Abstract
Fingerprints of the threedimensional organization of genomes have emerged using advances in HiC and imaging techniques. However, genome dynamics is poorly understood. Here, we create the chromosome copolymer model (CCM) by representing chromosomes as a copolymer with two epigenetic loci types corresponding to euchromatin and heterochromatin. Using novel clustering techniques, we establish quantitatively that the simulated contact maps and topologically associating domains (TADs) for chromosomes 5 and 10 and those inferred from HiC experiments are in good agreement. Chromatin exhibits glassy dynamics with coherent motion on micron scale. The broad distribution of the diffusion exponents of the individual loci, which quantitatively agrees with experiments, is suggestive of highly heterogeneous dynamics. This is reflected in the celltocell variations in the contact maps. Chromosome organization is hierarchical, involving the formation of chromosome droplets (CDs) on genomic scale, coinciding with the TAD size, followed by coalescence of the CDs, reminiscent of Ostwald ripening.
Introduction
The organization of chromosomes without topological entanglement or knot formation in the crowded tight space of the nucleus is remarkable. Understanding the structural organization and the dynamics of eukaryotic chromosomes and the mechanism of chromosome territories formation may hold the key to enunciating genome functions^{1,2}. Glimpses into the structures of the chromosomes have emerged, thanks to spectacular advances in chromosome conformation capture (3C, 4C, 5C, and HiC) experiments^{3,4,5}, from which the probability, P_{ij}, that any two loci i and j are in contact can be inferred. The set of P_{ij}, which is a twodimensional representation of the spatial organization, constitutes the contact map. More recently, imaging methods like FISH^{6} as well as superresolution technique^{7,8} have more directly determined the positions of the loci of single chromosomes, thus providing a muchneeded link to the indirect measure of spatial organization revealed through contact maps. The experiments by Zhuang and coworkers and others^{6,7,8} are of great value because the number of constraints needed to unambiguously infer structures from contact maps alone is very large^{9}.
Contact maps, constructed from HiC experiments, revealed that chromosome is organized into compartments on genomic length scales exceeding megabases (Mbps)^{4,5}. The partitioning of the structure into compartments are highly correlated with histone markers of the chromatin loci^{5}, implying that contacts are enriched within a compartment and depleted between different compartments. The loci associated with active histone markers and those associated with repressive histone markers localize spatially in different compartments. Higher resolution HiC experiments^{5} have also identified topologically associated domains (TADs) on scales on the order of hundreds of kilobases^{3}. The TADs are the square patterns along the diagonal of the contact maps in which the probability of two loci being in contact is more probable than between two loci belonging to distinct TADs. Multiplexed FISH experiments^{6} show most directly that TADs belonging to distinct compartments are spatially separated in the chromosomes.
The experimental studies have inspired a variety of polymer models^{10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}, which have provided insights into many aspects of chromosome organization. These studies are particularly important because finding a unique solution (if one exists) to the inverse problem of deriving spatial structures from contact maps is most challenging^{9}. Some of the features in the contact maps, such as the probability P(s) that two loci separated by a certain genomic distance (s) are in contact, may be computed using a homopolymer model^{4}, without accounting for the epigenetic states, whereas fine structures such as TADs and compartments require copolymer or heteropolymer models^{18,20,21,26}.
Biological functions, such as the search for genes by transcription factors or mechanism for DNA damage repair, not only depend on genome structure but also the associated dynamics. The use of polymer models in describing chromatin structure has a rich history^{10,11}. More recent studies show that polymer physics concepts have been most useful in predicting the probabilistic nature of chromosome organization inferred from HiC experiments^{12,13,14,15,16,17,18,19,20,21,22,23}. In contrast, the dynamic aspects of the interphase chromosome have received much less attention^{24,25,26,27,28,29}. Experiments have revealed that genomewide chromatin dynamics^{29,30,31,32} of chromatin fiber in mammalian cells exhibit heterogeneous subdiffusive behavior. Thus, it is important to understand how the slow dynamics of the individual locus and long length scale coherent collective motions emerge from the highly organized chromosomes.
Here, we develop a copolymer model to describe both the structure and dynamics of human interphase chromosomes based on the assumption that the largescale organization of human interphase chromosome is largely driven and maintained by the interactions between the loci of similar epigenetic states. Similar models, that differ greatly in details, have been developed to model the 3D structure of Drosophila chromosomes^{18,20}. Jost et al.^{18} used a heteropolymer model with four different types of monomers representing active, Polycomb, HP1 and black chromatin to describe the formation of TADs in Drosophila genome. Michieletto et al.^{26} constructed a heteropolymer with three epigenetic states (acetylated, methylated, and unmarked) to probe how the epigenetic states are maintained. A very different reverseengineering approach, with HiC contact maps as inputs, was used to construct an energy function with 27 parameters^{21}. We take a “bottomup” approach to incorporate the epigenetic states into the polymer model similar in spirit to the previous studies^{18,20,26}. We show that in order to capture the structural features faithfully, at least two types of beads, representing active and repressive loci are needed. Simulations of the resulting chromosome copolymer model (CCM) for human interphase chromosomes 5 and 10 show that the structural characteristics, such as the scaling of P(s) as a function of s, compartments, and TADs indicated in the HiC contact maps are faithfully reproduced. We use sophisticated clustering algorithms to quantitatively compare the simulated contact maps and those inferred from HiC experiments. The compartment feature noted in the HiC contact map is due to microphase separation between chromosome loci associated with different epigenetic states, implying that a copolymer model is needed for characterizing largescale genome organization. The TADs emerge by incorporating experimentally inferred positions of the loop anchors, whose formation is facilitated by CTCF motifs. The only free parameter in the CCM, the optimal loci–loci interaction strength between loci belonging to the same epigenetic states, is adjusted to give a good description of the HiC contact map. Using simulations based on the resulting CCM, we show that chromosome dynamics is highly heterogeneous and exhibits many of the characteristics of out of equilibrium glassy dynamics, with coherent motion on μm scale, including stretched exponential decay of the scattering function (F_{s}(k,t)), a nonmonotonicity behavior in the time dependence of the fourth order susceptivity associated with fluctuations in F_{s}(k,t). Of particular note is the remarkable celltocell and locitoloci variation in the time (t) dependence of the mean square displacement, Δ_{i}(t), of the individual loci. The distribution P(α) of the exponent associated with the increase in Δ_{i}(t) ~ t^{α} is broad. The simulated and experimentally measured P(α)s are in excellent agreement. Our work shows that chromosomes structures are highly dynamic exhibiting large celltocell variations in the contact maps and dynamics. The rugged chromosome energy landscape, with multiple minima separated by large barriers, is perhaps needed to achieve a balance between genomic conformational stability and dynamics for the execution of a variety of biological functions.
