Structural Learning of Genetic Regulatory Networks Based on Prior Biological Knowledge and Microarray Gene Expression Measurements

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Introduction

The Bayesian Network (BN) has been proven to be a useful and important tool in biomedical applications such as clinical decision support systems (Beinlich, Suermondt, Chavez & Cooper, 1989), information retrieval (Baeza-Yates & Ribeiro, 1999), and discovery of gene regulatory networks (Friedman, Linial, Nachman & Pe'er, 2000). Automatic learning of BNs from observational data has been an area of intense research for more than a decade, yielding practical algorithms and tools (Spirtes, Glymour & Scheines, 1993). The ability of the BN approach to reconstruct genetic networks from microarray gene expression data has been extensively evaluated.

Consider a set of microarray experiments that measures the expression of a set of N genes over M different conditions. We denote the gene expression values by an M × N matrix D = (d₁,…,d_n). The BN method discovers a directed acyclic graph (DAG) S such that the posterior probability P(S | X = D) is maximized. Here X = (X₁,…,X_N) denotes a set of random variables representing gene expression for genes i = 1,…,N. Let π_i be the set of parents of node i in an acyclic network S. Then, the probability P( X = D | S) can be decomposed into the product of local probabilities of nodes specified by the network structure S:

(1) where denotes the subset of variables corresponding to π_i and the corresponding observations. For ease of notation, we will omit the symbol X but use D indicating that X takes an observation D. The nodes in the learned network correspond to genes or their products and the edges correspond to direct probabilistic dependencies, such as causality, mediation, activation, or inhibition between the genes. The posterior probability P(S | D) is proportional to the product of the likelihood P(D | S) and the prior probability P(S) of network structure S based on prior knowledge, i.e.,P(S | D) ∝ P(S)P(D | S) (2)

The main approach to learning BNs from data is based on the strategy of search-and-score, which attempts to identify the most probable network S given the data D. This network has the highest posterior probability. Depending on assumptions, maximizing this probability corresponds to maximizing a score function. There are several ways to define the score. A straightforward definition is the likelihood P(D | S). For discrete data and multinomial distribution, the K2 score (Cooper & Herskovits, 1992) is often used to evaluate the networks generated. For a given network S, this score is defined as the likelihood:, (3) where N_ijk is the number of cases in D in which variable X_i has the kth value and the parent of i has the jth instantiation; q_i is the number of parents for i, and r_i the number of possible values of variable X_i. Thus,

. (4)

When the prior probability of S is considered, the K2 score for a network S can be modified by multiplying P(S) to P(D | S) in (3).