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. 2005 Dec 6;102(49):17559-64.
doi: 10.1073/pnas.0509033102. Epub 2005 Nov 28.

Reconstructing the Pathways of a Cellular System From Genome-Scale Signals by Using Matrix and Tensor Computations

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Free PMC article

Reconstructing the Pathways of a Cellular System From Genome-Scale Signals by Using Matrix and Tensor Computations

Orly Alter et al. Proc Natl Acad Sci U S A. .
Free PMC article

Abstract

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.

Figures

Fig. 1.
Fig. 1.
Discretized significant EVD subnetworks of the network â1 in the subsets of 150 correlations (red) and anticorrelations (green) largest in amplitude among all traditionally classified cell-cycle genes of â1, color-coded according to their cell-cycle classifications, M/G1 (yellow), G1 (green), S (blue), S/G2 (red), and G2/M (orange), and separately also according to their pheromone-response classifications, up-regulated (black) and down-regulated (gray). (a) The first subnetwork shows pheromone-response-dependent and cell-cycle-independent relations among the genes. (b) The second subnetwork shows pheromone-response- and cell-cycle-dependent relations. (c and d) The third and fourth subnetworks show cell-cycle-dependent relations that are orthogonal to each other.
Fig. 2.
Fig. 2.
Boolean AND intersections of the discretized EVD subnetworks of the pseudoinverse-projected â2 and â3, in the subsets of 200 correlations largest in amplitude among all traditionally classified cell-cycle genes of â2 and â3, respectively, with these of â1.(a) The first subnetwork of â2 AND fourth subnetwork of â1.(b) The second subnetwork of â2 AND third subnetwork of â1.(c) The subnetwork of â3 AND first subnetwork of â1.
Fig. 3.
Fig. 3.
Discretized significant HOEVD subnetworks of the series of networks {â1, â2, â3} and their couplings, in the subsets of 100 correlations largest in amplitude among all traditionally classified cell-cycle genes of {â1, â2, â3}. (a) The first subnetwork shows pheromone-response-dependent only relations among the genes. (b and c) The second and third subnetworks show orthogonal cell-cycle-dependent relations. (d and e) The couplings between the first and second, and first and third subnetworks, respectively, both show pheromone-response- and cell-cycle-dependent relations. (f) The coupling between the second and third subnetworks shows cell-cycle-dependent only relations.
Fig. 4.
Fig. 4.
Fractions of eigenexpression of the HOEVD subnetworks (a) and their couplings (b) in the individual networks â1 (red), â2 (blue), and â3 (green). The contributions of each coupling in each individual network cancel out in the overall network ââ1 + â2 + â3.

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