Results
Choosing the energy scale in the CCM
We fixed N, the size of the copolymer to N = 10,000, modeling a 12 Mbps (megabases) chromatin fiber, corresponding to a selected region of the Human Cell line GM12878 Chromosome 5 (Chr 5) from 145.87 Mbps to 157.87 Mbps. In the CCM (Fig. 1a and Supplementary Fig. 1), the only unknown parameter is \(\epsilon\), characterizing the strength of the interaction between the loci (Supplementary Table I). We chose a \(\epsilon\) value that reproduces the contact maps that is near quantitative agreement with the HiC data. As \(\epsilon\) increases, the structures of the chromosome are arranged in such a way that segments with small genomic distance s are more likely to be in spatial proximity (see the section “Chromosome Structures in terms of WLM” below). This is also illustrated in Supplementary Fig. 4, which shows that higher values of \(\epsilon\) lead to clearer segregation between the loci with different colors. The colors encode the genomic locations. The snapshots of the organized chromosome, the good agreement between the calculated and HiC contact maps, and the accurate description of the spatial organization as assessed by the ward linkage matrix (WLM) (Supplementary Note 9) confirm that \(\epsilon\) = 2.4k_{B}T produces the closest agreement with experiments. Increasing \(\epsilon\) beyond 2.4k_{B}T leads to a worse description of segregation between loci with distinct epigenetic states.
Furthermore, P(s) as a function of s obtained in simulations with \(\epsilon\) = 2.4k_{B}T is also consistent with experiments (see below). The sdependent contact probability, P(s) in Fig. 1b, shows that there are two scaling regimes. As \(\epsilon\) increases, the probability of shortrange (small s) increases by several folds, while P(s) for large s decreases by approximately an order of magnitude (Supplementary Fig. 13a). In particular, for \(\epsilon\) = 1.0k_{B}T, P(s), decreases much faster compared to experiments at small s. In contrast, we find that at \(\epsilon\) = 2.4k_{B}T, P(s) ~ s^{−0.75} for s < 0.5 Mbps and when s exceeds ~0.5 Mbps, P(s) ~ s^{−1.25} (red curve in Fig. 1b). Such a behavior, with P(s) exhibiting two distinct scaling regimes, agrees with experiments (black line in Fig. 1b). It is worth pointing out that the two scaling regimes in P(s) is a robust feature of all 23 human interphase chromosomes (Fig. 1c). It is clear the two scaling regimes in P(s) with a crossover from one to another at s ≈ 3 × 10^{5}–6 × 10^{5} bps is universally found in all the chromosomes. Interestingly, our simulation suggests that the crossover scale in P(s) coincides with the size of the chromosome droplets (see discussion).
Active and repressive loci microphase segregate
Comparison of the contact maps between simulations and experiments illustrates that compartment formation appearing as plaid or checkerboard patterns in Fig. 1d, shows good agreement with HiC data^{4,5}. The dashed rectangles mark the border of one such compartment enriched predominantly with interactions between loci of the same type, suggesting that compartments are formed through the clustering of the chromatin segments with the same epigenetic states. A previous experimental study suggests that the chromatin structuring in TADs is also driven by the epigenome feature^{33}. In order to make the comparison precise, we treated the contact maps as probabilistic matrices and used a variety of mathematical methods to quantitatively compare large matrices. First, the checkerboard pattern in the contact map is more prominent when illustrated using the Spearman correlation map (see Supplementary Note 7 and Supplementary Figs. 6 and 7). Second, to quantitatively compare the simulated results with experiments, we use the spectral coclustering algorithm^{34} to bicluster the computed Spearman correlation map (see Supplementary Note 8). Other methods, such as PCA^{4} and kmeans^{5}, have been used to extract the compartment features. The spectral coclustering deployed here, gives results that closely resemble those obtained using PCA (Supplementary Fig. 20). Finally, the similarity between the simulated and experimental data is assessed using the Adjusted Mutual Information Score (AMI) (Supplementary Note 8). The CCM model, based only on epigenetic information and the locations of the loop anchors, yields an AMI score that results in correctly reproducing ≈81% of the compartments obtained from the experimental data. In contrast, a pseudo homopolymer model with \(\epsilon _{{\mathrm{AA}}} = \epsilon _{{\mathrm{BB}}} = \epsilon _{{\mathrm{AB}}} = \epsilon\), which has the same loop anchors as the CCM, has an absolute AMI score that is 200 times smaller (Supplementary Fig. 9), and does not lead to the formation of compartments (correctly reproducing only ≈51% of the compartments, no better than random assignments). Thus, the CCM is the minimal model needed to reproduce the essential features found in the contact map.
