Likelihood Inference for Permuted Data With Application to Gene Regulation

Abstract

Abstract Given that all the cells of an individual have the same genetic information stored in their DNA, how can cells be as different as those of the retina and heart? Nature solves this problem through gene regulation, which often involves the binding of regulatory proteins to regulatory sites. These sites are short subsequences of 10 to 20 DNA base pairs whose pattern may be multinomially modeled. These sites usually occur “upstream” of the genes they regulate in a segment of a few hundred DNA base pairs called the promoter. But the positions of regulatory sites within promoters vary and are unobservable. This uncertainty in site position misaligns the data and renders the indices of the observations uncertain. Data with uncertain indices arise commonly in experimental biology whenever uncontrolled variability alters unobservable auxiliary identifying information. Current technology breaks the analysis of such data into two steps: alignment and analyses applied to the aligned data. This article proposes a methodology that combines these two steps and thus produces inferences that directly incorporate random alignment errors. The introduction of an index permutation indicator variable, which is treated as missing data, permits the formulation of these problems as novel finite mixtures. Using a missing information approach, we separate the likelihood into components representing variable uncertainty and index uncertainty. An EM algorithm to obtain the maximum likelihood estimates of the parameters for both of these components is also presented. Inferences specific to the index permutations stemming from index uncertainty are examined. An application to regulatory sites for a bacterial regulatory protein—cyclic adenosine monophosphate receptor protein (CRP)—is presented.