Parametric bootstrapping for biological sequence motifs

Patrick K. O’Neill, Ivan Erill

Biological sequence motifs drive the specific interactions of proteins and nucleic acids. Accordingly, the effective computational discovery and analysis of such motifs is a central theme in bioinformatics. Many practical questions about the properties of motifs can be recast as random sampling problems. In this light, the task is to determine for a given motif whether a certain feature of interest is statistically unusual among relevantly similar alternatives. Despite the generality of this framework, its use has been frustrated by the difficulties of defining an appropriate reference class of motifs for comparison and of sampling from it effectively. We define two distributions over the space of all motifs of given dimension. The first is the maximum entropy distribution subject to mean information content, and the second is the truncated uniform distribution over all motifs having information content within a given interval. We derive exact sampling algorithms for each. As a proof of concept, we employ these sampling methods to analyze a broad collection of prokaryotic and eukaryotic transcription factor binding site motifs. In addition to positional information content, we consider the informational Gini coefficient of the motif, a measure of the degree to which information is evenly distributed throughout a motif's positions. We find that both prokaryotic and eukaryotic motifs tend to exhibit higher informational Gini coefficients (IGC) than would be expected by chance under either reference distribution. As a second application, we apply maximum entropy sampling to the motif p-value problem and use it to give elementary derivations of two new estimators. Despite the historical centrality of biological sequence motif analysis, this study constitutes to our knowledge the first use of principled null hypotheses for sequence motifs given information content. Through their use, we are able to characterize for the first time differerences in global motif statistics between biological motifs and their null distributions. In particular, we observe that biological sequence motifs show an unusual distribution of IGC, presumably due to biochemical constraints on the mechanisms of direct read-out.