The inset in Fig. 2a, displaying a typical snapshot of the condensed chromosome, reveals that active (A, blue) and repressive (B, red) loci are clustered together, undergoing microphase separation (see Methods for definition of active and repressive loci). The tendency to segregate is vividly illustrated in the radial distribution functions g_{AA}(r), g_{BB}(r), and g_{AB}(r), which shows (Fig. 2a) that g_{AA}(r) and g_{BB}(r) have much higher values than g_{AB}(r) implying that active and repressive loci form the clusters of their own, and do not mix. Such a microphase separation between the Arich and Brich regions directly gives rise to compartments in the contact map. Interestingly, the normalized radial density (Fig. 2b) shows that active chromatin exhibits a peak at large radial distance, r, implying that the active loci localize on the periphery of the condensed chromosome whereas repressive chromatin is more homogeneously distributed. Visual inspection of the simulation trajectories also suggests that active and repressive chromatins are often separated in a polarized fashion, in accord with a recent experimental study^{6}, which shows that the two compartments are indeed similarly spatially arranged.
Spatial organization of the compact chromosome
In order to illustrate the spatial organization of the chromosome, we introduce the distance function,
where \(\langle \cdot \rangle\) denotes both an ensemble and time average. We calculated R(s), the mean endtoend distance between the loci, by constraining the genomic distance \(i  j_{}^{}\) to s. If the structured chromosome is maximally compact on all length scales, we expect R(s) ~ s^{1/3} for all s. However, the plot of R(s) on a loglog scale shows that in the range 10^{5} ≲ s ≲ 10^{6} bps, R(s) ~ s^{0.2}. The plateau at large s arises due to s reaching the boundary of the compact structure. The inset in Fig. 3a, comparing the simulation result and experimental data^{6}, both show the same scaling for R(s) as a function of s. Note that in ref. ^{6} spatial distances are measured between centroids of TADs domains rather than individual loci. We present the equivalence between Eq. (1) and the distances between the TAD centroids in Supplementary Note 6.
By a systematic analysis of the FISH data, Wang et al.^{6} established that the probability of contact formation between loci i and j, P_{ij}, is inversely proportional to a power of their mean spatial distance R_{ij} = \(\langle {\mathbf{r}}_i  {\mathbf{r}}_j\rangle\), with the latter providing a direct picture of the spatial organization. Similarly, in this work, we explored the relation between C_{ij} and R_{ij} where \(C_{ij}\left( { = P_{ij}\mathop {\sum}\nolimits_{i < j} C_{ij} \propto P_{ij}} \right)\) is the number of contacts between loci i and j that are recorded in the simulations. The heatmap of (1/C_{ij},R_{ij}) in Fig. 3b shows that the two matrices are proportional to each other. In accord with the FISH data^{6}, we find that \(1/C_{ij} \propto R_{ij}^\lambda\) where λ ≈ 4, suggesting that larger mean spatial distance between loci i and j implies smaller contact probability, which is the usual assumption when experimental HiC data are used to infer threedimensional chromosome organization. The decrease of C_{ij} with increasing R_{ij} with a large value of λ, is unexpected but is an important finding needed to link contact maps and spatial structures.
The slope of the dashed line in Fig. 3b obtained using the data in ref. ^{6}, is 4.1, which coincides with our simulation results. Meanfield arguments^{35} suggest that P(s) ~ R(s)^{−3}, which follows from the observation that the end of the chain is uniformly distributed over a volume R^{3}(s). This is neither consistent with our simulations nor with experiments, implying that the distribution of the chain ends is greatly skewed. Although both the simulated and experimental results establish a strong correlation between R(s) and P(s), such a correlation is only valid in an ensemble sense (see Supplementary Note 14 and Supplementary Fig. 19 for additional discussions as well as ref. ^{36}).
TADs and their shapes
Our model reproduces TADs, depicted as triangles along the diagonal in Fig. 1e, of an average length of 200 kbps along the diagonal of the contact map in which the interactions between the loci are greatly enhanced. It has been noted^{5} that in a majority of cases, boundaries of the TADs are marked by a pair of CTCF motifs with a high probability of interaction between them. They are visualized as peaks in the HiC map (Fig. 1e). To quantitatively detect the boundaries of the TADs, we adopt the procedure described in ref. ^{3} to identify the position of each TAD (see Supplementary Note 5 for a description of the Directionality Index method for identifying TADs). The boundaries of the TADs, shown in blue (HiC data) and green (simulations) are reproduced by the CCM (Fig. 1e).
To investigate the sizes and shapes of each individual TADs (defined as CTCF loops in the simulations), we calculated the radii of gyration, R_{g}, the relative shape anisotropies κ^{2}, as well as the shape parameters, S, for 32 TADs (see Supplementary Note 10 for details). These TADs are typical representations of all TADs. The genomic size of the 32 TADs is similar to the genomewide distribution (see Supplementary Fig. 2). The results are shown in Supplementary Fig. 10. The mean R_{g} for each individual TADs scales as their genomic length with exponent 0.27, which is an indicator of the compact structures for the TADs. However, unlike compact globular objects, their shapes are far from being globular and are much more irregular with smaller TADs adopting more irregular shapes compared to the larger TADs (see \(\langle \kappa ^2\rangle\) and \(\langle S\rangle\) as a function of TAD size in Supplementary Fig. 10). Such compact but irregularly shaped nature of TADs are vividly illustrated by typical snapshots for the two TADs (Fig. 1h, i). How can we understand this nontrivial highly aspherical shapes of the TADs when the chromosome is spherical on long length scales (several Mbps)? Since TADs are constrained by the CTCF loops, they may be viewed locally as ring polymers. Ring polymers in a melt are compact^{37} objects but adopt irregular shapes, consistent with our prediction for TADs.
