Fig 1.
(a) Workflow of the dictionary learning method. Multiway (multiplexed) chromatin interactions represented as hyperedges are clique expanded into standard networks and combined to create input networks for the algorithm. MCMC subnetwork sampling is then used to generate samples for initialization and online updates during iterative optimization of the objective function, resulting in convex dictionary elements. (b) Illustration of the clique expansion process. Hyperedges are subsets of indexed nodes shaded with the same color. (c) Illustration of clique expansion distortion. There is no hyperedge including nodes 3, 5, and 8 (colored red), and this 3-clique only exists due to shared nodes/edges of “real” hyperedges. Such distortion is negligible when the number of large hyperedges is limited. (d) Subnetwork sampling and the notion of a motif homomorphism. These correspond to subnetworks of the input network induced by a fixed number of nodes that contain a template motif topology. The set of homomorphisms for a network
and the template network F are defined in the Methods Section (Eq 7). Also depicted are an example homomorphism
and its induced adjacency matrix
for an input network
with 9 nodes. The template F is a star network on 4 nodes. In the adjacency matrix, a black field indicates 1, while a white field indicates 0. (e) Workflow of the MCMC sampling algorithm for path homomorphisms. Given a sample
at time t, obtained via a directed random walk from an initial state in the input network,
, we generate a sample
at time t + 1 by choosing uniformly at random a node v from the neighborhood of
(marked in green) and calculating a probability of acceptance β. If node v is accepted, we initiate a new directed random walk from v, otherwise, we restart a directed random walk from
to generate a new sample.
Fig 2.
(a) Organization of a dictionary comprising K dictionary elements that are convex combinations of real representative subnetworks. Each dictionary element itself is a sparse convex combination of a set of representatives which are small subnetworks of the input real-world network. In the example, there are 6 options for the representatives, and inclusion of a representative into a dictionary element is indicated by a colored entry in a 6-dimensional indicator column-vector. Each of the 6 representatives corresponds to a subnetwork of the input network with a fixed number of nodes (3 for our example). The dictionary element is generated by a convex combination of the corresponding adjacency matrices of its corresponding representative subnetworks. For the example, the resulting dictionary elements are 9 × 9 matrices. (b) Illustration of the representative region update. When an online data sample is observed, the distance of the sample to each of the current dictionary elements is computed and the sample is assigned to the representative region of the nearest dictionary element. From this expanded set of representatives, one representative is carefully selected for removal to improve the objective. The new dictionary element is then obtained as an optimized convex combination of the updated set of representatives.
Fig 3.
ChIA-Drop [17] adopts a droplet-based barcode-linked technique to reveal multiway chromatin interactions at a single molecule level. Chromatin samples are crosslinked and fragmented without a proximity ligation step. The samples are enriched for informative fragments through antibody pull-down.
Fig 4.
A dictionary element, represented as a matrix, consists of both proximal and distal interacting genomic regions.
The elements on the diagonal are not necessarily indexed by adjacent (consecutive) genomic fragments, as explained by the example in the second row. There, off-diagonal long-range interactions in the 9 × 9 matrix are included in a 3 × 3 dictionary element whose diagonal elements are not in consecutive order.
Fig 5.
Dictionary elements for Drosophila chromosomes 2L, 2R, 3L and 3R obtained using online cvxNDL.
Each subplot contains 25 dictionary elements for the corresponding chromosome and each block in the subplots corresponds to one dictionary element. The elements are ordered by their importance score. Note that the “diagonals” in the dictionary elements do not exclusively represent localized topologically associated domains (TADs) as in standard chromatin contact maps; instead, they can also capture long-range interactions. This is due to the fact that the indices of the dictionary element matrices represent genomic regions that may be far apart in the genome. In contrast, standard contact maps have indices that correspond to continuously ordered genomic regions, so that the diagonals truly represent TADs (see Fig 4). The color-code captures the actual locations of the genomic regions involved in the representatives and their dictionary elements. The most interesting dictionary elements are those that contain both dark blue, light blue/green, and red colors (since they involve long-range interactions). This is especially the case for chr3L and chr3R.
Fig 6.
Dictionary elements for Drosophila chromosome chr2L generated by (A) NMF, (B) online NDL, (C) CMF and (D) online cvxNDL.
NMF and CMF are learned off-line, using a total of 20, 000 samples. Note that these algorithms do not scale and cannot work with larger number of samples such as those used in online cvxNDL. The color-coding is performed in the same manner as for the accompanying online cvxNDL results. Columns of the dictionary elements in the second row are color-coded based on the genome locations of the representatives. As biologically meaningful locations can be determined only via convex methods, the top row corresponding to NMF and online NDL results is black-and-white.
Fig 7.
Original adjacency matrix and reconstructed adjacency matrices based on different DL methods, for an example Stochastic Block Model (SBM), including randomly selected dictionary elements.
Both the x and y axes in the figures index the nodes of the synthetic network generated by the stochastic block model (SBM). The nodes are reorganized to highlight the underlying community structure. For a more quantitative analytical accuracy comparisons, see Table 1.
Table 1.
Average Precision Recall for different DL methods, for all chromosomes as well as SBM synthetic datasets.
Methods that return interpretable dictionaries are indicated by the superscript i while methods that are scalable to large datasets are indicated by the superscript s. Online cvxNDL is both interpretable and scalable while maintaining performance on par with other noninterpretable and nonscalable methods.
Fig 8.
The 5 most enriched GO terms for genes covered by dictionary elements from chr3R.
Column ‘#’ indicates the number of dictionary elements that show enrichment for the given GO term. Also reported are up to 3 dictionary elements with the largest importance score in the dictionary, along with the “density” ρ of interactions in the dictionary element (defined in the Methods section) and median distance dmed of all adjacent pairs of nodes in its representatives.
Fig 9.
Empirical cumulative distribution functions (ECDF) of mean pairwise coexpressions of genes covered by random and online cvxNDL dictionary elements ((a) for chr2L, (b) for chr2R, (c) for chr3L and (d) for chr3R).
The results are based on the two-sample Kolmogorov-Smirnov test, and the null hypothesis described in the main text.
Fig 10.
Pairwise coexpression of genes covered by (a) the R1-R4 genomic regions, (b) the T1-T4 genomic regions, (c) an online cvxNDL dictionary element, and (d) a randomly constructed dictionary element.
We calculated the mean and standard deviation of absolute pairwise coexpression values, and the mean and standard deviation of coexpression values specifically for all positively correlated gene pairs. The mean coexpression values within TADs and dictionary elements are similar to each other and generally higher than those of randomly constructed dictionary elements. The x and y axis index genes that belong to the respective TAD regions or a specific dictionary element. Note that the plot (b) is of coarser resolution due to the small number of genes covered when compared to the cases (a), (c), (d).
Table 2.
Intersection between the set of genes within the R1-R4 genomic regions and the sets of genes covered by online cvxNDL dictionary elements for chr2L.
We determined the sizes of the intersections of the set of genes covered by each dictionary element and the genes in the R1-R4 genomic region and arranged them in decreasing order. The top 5 dictionary elements in this order cumulatively contain 38 out of the 85 genes within the R1-R4 genomic regions. This is in sharp contrast with randomly generated dictionary elements, where the top 5 elements with maximum intersection cover only 7 genes.
Table 3.
The mean pairwise co-expressions of genes (and their standard deviations) covered by all 25 online cvxNDL and CMF dictionary elements for each of the four chromosomes.