  The identification of transcription factor binding sites (TFBSs) on genomic
DNA is of crucial importance for understanding and predicting regulatory
elements in gene networks. TFBS motifs are commonly described by Position
Weight Matrices (PWMs), in which each DNA base pair independently contributes
to the transcription factor (TF) binding, despite mounting evidence of
interdependence between base pairs positions. The recent availability of
genome-wide data on TF-bound DNA regions offers the possibility to revisit this
question in detail for TF binding {\em in vivo}. Here, we use available fly and
mouse ChIPseq data, and show that the independent model generally does not
reproduce the observed statistics of TFBS, generalizing previous observations.
We further show that TFBS description and predictability can be systematically
improved by taking into account pairwise correlations in the TFBS via the
principle of maximum entropy. The resulting pairwise interaction model is
formally equivalent to the disordered Potts models of statistical mechanics and
it generalizes previous approaches to interdependent positions. Its structure
allows for co-variation of two or more base pairs, as well as secondary motifs.
Although models consisting of mixtures of PWMs also have this last feature, we
show that pairwise interaction models outperform them. The significant pairwise
interactions are found to be sparse and found dominantly between consecutive
base pairs. Finally, the use of a pairwise interaction model for the
identification of TFBSs is shown to give significantly different predictions
than a model based on independent positions.