We then wondered if TADs in each individual cells have similar sizes and shapes. We computed the dispersion in R_{g}, κ, and S (Fig. 3c and Supplementary Figs. 10 and 11) among different trajectories. Figure3c shows the \(P\left( {\langle R_{\mathrm{g}}^2\rangle /\overline {\langle R_{\mathrm{g}}^2\rangle } } \right)\), of the mean square radius of gyration \(\langle R_{\mathrm{g}}^2\rangle\) for the 32 Chr 5 TADs in each trajectory normalized by the average \(\overline {\langle R_{\mathrm{g}}^2\rangle }\) of each individual TAD. The bracket (bar) is the time (ensemble) average. The large dispersion in \(P\left( {\langle R_{\mathrm{g}}^2\rangle /\overline {\langle R_{\mathrm{g}}^2\rangle } } \right)\) (Fig. 3c) as well as \(P(\langle \kappa \rangle /\overline {\langle \kappa \rangle } )\) and \(P(\langle S\rangle /\overline {\langle S\rangle } )\) (Supplementary Fig. 11) suggest that TADs are fluctuating objects, which exhibit substantial celltocell variations. Our result supports the recent FISH^{38} and singlecell HiC experimental findings^{39,40}, showing that individual TAD compaction varies widely from highly extended to compact states among different cells. To decipher how the variation of the structure of the chromosome changes as a function of s, we calculated the coefficient of variation, δR(s) = \(\left( {\left\langle {R_s^2} \right\rangle  \left\langle {R_s} \right\rangle ^2} \right)^{1/2}/\left\langle {R\left( s \right)} \right\rangle\) (see Supplementary Note 11 for details). Interestingly, δR(s) first increases with s up to s ≈ 10^{5}–10^{6} bps and then decreases as s further increases (Fig. 3d). Analysis of the experimental data from ref. ^{6} shows a similar decreasing trend for s > 10^{5} bps (Supplementary Fig. 12c). Higher resolution experiments are needed to resolve the variance for s < l0^{5} bps. The predicted nonmonotonic dependence of δR(s) on s is amenable to experimental test.
Chromosome structures in terms of the WLM
To quantitatively analyze the spatial organization of the compact chromosome, we use the unsupervised agglomerative clustering algorithm to reveal the hierarchy organization on the different length scales. A different method, which is also based on clustering techniques, has recently been applied to HiC contact map^{41}. We use the WLM (see Supplementary Note 9 for details), which is directly applicable to the spatial distance matrix, R in which the element, R_{ij} = \(\langle {\mathbf{r}}_i  {\mathbf{r}}_j\rangle\), is the mean spatial distance between the loci i and j. We also constructed the experimental WLM by converting the HiC contact map to a distance map by exploiting the approximate relationship between R_{ij} and P_{ij} \(\left( { \propto R_{ij}^{  4.1}} \right)\) discussed previously (also see Fig. 3b). The advantages of using distance matrices instead of contact maps are two folds. First, matrix R is a direct depiction of the threedimensional organization of the chromosome. The WLM, constructed from R is a cophenetic matrix, which can be used to reveal the hierarchical nature of the chromosome organization. Second, the contact map matrix elements do not obey triangle inequality. Therefore, it is not a good indicator of the actual 3D spatial arrangement of the loci. We show the comparison between simulated WLMs and experimentally inferred WLM for ε=(1.0,2.4)k_{B}T (Fig. 4). Visual inspection of the WLMs for \(\epsilon\) = 2.4k_{B}T shows distinct segregation in the spatial arrangement of the loci. It is clear from Fig. 4 that the experimentally inferred WLM, constructed from HiC data, and simulations result with \(\epsilon\) = 2.4k_{B}T are almost identical. From the WLMs for both \(\epsilon\) = 1.0k_{B}T and \(\epsilon\) = 2.0k_{B}T (Supplementary Fig. 4), we surmise that loci with large genomic separation s are in spatial proximity, which is inconsistent with the experimental WLM. The Pearson correlation coefficient between experimental result and CCM using \(\epsilon\) = 2.4k_{B}T is 0.96 (0.53 for \(\epsilon\) = 1.0k_{B}T, 0.84 for \(\epsilon\) = 2.0k_{B}T and 0.75 for \(\epsilon\) = 2.7k_{B}T). Thus, the poorer agreement between the simulated WLM (Supplementary Fig. 4) as well as Spearman correlation matrix (Supplementary Fig. 6) using \(\epsilon\) = (1.0, 2.0, 2.7)k_{B}T and experiments, compared to \(\epsilon\) = 2.4k_{B}T, further justifies the latter as the optimum value in the CCM. We find it remarkable that the CCM, with only one adjusted energy scale (\(\epsilon\)) is sufficient to produce such a robust agreement with experiments.
Celltocell variations in the WLM
To assess the large structural variations between cells, we calculated the WLM for individual cells. We obtain the singlecell WLM using time averaged distance map of individual trajectories. Figure 5 shows that there are marked differences between the WLM for individual cells, with the ensemble average deviating greatly from the patterns in individual cells. Thus, the chromosome structure is highly heterogeneous. These findings are reflected in the small mean value of Pearson correlation coefficients ρ between all pairs of cells (Fig. 5b). The distribution P(ρ) has mean \(\bar \rho\) = 0.2 with a narrow shape, implying little overlap in the WLMs between any two cells.
In order to make quantitative comparisons to experimental data, with the goal of elucidating largescale variations in the spatial organizations of human interphase chromosomes, we constructed singlecell WLMs for Chr 21 using the spatial distance data provided in ref. ^{6} and computed the corresponding P(ρ) (Fig. 5b). The results show that the experimentally organization of Chr 21 in vivo also exhibits large variations manifested by the distribution P(ρ) covering a narrow range of low values of ρ with a small mean \(\bar \rho\) = 0.25. Comparison to simulated result suggest that Chr 21 shows a slightly lower degree of structural heterogeneity (with a modestly larger mean \(\bar \rho\) = 0.25) compared to Chr 5 investigated using CCM. Nevertheless, both the simulated and experimental results indicate that human interphase chromosomes do not have any welldefined “native structure”. To investigate whether Chr 5 has a small number of spatially distinct structures, we show twodimensional tSNE (tdistributed stochastic neighboring embedding) representation of 90 individual WLMs of the metric \(\sqrt {1  \rho }\) (Fig. 5c). It is clear that there is no dominant cluster, indicating that each Chr 5 in single cells is organized differently rather than belonging to a small subset of conformational states. Such large celltocell variations in the structures, without a small number of welldefined states, is another hallmark of glasses, which are also revealed in recent experiments^{40,42}. The presence of multiple organized structures has profound consequences on the chromosome dynamics (see below).
