Examples

The segmentation algorithm was run on high quality intra-chromosomal Hi-C heatmaps with 40kb bins for cell types IMR90 (lung fibroblast) and H1 (human embryonic stem cell), generated from the raw data of and and processed using the method of Imakaev . The figure below shows that by increasing the resolution parameter \(\gamma\), one obtains a monotonic increase in the number of segments (equivalently, the number of boundary nodes) in the optimal segmentation. This results in a characteristic ``boundary saturation’’ curve for each heatmap (see Figure ).

Note that our formulation of the segmentation problem did not explicity introduce “gap” segments that occur between high scoring “domain” segments. Instead, regions with low scores accumulate sequences of consecutive boundary nodes that imply a lack of discernable domain structure, as far as the resolution of the data allows.

The y-axis gives the number fraction of nodes \(\theta\) that are the starting boundaries of a segment. Because in the Potts model, the expected number of communities found scales as \(\sqrt{\gamma m}\) , the values on the x-axis are rescaled to make the saturation curves from different chromsome heatmaps comparable. Interestingly, the curves for most chromosomes collapse near one another, with characteristic profiles for the two cell types.

optimal segmentation (blue triangles) at resolution \(\gamma=4\), IMR90 chr22, pixels are log-transformed Hi-C data

boundary saturation curves as function of resolution parameter \(\gamma\) (optimal segmentations, i.e. zero temperature)

marginal boundary distributions at different temperatures (\(1/\beta\))

upper triangle: marginal boundary co-occurrence distribution at finite temperature

upper triangle: (log) marginal segment distribution \(\gamma = 8\) at finite temperature

upper triangle: marginal co-segmentation distribution \(\gamma = 6\) at finite temperature