Inference and Experimental Design for the Discovery of Genetic Regulatory Structure through Experimental Interventions : Statistical Realism and Combinatorial Complexity
Motivation. One strategy for understanding gene regulation involves the differential measurement of expression in wild type versus strains in which expression of specific genes has been suppressed or enhanced. Various proposals have been made for optimizing the information obtained from such experiments and for determining the most informative next experiment in a sequence. At issue is the correctness of these algorithms, the possibility of improved algorithms, and the combinatorial complexities involved. Results. We show that two proposed algorithms for determining regulatory networks from expression differences in wild and perturbed genomes are incorrect. We generalize and exploit the main idea of one of these procedures to present a correct, feasible algorithm for finding a set of informative, alternative networks to explain the data, but each alternative is possibly—and typically—incomplete. We describe a correct but exponential algorithm for computing the possible extensions of these networks, consistent with the data, and a heuristic for choosing the next most informative experiment. Using data from systematic experiments perturbing the yeast genome, we illustrate these algorithms, and the False Discovery Rate procedure for making simultaneous decisions about significant differential expression of genes in pairs of experiments,. Our conclusions are largely negative: (i) the set of significant differences of expression are very sensitive to how the statistical tests are constructed; (ii) the set of alternative initial networks is very large; (iii) correct algorithms for finding all regulatory networks consistent with the data, and reasonable heuristics for computing the most informative next experiment, are infeasible in most realistic cases. Contact: ddanks@ai.uwf.edu 1. Problems with Current Methods. The development of techniques for the simultaneous measurement of mRNA transcripts from thousands of genes has prompted many proposals (Akutsu, 1998; D’hasseleer, 2000; Friedman, 2000; Hartemink, 2000) of statistical and machine learning methods to infer the network of regulatory relations among genes of a given species. Many of these techniques attempt to exploit, implicitly or explicitly, the conditional associations of measured gene expressions (that is, associations of expression levels of two genes when each of a set of other genes does not vary in expression) in repeated experimental trials. These covariational methods contrast with difference methods that measure expression in a “wild cell type,” and then experimentally enhance or suppress the expression of a particular gene or genes and measure which other genes have their expression altered (relative to the wild type) in consequence. Difference methods are essentially variants of standard causal inferences from experimental interventions and controls, supplemented with algorithms that attempt to extract maximal information from the data. There are general theoretical grounds (Chu, et al., 2002), as well as simulation results (Wimberly, et al., 2002) for doubting that algorithms of the covariational kind will be useful in unraveling regulatory structure from microarray measurements of mRNA hybridization. It is important, therefore, to investigate the optimization of inferences that can be made by difference methods, and the adequacy of the requisite statistical inferences to differences of means from small sample sizes with many variables. 2. The DBRF Method. Onami, et al. (2001) describe what they call the Difference-Based Regulation Finding (DBRF) method: Given measurements of n genes in k experiments in each of which one gene (or none – i.e., the wild type) is enhanced or suppressed, form the directed graph whose nodes or vertices represent genes, and add a directed edge from vertex i to vertex j if and only if gene j changes in comparison with the wild-type strain when expression of gene i is experimentally suppressed or enhanced. Each directed edge is marked ‘+’ if overexpression produces overexpression or if suppression produces suppression, and otherwise marked ‘-’. (The graph produced by this process includes edges corresponding to indirect causal influences, e.g., if gene i affects gene j only through gene k, there is still an i j edge). Then, for each pair of vertices, i, j, consider the set of all paths from i to j (presumably acyclic paths; the authors do not say, but they explicitly allow cyclic graphs). Let the parity of each path be the ordinary sign of the product of signs over all directed edges in the path. Among all paths between i, j of the same parity, eliminate all of the paths except the longest. Because the DBRF algorithm keeps only the longest chains, it can only find the correct structure if all upregulatory (and, respectively, downregulatory) pathways from gene i to gene j are of the same length (i.e., involve the same number of intermediate genes), There is no biological evidence for such an assumption. Furthermore, the restriction to single knockouts amounts to an assumption, again without biological basis, that two or more genes cannot regulate a third through a disjunctive dependency, so that, for example, the third is upregulated by either of the two regulators. The authors suggest that their algorithm can be extended to multiple