Reconstructing the pathways of a cellular system from genome-scale signals by using matrix and tensor computations - PubMed (original) (raw)

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. 2005.

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.

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Figures

Fig. 1.

Fig. 1.

Discretized significant EVD subnetworks of the network _â_1 in the subsets of 150 correlations (red) and anticorrelations (green) largest in amplitude among all traditionally classified cell-cycle genes of _â_1, 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 _â_2 and _â_3, in the subsets of 200 correlations largest in amplitude among all traditionally classified cell-cycle genes of _â_2 and _â_3, respectively, with these of _â_1.(a) The first subnetwork of _â_2 AND fourth subnetwork of _â_1.(b) The second subnetwork of _â_2 AND third subnetwork of _â_1.(c) The subnetwork of _â_3 AND first subnetwork of _â_1.

Fig. 3.

Fig. 3.

Discretized significant HOEVD subnetworks of the series of networks {_â_1, _â_2, _â_3} and their couplings, in the subsets of 100 correlations largest in amplitude among all traditionally classified cell-cycle genes of {_â_1, _â_2, _â_3}. (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 _â_1 (red), _â_2 (blue), and _â_3 (green). The contributions of each coupling in each individual network cancel out in the overall network â ≡ _â_1 + _â_2 + _â_3.

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