Chromosome dynamics is glassy
We probe the dynamics of the organized chromosome with \(\epsilon\) = 2.4k_{B}T, a value that yields the best agreement with the experimental HiC contact map. We first calculated the incoherent scattering function, F_{s}(k,t) = (1/N)\(\left\langle {\mathop {\sum}\nolimits_{j = 1}^N {\mathrm{e}}^{{\mathrm{i}}{\mathbf{k}}({\mathbf{r}}_j(t)  {\mathbf{r}}_j(0))}} \right\rangle\) where r_{j}(t) is the position of jth loci at time t. The decay of F_{s}(k,t) (orange line in Fig. 6a) for k ~ 1/r_{s} (r_{s} is the position of the first peak in the radial distribution function (g_{AA}(r) and g_{BB}(r)) (Fig. 2a)) is best fit using the stretched exponential function, \(F_{\mathrm{s}}(k,t) \sim e^{  (t/\tau _\alpha )^\beta }\) with a small stretching coefficient, β ≈ 0.27. The stretched exponential decay with small β is another hallmark of glassy dynamics. For comparison, F_{s}(k,t) decays exponentially for \(\epsilon\) = 1.0k_{B}T, implying liquidlike dynamics (blue line in Fig. 6a).
In the context of relaxation in supercooled liquids, it has been shown that the fourth order susceptibility^{43}, χ_{4}(k,t) = \(N[\langle F_{\mathrm{s}}(k,t)^2\rangle  \langle F_{\mathrm{s}}(k,t)\rangle ^2]\) provides a unique way of distinguishing between fluctuations in the liquid and frozen states. As in structural glasses, the value of χ_{4}(k,t) increases with t reaching a peak at t = t_{M} and decays at longer times. The peak in the χ_{4}(k,t) is an indication of dynamic heterogeneity, which in the chromosome is manifested as dramatic variations in the loci dynamics (see below). For \(\epsilon\) = 2.4k_{B}T, χ_{4}(k,t) reaches a maximum at t_{M} ≈ 1 s (Fig. 6b), which surprisingly, is the same order of magnitude (~5 s) in which chromatin movement was found to be coherent on a length scale of ≈1 μm^{31}. The dynamics in F_{s}(k,t) and χ_{4}(k,t) together show that human interphase chromosome dynamics is glassy^{27}, and highly heterogeneous. F_{s}(k,t) and χ_{4}(k,t) at smaller values of k also show that at longer length scale, chromosome exhibits glassy dynamics (Supplementary Fig. 18c, d).
Single loci mean square displacements are heterogeneous
In order to ascertain the consequences of glassy dynamics at the microscopic level, we plot the MSD, Δ(t) = \(\frac{1}{N}\left\langle {\mathop {\sum}\nolimits_{j = 1}^N (({\mathbf{r}}_j(t)  {\mathbf{r}}_{{\mathrm{com}}}(t))  ({\mathbf{r}}_j(0)  {\mathbf{r}}_{{\mathrm{com}}}(0)))^2} \right\rangle\) in Fig. 7 where r_{com} is the position of center of mass of the whole chromosome, from which a few conclusions can be drawn.

1.
Because of the polymeric nature of the chromosome, the maximum excursion in \(\Delta (t \to \infty ) = 2R_{\mathrm{g}}^2\), where R_{g} ≈ 0.7 μm is the radius of gyration of Chr 5. Consequently, for both \(\epsilon\) = 1.0k_{B}T and \(\epsilon\) = 2.4k_{B}T, Δ(t) in the long time limit is smaller than \(2R_{\mathrm{g}}^2\) (Fig.7a). For \(\epsilon\) = 2.4k_{B}T, Δ(t) shows a crossover at t ≈ 10^{−2} s from slow to a faster diffusion, another indicator of glassy dynamics^{44}. The slow diffusion is due to caging by neighboring loci, which is similar to what is typically observed in glasses. The plateau in Δ(t) (Fig. 7a) is not pronounced, suggesting that the compact chromosome is likely on the edge of glassiness. The crossover is more prominent in the time dependence of the mean squared displacement of single loci (see below). The slow diffusion predicted from the CCM is in accord with a number of experiments (Supplementary Fig. 21). In contrast, diffusion coefficients measured in experiments are one or two orders of magnitude smaller than the system exhibiting liquidlike behavior, which further supports the glassy dynamics for mammalian chromosomes predicted here.

2.
The two dashed lines in Fig. 7a show Δ(t) ~ t^{α} with α = 0.45. The value of α is close to 0.4 for the condensed polymer, which can be understood using the following arguments. The total friction coefficient experienced by the whole chain is the sum of contributions from each of the N monomers, ξ_{T} = Nξ. The time for the chain to move a distance ≈R_{g} is \(\tau _{\mathrm{R}} = R_{\mathrm{g}}^2/D_{\mathrm{R}} \sim N^{2{\nu} + 1}\). Let us assume that the diffusion of each monomer scales as Dt^{α}. If each monomer moves a distance on the order of R_{g} then the chain as a whole will diffuse by R_{g}. Thus, by equating \(D\tau _{\mathrm{R}}^{\alpha} \sim R_{\mathrm{g}}^2\), we get α = 2ν/(2ν + 1). For an ideal chain ν = 0.5, which recovers the prediction by Rouse model, α = 0.5. For a selfavoiding chain, ν ≈ 0.6, we get α ≈ 0.54. For a condensed chain, ν = 1/3, we get α = 0.4, thus rationalizing the findings in the simulations. Similar arguments have been reported recently for dynamics associated with fractal globule^{28} and for the β^{−}polymer model^{29}. Surprisingly, α = 0.45 found in simulations is in good agreement with recent experimental findings^{32}. We also obtained a similar result using a different chromosome model^{45}, when the dynamics were examined on a longer length scale. The finding that there is no clear Rouse regime (α = 0.5) is also consistent with several other experimental results (Supplementary Fig. 21). We should note that distinguishing between the difference, 0.4 and 0.5, in the diffusion exponent is subtle. Additional experiments are needed to determine the accurate values of the diffusion exponents of Human interphase chromatin loci in different time regimes.

