Sequence-based multiscale modeling for high-throughput chromosome conformation capture (Hi-C) data analysis

Hodge theory-based biomolecular data analysis

Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.

BHi-Cect: a top-down algorithm for identifying the multi-scale hierarchical structure of chromosomes

High-throughput chromosome conformation capture (Hi-C) technology enables the investigation of genome-wide interactions among chromosome loci. Current algorithms focus on topologically associating domains (TADs), that are contiguous clusters along the genome coordinate, to describe the hierarchical structure of chromosomes. However, high resolution Hi-C displays a variety of interaction patterns beyond what current TAD detection methods can capture. Here, we present BHi-Cect, a novel top-down algorithm that finds clusters by considering every locus with no assumption of genomic contiguity using spectral clustering. Our results reveal that the hierarchical structure of chromosome is organized as ‘enclaves’, which are complex interwoven clusters at both local and global scales. We show that the nesting of local clusters within global clusters characterizing enclaves, is associated with the epigenomic activity found on the underlying DNA. Furthermore, we show that the hierarchical nesting that links different enclaves integrates their respective function. BHi-Cect provides means to uncover the general principles guiding chromatin architecture.