Fig. 1
From: DC3 is a method for deconvolution and coupled clustering from bulk and single-cell genomics data
Overview of the DC3 method. a DC3 performs joint analysis using three types of data from separate samples from the same cell population: scRNA-seq, scATAC-seq, bulk HiChIP. E denotes the genes expression level in each cell measured in scRNA-seq; O denotes enhancer chromatin accessibilities in each cell measured in scATAC-seq; C denotes the enhancer-promoter interactions strength (loop counts) between each gene and each enhancer measured in bulk HiChIP. b A graphical example for simultaneously decomposing E, O, C to get the underlying clusters and cluster-specific HiChIP in K = 3 case: (1) \(\left\| {E - W_1H_1} \right\|_F^2\): wik1gives the mean gene expression for the i-th gene in the the k-th cluster of cells, while \(h_{kj}^1\) gives the assignment weights of the j-th cell to the k-th cluster; (2) \(\left\| {O - W_2H_2} \right\|_F^2:w_{ik}^2\) gives the mean chromatin accessibility for the i-th enhancer in the k-th cluster of cells, while the j-th column of H2 gives the assignment weights of the j-th cell to the different clusters; (3) \(\hat C = \alpha D \odot \left( {W_1\Lambda W_2^T} \right)\): each enhancer–promoter interaction cij can be decomposed into subpopulation-specific interactions, i.e. \(c_{ij} = \lambda _k\mathop {\sum }\limits_k c_{ijk}\), where cijk is the interaction strength in the k-th subpopulation and λk is proportional to the size of the subpopulation; Λ is a K by K diagonal matrix [λ1, λ2, …λK]. Within each subpopulation, following the assumption that an enhancer-promoter interaction is proportional to the product of accessibilities of the corresponding enhancer and promoter, we model cij as \(c_{ij} = {\mathrm{\alpha }}d_{ij}\mathop {\sum }\limits_k \lambda _kw_{ik}^1w_{jk}^2,\) where dij is a set of indicators selecting the enhancer-promoter pair to be modeled. Therefore, cluster-specific HiChIP interactions of k-th subpopulation can be obtained from the k-th column of W1 multiple the transposition the k-th column of \(W_2:\alpha D \odot \left( {\lambda _kW_1^{\left( { \cdot k} \right)}W_2^{\left( { \cdot k} \right)T}} \right)\)