3.
We also calculated the diffusion of a single locus (sMSD) defined as Δ_{i}(t) = \(\left\langle {({\mathbf{r}}_i(t_0 + t)  {\mathbf{r}}_i(t_0))^2} \right\rangle _{t_0}\), where \(\langle \cdot \rangle _{t_0}\) is the average over the initial time t_{0}. Distinct differences are found between the polymer exhibiting liquidlike and glassylike dynamics. The variance in single loci MSD is large for \(\epsilon\) = 2.4k_{B}T, illustrated in Fig. 7b, which shows 10 typical trajectories for \(\epsilon\) = 1.0k_{B}T and \(\epsilon\) = 2.4k_{B}T each. For glassy dynamics, we found that the loci exhibiting high and low mobilities coexist in the chromosome, with orders of magnitude difference in the values of the effective diffusion coefficients, obtained by fitting Δ_{i}(t) = \(D_\alpha t^{\alpha _i}\). Caging effects are also evident on the time scale as long as seconds. Some loci are found to exhibit caginghopping diffusion, which is a hallmark in glassforming systems^{46,47}. Interestingly, such caginghopping process has been observed in human cell some time ago^{48}.

4.
The large variance in sMSD has been found in the motion of chromatin loci in both E.coli and human cells^{49,50,51,52,53}. To further quantify heterogeneities in the loci mobilities, we calculated the Van Hove function P(Δx), P(ΔxΔt) = \(\left\langle {(1/N)\mathop {\sum}\nolimits_{i = 1}^N \delta (\Delta x  [x_i(\Delta t)  x_i(0)])} \right\rangle\). Figure 7c and d shows the P(ΔxΔt) and normalized P(Δx/σΔt) for \(\epsilon\) = 2.4k_{B}T at different lag times Δt. For \(\epsilon\) = 1.0k_{B}T, Van Hove function is well fit by a Gaussian at different lag times Δt (Supplementary Fig. 14). In contrast, for chromosome with glassy dynamics, all the P(ΔxΔt) exhibit fat tail, which is well fit by an exponential function at large values of Δx (Fig. 7c, d) at all δt values, suggestive of the existence of fast and slow loci^{47}.

5.
The results in Fig. 7 allow us to make direct comparisons with experimental data to establish signatures of dynamic heterogeneity. We calculated the distribution of effective diffusion exponent α_{i}, P(α), where α_{i} is obtained by fitting the sMSD to \(\sim t^{\alpha _i}\) within some lag time (Δt) range. Figure 7e shows that P(α) calculated from simulations is in good agreement with experiments^{54} in the same lag time range (0.42 s < Δt < 10 s). The P(α) distribution in the range 10^{−6} s < Δt < 0.42 s shows two prominent peaks, further validating the picture of coexisting fast and slow moving loci. The good agreement between the predictions of the CCM simulations with data, showing large variations of mobilities among individual loci in vivo, further supports our conclusion that organized chromosome dynamics is glassy. Interestingly, a recent computational study in which Human interphase chromosomes are modeled as a generalized Rouse chain suggests that the heterogeneity of the loci dynamics measured in live cell imaging is due to the large variation of crosslinking sites from celltocell^{24}. Our model implies a different mechanism that the heterogeneity observed is a manifesto of the intrinsic glassy dynamics of chromosomes.
Active loci has higher mobility
Figure 8a shows MSD for active and repressive loci. For \(\epsilon\) = 1.0k_{B}T, there is no difference between active and repressive loci in their mobilities. However, in the glassy state active loci diffuses faster than the repressive loci. The ratio between the effective diffusion coefficients (the slope of the dashed line) of the active and repressive loci is 0.0116/0.008 \(\simeq\) 1.45, in good agreement with experimental estimate 0.018/0.013 \(\simeq\) 1.38^{32}. Such a difference in their mobilities is also confirmed by F_{s}(k,t) and χ_{4}(t) (Supplementary Note 13 and Supplementary Fig. 18a, b). These variations are surprising since the parameters characterizing the A–A and B–B interactions are identical. To investigate the origin of the differences between the dynamics of A and B loci, we plot the displacement vectors of the loci across the crosssection of the condensed chromosome (Fig. 8b) for a time window Δt = 0.1 s. The loci on the periphery have much greater mobility compared to the ones in the interior. In sharp contrast, the fluidlike state exhibits no such difference in the mobilities of A and B (Fig. 8d). To quantify the dependence of the mobility on the radial position of the loci, we computed the amplitude of the displacement normalized by its mean, as a function of the radial position of the loci, r (Fig. 8c). For the chromosome exhibiting glasslike behavior, the mobility increases sharply around r ≈ 0.7 μm whereas it hardly changes over the entire range of r in the fluidlike system. Because the active loci are mostly localized on the periphery and the repressive loci are in the interior (Fig. 2b), the results in Fig. 8 suggest that the differences in the mobilities of the loci with different epigenetic states are due to their preferred locations in the chromosome. It is intriguing that glassy behavior is accompanied by a positiondependent mobility, which can be understood by noting that the loci in the interior are more caged by the neighbors, thus restricting their movement. In a fluidlike system, the cages are so shortlived that the apparent differences in the environments the loci experience are averaged out on short timescales. Note that in the experimental result^{32} comparison is made between the loci in the periphery and interior of the nucleus. It is well known that the nucleus periphery is enriched with heterochromatin (repressive loci) and the interior is enriched with euchromatin (active loci). However, for an individual chromosome, singlecell HiC study^{40} and other experimental studies^{55,56,57,58} suggest that the active loci are preferentially localized at the surface of the chromosome territory.
