Supplemental material for O. Alter and G. H. Golub, "Reconstructing the Pathways of a Cellular System from Genome-Scale Signals by Using Matrix and Tensor Computations," Proceedings of the National Academy of Sciences (PNAS) USA 102 (49), pp. 17559–17564 (December 2005); doi: 10.1073/pnas.0509033102.
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 × genes network as a linear superposition of genes × 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.

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Yeast RNA Expression Dataset

Reproduced from Spellman et al.

Classification List of Yeast Cell Cycle- and Pheromone-Regulated Genes

Reproduced from Roberts et al.

Yeast Proteins' DNA-Binding Datasets

Reproduced from Lee et al.