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In 2006, Valiant introduced a variation to his celebrated PAC (Probably Approximately Correct) model to biology, by which he wished to explain how, with two simple mechanisms—random variat...

Genome-wide association studies (GWAS) have discovered numerous loci involved in genetic traits. Virtually all studies have reported associations between individual single nucleotide polymorphisms (SNPs) and traits. However, it is likely that complex traits are influenced by interaction of multiple SNPs. One approach to detect interactions of SNPs is the brute force approach which performs a pairwise association test between a trait and each pair of SNPs. The brute force approach is often computationally infeasible because of the large number of SNPs collected in current GWAS studies. We propose a two-stage model, Threshold-based Efficient Pairwise Association Approach (TEPAA), to reduce the number of tests needed while maintaining almost identical power to the brute force approach. In the first stage, our method performs the single marker test on all SNPs and selects a subset of SNPs that achieve a certain significance threshold. In the second stage, we perform a pairwise association test between traits and pairs of the SNPs selected from the first stage. The key insight of our approach is that we derive the joint distribution between the association statistics of a single SNP and the association statistics of pairs of SNPs. This joint distribution allows us to provide guarantees that the statistical power of our approach will closely approximate the brute force approach. We applied our approach to the Northern Finland Birth Cohort data and achieved 63 times speedup while maintaining 99% of the power of the brute force approach.

Sequence alignment (the grouping of homologous bases into one column) is fundamental to almost any task in comparative genomics. This translates to positing gaps in the genomic sequences to account for events of insertions and deletions (indels). The interrelationship between sequence alignment and phylogenetic reconstruction has drawn substantial attention recently with works showing the significance of differences in alignments. One of the plausible approaches in this direction is to grade the suitability of a tree to an associated alignment and vice verse. We here present a combinatorial (as opposed to statistical) approach based on the indel history. We show--both by simulations and by using real biological data from the Encyclopedia of DNA Elements (ENCODE)--that this criterion is sound. The novelty of our approach is the distinguishing between insertions and deletions, and augmenting the analysis with a dimension of "depth," extending it from the sequence space to the phylogenetic space. Using this approach, we perform a comprehensive study of indel characteristic behavior among mammals in both coding and non-coding regions. Our results show significant differences in indel patterns between coding and non-coding regions. We also show other characteristic patterns of indel evolution in the depth of the underlying phylogeny.

Horizontal transfer (HT) is the event of a DNA sequence being transferred between species not by inheritance. This phenomenon violates the tree-like evolution of the species under study turning the trees into networks. At the sequence level, HT offers basic characteristics that enable not only clear identification and distinguishing from other sequence similarity cases but also the possibility of dating the events. We developed a novel, self-contained technique to identify relatively recent horizontal transfer elements (HTEs) in the sequences. Appropriate formalism allows one to obtain confidence values for the events detected. The technique does not rely on such problematic prerequisites as reliable phylogeny and/or statistically justified pairwise sequence alignment. In conjunction with the unique properties of HT, it gives rise to a two-level sequence similarity algorithm that, to the best of our knowledge, has not been explored. From evolutionary perspective, the novelty of the work is in the combination of small scale and large scale mutational events. The technique is employed on both simulated and real biological data. The simulation results show high capability of discriminating between HT and conserved regions. On the biological data, the method detected documented HTEs along with their exact locations in the recipient genomes. Supplementary Material is available online at www.libertonline.com/cmb.

Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM), are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model--three taxa, two state characters, under a molecular clock. Four taxa rooted trees have two topologies--the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). In a previous work, we devised a closed form analytic solution for the ML molecular clock fork. In this work, we extend the state of the art in the area of analytic solutions ML trees to the family of all four taxa trees under the molecular clock assumption. The change from the fork topology to the comb incurs a major increase in the complexity of the underlying algebraic system and requires novel techniques and approaches. We combine the ultrametric properties of molecular clock trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations. We finally use tools from algebraic geometry (e.g., Gröbner bases, ideal saturation, resultants) and employ symbolic algebra software to obtain analytic solutions for the comb. We show that in contrast to the fork, the comb has no closed form solutions (expressed by radicals in the input data). In general, four taxa trees can have multiple ML points. In contrast, we can now prove that under the molecular clock assumption, the comb has a unique (local and global) ML point. (Such uniqueness was previously shown for the fork.).