Discussion
In order to demonstrate the transferability of the CCM, we simulated Chr 10 using exactly the same parameters as for Chr 5 (Supplementary Note 12). Supplementary Fig. 16 compares the WLM obtained from simulations for different \(\epsilon\) values and the computed WLM using the HiC contact map. The contact map is translated to the distance R_{ij} by assuming that \(P_{ij} \propto R_{ij}^{  4.1}\) holds for Chr 10 as well. It is evident that the CCM nearly quantitatively reproduces the spatial organization of Chr 10 (Supplementary Fig. 16). Thus, it appears that the CCM could be used for simulating the structures and dynamics of other chromosomes as well.
Two scaling regimes in P(s) is suggestive of scaledependent folding of genome. In order to reveal how chromosome organizes itself and to link these processes to the experimentally measurable P(s), we calculated the timedependent change in P(s) as a function of t. At scales less (above) than s* ≈ 5×10^{5}bps, P(s) decreases (increases) as the chromosome becomes compact. The P(s) ~ s^{−0.75} scaling for s < s* (see also Fig. 1b) is the result of organization on the small genomic scale during the early stage of chromosome condensation (Fig. 9a). In the initial stages, compaction starts by forming ≈s* sized chromosome droplets (CDs) as illustrated in Fig. 9a. In the second scaling regime, P(s) ~ s^{−1.25}, global organization occurs by coalescence of the CDs (Fig. 9a). Thus, our CCM model, which suggests a hierarchical chromosome organization on two distinct scales, also explains the two scaling in P(s).
The hierarchical nature of the structural organization is further illustrated using A(s), the number of contacts that a ssized subchain forms with the rest of the chromosome. For a compact structure, A(s) ~ s^{2/3} and A(s) ~ s for an ideal chain. Supplementary Fig. 17 shows that A(s) computed using the HiC data (black square line) varies as s^{2/3}, suggesting that chromosome is compact on all length scales. We also find that upon increasing ε, the range of A(s) ~ s^{2/3} expands. The pictorial view of chromosome organization (Fig. 9a) and the A(s) scaling show that chromosome structuring occurs hierarchically with the formation of CDs and subsequent growth of the large CDs at the expense of smaller ones. We quantitatively monitored the growth of CDs during the condensation process and found that the size of CD grows linearly with time during the intermediate stage (Fig. 9b). Such a condensation process is reminiscent of the Lifshitz–Slazov mechanism^{59} used to describe Ostwald ripening.
Our simulations show that the average TAD size and the crossover scale (s*) the dependence of P(s) on s coincide. In addition, the size of the CDs is also on the order of s*, which is nearly the same for all the chromosomes (Fig. 1c). We believe that this is a major result. The coincidence of these scales suggests that both from the structural and dynamical perspective, chromosome organization takes place by formation of TADs, which subsequently arrange to form structures on larger length scales. Because gene regulation is likely controlled by the TADs, it makes sense that they are highly dynamic. We hasten to add that the casual connection between TAD size and s* as well as the CDs size has to be studied further. If this picture is correct then chromosome organization, at length scales exceeding about 100 kbps, may be easy to describe.
In summary, we developed the CCM, a selfavoiding polymer with two epigenetic states and with fixed loop anchors whose locations are obtained from experiment to describe chromosome dynamics. The use of rigorous clustering techniques allowed us to demonstrate that the CCM nearly quantitatively reproduces HiC contact maps, and the spatial organization gleaned from superresolution imaging experiments. It should be borne in mind that contact maps are probabilistic matrices that are a low dimensional representation of the threedimensional organization of genomes. Consequently, many distinct copolymer models are likely to reproduce the probability maps encoded in the HiC data. In other words, solving the inverse problem of going from contact maps to an energy function is not unique^{9}.
Chromosome dynamics is glassy, with correlated dynamics on scale ≈ 1 μm, implying that the free energy landscape has multiple equivalent minima. Consequently, it is likely that in genomes only the probability of realizing these minima is meaningful, which is the case in structural glasses. The presence of multiple minima also leads to celltocell heterogeneity with each cell exploring different local minimum in the free energy landscape. We speculate that the glasslike landscape might also be beneficial in chromosome functions because only a region on size ~s* needs to be accessed to carry out a specific function, which minimizes largescale structural fluctuations. In this sense, chromosome glassiness provides a balance between genomic conformational stability and mobility.
Methods
Construction of the CCM
Contact maps^{4,5} of interphase chromosomes show that they are partitioned into genomewide compartments, displaying plaid (checkerboard) patterns. If two loci belong to the same compartment they have the higher probability to be in contact than if they are in different compartments. Although finer classifications are possible, compartments^{4} can be categorized broadly into two (open (A) and closed (B)) classes associated with distinct histone markers. Open compartment is enriched with transcriptional activityrelated histone markers, such as H3K36me3, whereas the closed compartment is enriched with repressive histone markers, such as H3K9me3. Chromatin segments with repressive histone markers have effective attractive interactions, which models HP1 proteinregulated interactions between heterochromatin regions^{60,61}. We assume that chromatin fiber, with active histone markers, also has such a similar attraction. From these considerations, it follows that the minimal model for human chromosome should be a copolymer where the two types of monomers represent active and repressive chromatin states. To account for the two states, we introduce the CCM as a selfavoiding polymer with two kinds of beads. Similar genre of models have been proposed in several recent studies^{18,19,20,21,26} to successfully decipher the organization of genomes.
