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Motivation: A common class of behaviour encountered in the biological sciences involves branching and recombination. During branching, a statistical process bifurcates resulting in two or more potentially correlated processes that may undergo further branching; the contrary is true during recombination, where two or more statistical processes converge. A key objective is to identify the time of this bifurcation (branch or recombination time) from time series measurements, e.g. by comparing a control time series with perturbed time series. Gaussian processes (GPs) represent an ideal framework for such analysis, allowing for nonlinear regression that includes a rigorous treatment of uncertainty. Currently, however, GP models only exist for two-branch systems. Here, we highlight how arbitrarily complex branching processes can be built using the correct composition of covariance functions within a GP framework, thus outlining a general framework for the treatment of branching and recombination in the form of branch-recombinant Gaussian processes (B-RGPs)., Results: We first benchmark the performance of B-RGPs compared to a variety of existing regression approaches, and demonstrate robustness to model misspecification. B-RGPs are then used to investigate the branching patterns of Arabidopsis thaliana gene expression following inoculation with the hemibotrophic bacteria, Pseudomonas syringae DC3000, and a disarmed mutant strain, hrpA. By grouping genes according to the number of branches, we could naturally separate out genes involved in basal immune response from those subverted by the virulent strain, and show enrichment for targets of pathogen protein effectors. Finally, we identify two early branching genes WRKY11 and WRKY17, and show that genes that branched at similar times to WRKY11/17 were enriched for W-box binding motifs, and overrepresented for genes differentially expressed in WRKY11/17 knockouts, suggesting that branch time could be used for identifying direct and indirect binding targets of key transcription factors., Availability and Implementation: https://github.com/cap76/BranchingGPs., Supplementary Information: Supplementary data are available at Bioinformatics online.

Background: Classically, models of DNA-transcription factor binding sites (TFBSs) have been based on relatively few known instances and have treated them as sites of fixed length using position weight matrices (PWMs). Various extensions to this model have been proposed, most of which take account of dependencies between the bases in the binding sites. However, some transcription factors are known to exhibit some flexibility and bind to DNA in more than one possible physical configuration. In some cases this variation is known to affect the function of binding sites. With the increasing volume of ChIP-seq data available it is now possible to investigate models that incorporate this flexibility. Previous work on variable length models has been constrained by: a focus on specific zinc finger proteins in yeast using restrictive models; a reliance on hand-crafted models for just one transcription factor at a time; and a lack of evaluation on realistically sized data sets., Results: We re-analysed binding sites from the TRANSFAC database and found motivating examples where our new variable length model provides a better fit. We analysed several ChIP-seq data sets with a novel motif search algorithm and compared the results to one of the best standard PWM finders and a recently developed alternative method for finding motifs of variable structure. All the methods performed comparably in held-out cross validation tests. Known motifs of variable structure were recovered for p53, Stat5a and Stat5b. In addition our method recovered a novel generalised version of an existing PWM for Sp1 that allows for variable length binding. This motif improved classification performance., Conclusions: We have presented a new gapped PWM model for variable length DNA binding sites that is not too restrictive nor over-parameterised. Our comparison with existing tools shows that on average it does not have better predictive accuracy than existing methods. However, it does provide more interpretable models of motifs of variable structure that are suitable for follow-up structural studies. To our knowledge, we are the first to apply variable length motif models to eukaryotic ChIP-seq data sets and consequently the first to show their value in this domain. The results include a novel motif for the ubiquitous transcription factor Sp1.