Alter, Orly (Univ. of Texas at Austin), Golub, Gene H. (Stanford Univ.)

Matrix and Tensor Computations for Reconstructing the Pathways of a Cellular System from Genome-Scale Signals

Scheduled for presentation during the Mini-Symposium "Genomic Signals and Systems" (TuP10), Tuesday, July 25, 2006, 15:45−16:10, Room 103

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on May 27, 2023

Abstract

DNA microarrays make it possible to record the complete genomic signals that are generated and sensed by cellular systems. The underlying genome-scale networks of relations among all genes of the cellular systems can be computed from these signals. These relations are known to be pathway-dependent, i.e., conditioned by the biological and experimental settings in which they are observed. We describe the use of the matrix eigenvalue decomposition (EVD) and pseudoinverse projection and a tensor higher-order EVD (HOEVD) in reconstructing the pathways, or genome-scale pathway-dependent relations among the genes of a cellular system, from nondirectional networks of correlations, which are computed from measured genomic signals and tabulated in symmetric matrices [Alter & Golub, PNAS 2005]. EVD formulates a genes x genes network, computed from a "data" signal, as a linear superposition of genes x genes decorrelated and decoupled rank-1 subnetworks. Significant EVD subnetworks might represent 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, i.e., pseudoinverse projection filters off the network the subnetworks that are exclusive to the data signal. The pseudoinverse-projected network simulates observation of only the pathways that are manifest under both sets of conditions where the data and basis signals are measured. We define a comparative HOEVD, that formulates a series of networks computed from a series of signals as linear superpositions of decorrelated rank-1 subnetworks and the rank-2 couplings among these subnetworks. Significant HOEVD subnetworks and couplings might represent independent pathways or transitions among them common to all or exclusive to a subset of the signals. We illustrate the EVD, pseudoinverse projection and HOEVD of genomic networks with analyses of yeast mRNA expression and transcription factors' DNA-binding data. Boolean functions of the discretized subnetworks and couplings highlight known and novel differential, i.e., pathway-dependent relations among genes.