The energy function in the CCM is, E = U_{C} + U_{LJ}, where U_{C} contains bond potential (U^{S}) and loop interaction (U^{L}), and U_{LJ} is the Lennard–Jones pairwise interaction between the monomers (see Supplementary Note 1 for details). If two monomers belong to type A (B), the interaction strength is \(\epsilon _{{\mathrm{AA}}}\)\(\left( {\epsilon _{{\mathrm{BB}}}} \right)\). The interaction strength between A and B is \(\epsilon _{{\mathrm{AB}}}\). Each monomer represents 1200 base pairs (bps), with six nucleosomes connected by six linker DNA segments. The size of each monomer, σ, is estimated by considering two limiting cases. If we assume that nucleosomes are compact then the value of σ may be obtained by equating the volume of the monomer to 6v where v is the volume of a single nucleosome. This leads to σ ≈ 6^{1/3}R_{N} ≈ 20 nm where R_{N} ≈ 10 nm is the size of each nucleosome^{62}. Another limiting case may be considered by treating the six nucleosome array as a wormlike chain. The persistence length of the chromatin fiber is estimated to be ~1000 bps^{63}, which is about the size of one monomer. The mean endtoend distance of a wormlike chain whose persistence length is comparable to the contour length L is \(R \approx L\sqrt {2/e}\). The value of L for a six nucleosome array is 6(16.5 + R_{N})nm where the length of a single linker DNA is 16.5 nm. This gives us the upper bound of σ to be 130 nm. Thus, the two limiting values of σ are 20 nm and 130 nm. We assume that the value of σ is an approximate mean, yielding σ = 70 nm.
The type of monomer is determined using the Broad ChromHMM track^{64}. There are totally 15 chromatin states, out of which the first eleven are related to gene activity. Thus, we consider state 1 to state 11 as a single active state (A) and states 12–15 as a single repressive state (B). For the genome range 146–158 Mbps in the Chromosome 5 in Human GM12878 cell, which is investigated mainly in this work, the numbers of active and repressive loci are 2369 and 7631, respectively. The positions of the loop anchors in CCM are determined from HiC experiment^{5}. Details of the assignment of monomer type and loop anchors are given in the Supplementary Note 2.
Simulations
As detailed in the Supplementary Notes 3 and 4, we performed simulations using both Langevin dynamics (low friction) and Brownian dynamics (high friction) using a custommodified version of the molecular dynamics package LAMMPS. The use of Langevin dynamics accelerates the sampling of the conformational space^{65}, needed for reliable computation of static properties. Realistic value of the friction coefficient is used in Brownian dynamics simulations to investigate chromosome dynamics, thus allowing us to make direct comparisons with experiments.
We varied the values of \(\epsilon _{{\mathrm{AA}}}\), \(\epsilon _{{\mathrm{BB}}}\) and \(\epsilon _{{\mathrm{AB}}}\) to investigate the effect of interaction strength on the simulation results. For simplicity, we set \(\epsilon _{{\mathrm{AA}}} = \epsilon _{{\mathrm{BB}}} \equiv \epsilon\). By fixing the ratio \(\epsilon {\mathrm{/}}\epsilon _{{\mathrm{AB}}}\) to 11/9, \(\epsilon\) is the only relevant energy scale in the CCM. The results presented in the main text are obtained with \(\epsilon\) = 2.4k_{B}T unless stated otherwise. The contacts between loci in the simulation are determined by the threshold distance r_{c} = 2σ where σ = 70 nm. We should note that by independently tuning three parameters \(\epsilon _{{\mathrm{AA}}}\), \(\epsilon _{{\mathrm{BB}}}\), and \(\epsilon _{{\mathrm{AB}}}\) separately, the model could be further optimized in the comparison with experiment HiC data. For the simplicity and because the errors in HiC data are hard to quantify, we make the assumption that \(\epsilon _{{\mathrm{AA}}} = \epsilon _{{\mathrm{BB}}} \equiv \epsilon\) and fix the ratio \(\epsilon _{{\mathrm{AB}}}{\mathrm{/}}\epsilon\), resulting in only one free parameter, either \(\epsilon _{{\mathrm{AB}}}\) or \(\epsilon\). In practice, we varied \(\epsilon\) while keeping the ratio \(\epsilon _{{\mathrm{AB}}}{\mathrm{/}}\epsilon\) a constant. Our results suggest that even with this simplification, the results based on the CCM produces near quantitative agreement with HiC data.
The folding of chromatin is simulated starting from extended conformations (Supplementary Note 4). Due to the slow relaxation process, theoretically predicted in a previous study^{27}, and topological constraints^{14}, long polymers such as human interphase chromosomes are unlikely to come to equilibrium even on the time scale of a singlecell cycle. Thus, the initial conformations could in principle affect the organization of genomes. Although the folding from an extended conformation is unlikely to occur for chromosome as a whole in vivo, we believe that the folding process investigated in this work provides insights into gene activation because it involves only local folding or unfolding^{66,67,68}.
Data availability
The data that support the findings of this study are available from the authors upon reasonable request.
Code availability
The custom LAMMPS code used in this study for BD simulations are provided in the public GitHub repository https://github.com/anyuzx/Lammps_brownian.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
We want to thank Abdul N MalmiKakkada and Xin Li for discussions and comments on the manuscript. We are grateful to the National Science Foundation (CHE 1636424) and the CollieWelch Regents Chair (F0019) for supporting this work.
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Affiliations
Biophysics Program, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA
 Guang Shi
Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea
 Lei Liu
 & Changbong Hyeon
Department of Chemistry, University of Texas at Austin, Austin, TX, 78712, USA
 D. Thirumalai
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Contributions
G. S. and D. T. designed research, G.S., C.H. and D.T. performed research, G.S., L.L., and D.T. analyzed data, G.S., C.H., and D.T. wrote the paper.
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The authors declare no competing interests.
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Correspondence to D. Thirumalai.
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