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We describe the use of a higher-order singular value decomposition (HOSVD) in transforming a data tensor of genes x 'x-settings,' that is, different settings of the experimental variable x x 'y-settings,' which tabulates DNA microarray data from different studies, to a 'core tensor' of 'eigenarrays' x 'x-eigengenes' x 'y-eigengenes.' Reformulating this multilinear HOSVD such that it decomposes the data tensor into a linear superposition of all outer products of an eigenarray, an x- and a y-eigengene, that is, rank-1 'subtensors,' we define the significance of each subtensor in terms of the fraction of the overall information in the data tensor that it captures. We illustrate this HOSVD with an integration of genome-scale mRNA expression data from three yeast cell cycle time courses, two of which are under exposure to either hydrogen peroxide or menadione. We find that significant subtensors represent independent biological programs or experimental phenomena. The picture that emerges suggests that the conserved genes YKU70, MRE11, AIF1, and ZWF1, and the processes of retrotransposition, apoptosis, and the oxidative pentose phosphate pathway that these genes are involved in, may play significant, yet previously unrecognized, roles in the differential effects of hydrogen peroxide and menadione on cell cycle progression. A genome-scale correlation between DNA replication initiation and RNA transcription, which is equivalent to a recently discovered correlation and might be due to a previously unknown mechanism of regulation, is independently uncovered. cell cycle | DNA replication initiation | N-mode singular value decomposition | oxidative stress | yeast Saccharomyces cerevisiae

We describe the singular value decomposition (SVD) of yeast genome-scale mRNA lengths distribution data measured by DNA microarrays. SVD uncovers in the mRNA abundance levels data matrix of genes x arrays, i.e., electrophoretic gel migration lengths or mRNA lengths, mathematically unique decorrelated and decoupled 'eigengenes.' The eigengenes are the eigenvectors of the arrays x arrays correlation matrix, with the corresponding series of eigenvalues proportional to the series of the 'fractions of eigen abundance.' Each fraction of eigen abundance indicates the significance of the corresponding eigengene relative to all others. We show that the eigengenes fit 'asymmetric Hermite functions,' a generalization of the eigenfunctions of the quantum harmonic oscillator and the integral transform which kernel is a generalized coherent state. The fractions of eigen abundance fit a geometric series as do the eigenvalues of the integral transform which kernel is a generalized coherent state. The 'asymmetric generalized coherent state' models the measured data, where the profiles of mRNA abundance levels of most genes as well as the distribution of the peaks of these profiles fit asymmetric Gaussians. We hypothesize that the asymmetry in the distribution of the peaks of the profiles is due to two competing evolutionary forces. We show that the asymmetry in the profiles of the genes might be due to a previously unknown asymmetry in the gel electrophoresis thermal broadening of a moving, rather than a stationary, band of RNA molecules. DNA microarrays | yeast Saccharomyces cerevisiae | Hermite functions | generalized coherent states | evolutionary forces

We describe the use of the matrix eigenvalue decomposition (EVD) and pseudoinverse projection and a tensor higher-order EVD (HOEVD) in reconstructing the pathways that compose a cellular system from genome-scale nondirectional networks of correlations among the genes of the system. The EVD formulates a genes x genes network as a linear superposition of genes x genes decorrelated and decoupled rank-1 subnetworks, which can be associated with functionally independent pathways. The integrative pseudoinverse projection of a network computed from a 'data' signal onto a designated 'basis' signal approximates the network as a linear superposition of only the subnetworks that are common to both signals and simulates observation of only the pathways that are manifest in both experiments. We define a comparative HOEVD that formulates a series of networks as linear superpositions of decorrelated rank-1 subnetworks and the rank-2 couplings among these subnetworks, which can be associated with independent pathways and the transitions among them common to all networks in the series or exclusive to a subset of the networks. Boolean functions of the discretized subnetworks and couplings highlight differential, i.e., pathway-dependent, relations among genes. We illustrate the EVD, pseudoinverse projection, and HOEVD of genome-scale networks with analyses of yeast DNA microarray data. DNA microarrays | eigenvalue decomposition | higher-order eigenvalue decomposition | pseudoinverse projection | yeast Saccharomyces cerevisiae cell cycle and mating

We describe an integrative data-driven mathematical framework that formulates any number of genome-scale molecular biological data sets in terms of one chosen set of data samples, or of profiles extracted mathematically from data samples, designated the 'basis' set. By using pseudoinverse projection, the molecular biological profiles of the data samples are least-squares-approximated as superpositions of the basis profiles. Reconstruction of the data in the basis simulates experimental observation of only the cellular states manifest in the data that correspond to those of the basis. Classification of the data samples according to their reconstruction in the basis, rather than their overall measured profiles, maps the cellular states of the data onto those of the basis and gives a global picture of the correlations and possibly also causal coordination of these two sets of states. We illustrate this framework with an integration of yeast genome-scale proteins' DNA-binding data with cell cycle mRNA expression time course data. Novel correlation between DNA replication initiation and RNA transcription during the yeast cell cycle, which might be due to a previously unknown mechanism of regulation, is predicted. Singular value decomposition | generalized singular value decomposition | DNA microarrays | yeast Saccharomyces cerevisiae cell cycle