Fig 1.
Three examples of nestedness (Nij) in a simple bipartite network.
The networks in panels (a)–(c) have the same number of nodes in each layer—18 domains (small circles) and two communities (i and j, large circles)—and the same number of links (18) but connected differently to achieve varying nestedness. Below each network, we illustrate how we calculate Nij using Eqs (4)–(7). On the horizontal k-axis, we indicate the number of shared nodes Sij for the community pair and the expected overlap μij calculated from Eq (3). The panels (a)–(c) show three essential Nij regimes (μij = 4 for all of the cases). (a) Mostly segregated (Sij < μij, Nij = −0.5). Because Sij = 2 and μij = 4, the i and j communities are half-way to full segregation. We illustrate this with a dark-blue stripe covering half the 0 ≤ k ≤ 4 = μij range. (b) Random overlap (Sij = μij, Nij = 0). The number of shared nodes equals the random expectation. (c) Mostly nested (Sij > μij, Nij = 0.5). Here i and j share one domain more than expected (Sij = 5). This yields Nij = 0.5 because their overlap is at the midpoint between the random and the maximum overlap (Sij = 6) that would result in ideal nesting (Nij = 1). We illustrate this with the orange stripe spanning half of the range μij(= 4) ≤ k ≤ 6. This example shows that Nij measures the relative overlap compared to what is achievable given the link density rather than absolute numbers.
Fig 2.
Hi-C maps, 3D communities, and domains.
(a) Hi-C maps where the red-to-blue pixel colors are a proxy for short-to-long 3D distances. The squares decorating the map’s diagonals represent GenLouvain-derived 3D communities for three γ values (0.5, 0.6, and 0.7). Above each map, we show the community coverage as a colored stripe. Having unique colors, we observe that the communities comprise scattered linear DNA segments. The white cross shows the centromere. (b) Community borders and coverage across chromosome 10 for 16 γ values. The upper turquoise stripe shows DNA stretches that never split for 0 < γ ≲ 1. We refer to these indivisible regions as domains. The smallest 3D domain is 100 kb long, which is the resolution limit of the Hi-C data set we use.
Fig 3.
Cross-scale community organization in chromosome 10.
(a) Circular tree showing how domains (filled circles on the outer rim) merge into larger and larger 3D structures (filled circles on the inner rings). Each ring represents one value of GenLouvain’s resolution parameter γ, and the diameters of the filled circles are associated with their DNA length (measured by the number of Hi-C bins). The red circles mark delocalized 3D structures forming a single 3D community at γ = 0.9 (denoted 100.9). The dark links show folding trajectories for the domains passing through 100.9 towards the root. The left panel shows a two-domain folding path and defines our label convention. We plotted the tree using RAW Graphs [37]. (b) Joining and splitting of the 13 domains belonging to the community 100.9. These domains (filled dark-blue circles) pass through the 3D communities (open circles), joining other domains (filled light-blue circles). The edges connect 3D communities with dark-blue domains. We also highlighted these folding pathways in (a) (dark links).
Fig 4.
Nestedness of community pairs in human chromosomes 3, 5, 10, and 22.
(a) Schematic community-domain overlap in three cases: fully segregated (Nij = −1, violet), random (Nij = 0, red), and fully nested (Nij = 1, yellow). Layer 1 contains communities belonging to different γ values (dotted lines). The bottom layer shows the irreducible domains, and the edges indicate community memberships. (b) Nestedness histogram (Nij) for chromosomes 3, 5, 10, and 22. For each chromosome, we derived communities from 16 γ values. The peaks at ±1 suggest that several communities are segregated (−1) and nested (+1). However, there also exist significant intermediate levels of nestedness. The stripe overlaying the histogram indicates what we classify as fully segregated or nested according to the nestedness metric outlined in Methods: Network nestedness. We show the nestedness of individual chromosomes in S3 Fig, and we visualize Nij distributions for individual γ-pairs in Supplementary Material, S5 Fig (chromosome 10). (c) Significant versus random community nestedness. As outlined in Methods: Network nestedness, we filter community overlaps having p-values ≤ 0.025 and show the relative proportions of significant and random overlap associated with the Nij histogram in (a). The colors indicate significant (orange) and random overlaps (blue-green).
Fig 5.
Hierarchical and semi-hierarchical models of chromatin folding for human chromosome 10.
(a) Ideal hierarchical folding (Q = 0). Filled circles on the outer rim represent domains; the root symbolizes the entire chromosome. We align the domain aggregates (superstructures) with the inner tree rings, each defining a scale of organization. We select a few domains (red-filled circles) and show their domain-to-root paths with thick edges. These domains assemble into yet larger structures (violet) at every inner ring. As soon as the domains merge into a superstructure labeled ‘453’ (dark violet), they never split apart. (b) Semi-hierarchical folding (Q = 0.30). As in (a), we color the domains in red that merge into a superstructure ‘453’ and highlight their folding paths with thick edges going from the outer rim to the root. Unlike (a), node ‘453’ is scattered across seven tree branches. Thus, ‘453’ only partially nests into larger structures and the domains split and reunite when approaching the root. (c) Nestedness histogram when Q = 0 (ideal hierarchy, red bars) and Q = 1 (random nesting, open bars). When Q = 0, we see two peaks at Nij ± 1, indicating complete segregation and full nestedness. When Q = 1, the domains are fully randomized between the superstructures. While there is still perfect nesting and segregation (as we expect from the random null hypothesis in Methods: Network nestedness), there is also partial overlap for −0.8 < Nij < 0.8. (d) Nestedness histogram with some randomness (Q = 0.30, light-blue bars) overlaying the actual GenLouvain-derived data for chromosome 10 (dark-grey bars). We produced (a) and (b) using RAW Graphs [37].
Fig 6.
Chromatin type and cross-scale nestedness between community pairs in chromosome 10.
(a, left) Each chromatin cross pair (AB, AC, AD, etc.) has three community types. For example, these may enrich A (“Promoters”), B (“Enhancers”), or A and B. The large dashed circles represent the Venn diagram of all A or B community types (small filled circles). The set illustrates that one community can be in one of three categories: enriched with A (upper), B (lower), or both (intersection). (a, right) Schematic illustrating the color codes used in the nestedness histograms associated: complete overlap (dark blue), the difference (dark red), and the intersection (light blue). In pale blue, we indicate the chromosome-wide average of chromosome 10. (b) Nestedness distributions (Nij) for 10 combinations of chromatin types A–D. The diagonal panels show the nestedness histograms for community pairs belonging to the same chromatin type (AA, BB, etc.). The off-diagonal panels show the other six paired combinations (AB, AC, AD, etc.); see panel (a) for detailed descriptions. The faint pale blue background in all histograms portrays the complete nestedness histogram from chromosome 10 (like Fig 4).
Fig 7.
Median community modularity (rescaled with community size) for four chromatin groups A—D (solid lines) across different scale parameters γ.
The dashed line shows the median for the entire network. A, B, and C communities have higher modularity than D and the whole network.