Probing long-range interactions by extracting free energies from genome-wide chromosome conformation capture data

Abstract

Background: A variety of DNA binding proteins are involved in regulating and shaping the packing of chromatin. They aid the formation of loops in the DNA that function to isolate different structural domains. A recent experimental technique, Hi-C, provides a method for determining the frequency of such looping between all distant parts of the genome. Given that the binding locations of many chromatin associated proteins have also been measured, it has been possible to make estimates for their influence on the long-range interactions as measured by Hi-C. However, a challenge in this analysis is the predominance of non-specific contacts that mask out the specific interactions of interest.

Results: We show that transforming the Hi-C contact frequencies into free energies gives a natural method for separating out the distance dependent non-specific interactions. In particular we apply Principal Component Analysis (PCA) to the transformed free energy matrix to identify the dominant modes of interaction. PCA identifies systematic effects as well as high frequency spatial noise in the Hi-C data which can be filtered out. Thus it can be used as a data driven approach for normalizing Hi-C data. We assess this PCA based normalization approach, along with several other normalization schemes, by fitting the transformed Hi-C data using a pairwise interaction model that takes as input the known locations of bound chromatin factors. The result of fitting is a set of predictions for the coupling energies between the various chromatin factors and their effect on the energetics of looping. We show that the quality of the fit can be used as a means to determine how much PCA filtering should be applied to the Hi-C data.

Conclusions: We find that the different normalizations of the Hi-C data vary in the quality of fit to the pairwise interaction model. PCA filtering can improve the fit, and the predicted coupling energies lead to biologically meaningful insights for how various chromatin bound factors influence the stability of DNA loops in chromatin.

Figure 1

Average free energy of interaction. The genome-wide average free energy, ${\overline{F}}_k$, as a function of genomic separation (a 600 kb window at 10 kb resolution) for free energies derived from three contact matrices (shown in legend). All show that the average free energy cost associated with forming a loop grows with the linear separation between genomic bins. Fitting a polymer model, ${\overline{F}}_k\sim \alpha log\left|k\right|$ (see Methods) gives α=1.09, 1.085 and 1.12 for the raw, raw + ICE and hierarchical matrices.

Figure 2

Free energy principal components and chromatin-binding profiles. Shown are the first four principal components (A, B, C, D) calculated genome-wide from the F _i,j matrix created from the raw + ICE contact matrix (top plots). Below each free energy profile are heat maps of the genome-wide average binding profiles for the selected chromatin factors (see Text). The top heat map corresponds to the positive free energy interaction profile (blue curve), and the bottom heat map for that of the inverse profile (red curve). Red regions in the heat maps represent locations of higher occupancy and blue regions represent lower occupancy. The range of the heat maps goes from 0.0 (blue) to 1.0 (red).

Figure 3

Specific energies of interaction and associated chromatin factor contacts. Shown in (A, B, C) are the energies of interaction $\delta F_{i,j}=F_{i,j}-{\overline{F}}_{j-i}$ of a portion of chromosome 2L for three different F _i,j matrices: A) raw, B) raw + PC filtered and hierarchical. (The first 35 PCs were used in reconstructing the raw + PC free energies). All have been aligned so that the zeroth column corresponds to i=j. Blue regions correspond to attractive interactions (negative) and red regions to effective repulsive interactions (positive). Figures (D, E) show the locations of pairwise self contacts, $S_i^{\mu }{S}_j^{\mu }$ for the insulator factor BEAF and the polycomb group protein Pc (blue corresponds to $S_i^{\mu }{S}_j^{\mu }=0$ and red to $S_i^{\mu }{S}_j^{\mu }=1$). Comparing the interaction energies (A, B, C) with the locations of pairwise contacts (D, E) highlights how these contacts could be generating the observed interactions.

Figure 4

PCA filtering can improve fit to interaction model.A) There is an optimal number of PCs to use in reconstructing the energies of interaction δ F _i,j. Shown are the Pearson correlation coefficient and reduced χ ² of the genome-wide fits of the given data (see legend) to the model using Eq. 5 as a function of the number of PCs used in the filtering. For the energies derived from raw matrix, PC1 was excluded as it is simply a DC offset. (B, C) show the best fit results for the various datasets by chromosome. PCA filtering for the raw matrix leads to the best overall results.

Figure 5

Chromatin factor coupling energies from fitting. The fitted coupling energies, J _μ,ν, between chromosome associated factors. The left heat maps show the chromosomal average J’s, and the right heat map the associated standard deviations in the average values. The following free energy matrices were used: A) raw, B) raw + PC filtering (optimal number of PCs used was 35) and C) hierarchical.