Discovering local structure in gene expression data: the order-preserving submatrix problem

This paper concerns the discovery of patterns in gene expression matrices, in which each element gives the expression level of a given gene in a given experiment. Most existing methods for pattern discovery in such matrices are based on clustering genes by comparing their expression levels in all experiments, or clustering experiments by comparing their expression levels for all genes. Our work goes beyond such global approaches by looking for local patterns that manifest themselves when we focus simultaneously on a subset G of the genes and a subset T of the experiments. Specifically, we look for order-preserving submatrices (OPSMs), in which the expression levels of all genes induce the same linear ordering of the experiments (we show that the OPSM search problem is NP-hard in the worst case). Such a pattern might arise, for example, if the experiments in T represent distinct stages in the progress of a disease or in a cellular process, and the expression levels of all genes in G vary across the stages in the same way.We define a probabilistic model in which an OPSM is hidden within an otherwise random matrix. Guided by this model we develop an efficient algorithm for finding the hidden OPSM in the random matrix. In data generated according to the model the algorithm recovers the hidden OPSM with very high success rate. Application of the methods to breast cancer data seems to reveal significant local patterns.Our algorithm can be used to discover more than one OPSM within the same data set, even when these OPSMs overlap. It can also be adapted to handle relaxations and extensions of the OPSM condition. For example, we may allow the different rows of G x T to induce similar but not identical orderings of the columns, or we may allow the set T to include more than one representative of each stage of a biological process.

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