Dissecting the molecular basis of quantitative traits is a significant challenge and is essential for understanding complex diseases. Even in model organisms, precisely determining causative genes and their interactions has remained elusive, due in part to difficulty in narrowing intervals to single genes and in detecting epistasis or linked quantitative trait loci. These difficulties are exacerbated by limitations in experimental design, such as low numbers of analyzed individuals or of polymorphisms between parental genomes. We address these challenges by applying three independent high-throughput approaches for QTL mapping to map the genetic variants underlying 11 phenotypes in two genetically distant Saccharomyces cerevisiae strains, namely (1) individual analysis of >700 meiotic segregants, (2) bulk segregant analysis, and (3) reciprocal hemizygosity scanning, a new genome-wide method that we developed. We reveal differences in the performance of each approach and, by combining them, identify eight polymorphic genes that affect eight different phenotypes: colony shape, flocculation, growth on two nonfermentable carbon sources, and resistance to two drugs, salt, and high temperature. Our results demonstrate the power of individual segregant analysis to dissect QTL and address the underestimated contribution of interactions between variants. We also reveal confounding factors like mutations and aneuploidy in pooled approaches, providing valuable lessons for future designs of complex trait mapping studies.
Context dependence is central to the description of complexity. Keying on the pairwise definition of "set complexity," we use an information theory approach to formulate general measures of systems complexity. We examine the properties of multivariable dependency starting with the concept of interaction information. We then present a new measure for unbiased detection of multivariable dependency, "differential interaction information." This quantity for two variables reduces to the pairwise "set complexity" previously proposed as a context-dependent measure of information in biological systems. We generalize it here to an arbitrary number of variables. Critical limiting properties of the "differential interaction information" are key to the generalization. This measure extends previous ideas about biological information and provides a more sophisticated basis for the study of complexity. The properties of "differential interaction information" also suggest new approaches to data analysis. Given a data set of system measurements, differential interaction information can provide a measure of collective dependence, which can be represented in hypergraphs describing complex system interaction patterns. We investigate this kind of analysis using simulated data sets. The conjoining of a generalized set complexity measure, multivariable dependency analysis, and hypergraphs is our central result. While our focus is on complex biological systems, our results are applicable to any complex system.
Information theory is valuable in multiple-variable analysis for being model-free and nonparametric, and for the modest sensitivity to undersampling. We previously introduced a general approach to finding multiple dependencies that provides accurate measures of levels of dependency for subsets of variables in a data set, which is significantly nonzero only if the subset of variables is collectively dependent. This is useful, however, only if we can avoid a combinatorial explosion of calculations for increasing numbers of variables. The proposed dependence measure for a subset of variables, τ, differential interaction information, Δ(τ), has the property that for subsets of τ some of the factors of Δ(τ) are significantly nonzero, when the full dependence includes more variables. We use this property to suppress the combinatorial explosion by following the “shadows” of multivariable dependency on smaller subsets. Rather than calculating the marginal entropies of all subsets at each degree level, we need to consider only calculations for subsets of variables with appropriate “shadows.” The number of calculations for n variables at a degree level of d grows therefore, at a much smaller rate than the binomial coefficient (n, d), but depends on the parameters of the “shadows” calculation. This approach, avoiding a combinatorial explosion, enables the use of our multivariable measures on very large data sets. We demonstrate this method on simulated data sets, and characterize the effects of noise and sample numbers. In addition, we analyze a data set of a few thousand mutant yeast strains interacting with a few thousand chemical compounds.
The complex of central problems in data analysis consists of three components: (1) detecting the dependence of variables using quantitative measures, (2) defining the significance of these dependence measures, and (3) inferring the functional relationships among dependent variables. We have argued previously that an information theory approach allows separation of the detection problem from the inference of functional form problem. We approach here the third component of inferring functional forms based on information encoded in the functions. We present here a direct method for classifying the functional forms of discrete functions of three variables represented in data sets. Discrete variables are frequently encountered in data analysis, both as the result of inherently categorical variables and from the binning of continuous numerical variables into discrete alphabets of values. The fundamental question of how much information is contained in a given function is answered for these discrete functions, and their surprisingly complex relationships are illustrated. The all-important effect of noise on the inference of function classes is found to be highly heterogeneous and reveals some unexpected patterns. We apply this classification approach to an important area of biological data analysis—that of inference of genetic interactions. Genetic analysis provides a rich source of real and complex biological data analysis problems, and our general methods provide an analytical basis and tools for characterizing genetic problems and for analyzing genetic data. We illustrate the functional description and the classes of a number of common genetic interaction modes and also show how different modes vary widely in their sensitivity to noise.
Abstract:Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence-a function central to information theory-and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric.
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