations and explores multiple modes efficiently. The GOP algorithm is well-suited for parallel computations in the key optimization steps. "

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