Polymer Model of Loop Extrusion.

To investigate the interplay of loop extrusion with compartmentalization, we simulate the chromatin fiber as a polymer subject to loop extrusion and compartmental segregation (Fig. 1). LEFs can attach to the chromatin polymer at random positions and extrude loops bidirectionally until they either fall off, bump into each other, or encounter an extrusion barrier. When blocked on one side, they continue extruding unidirectionally. LEFs are characterized by three parameters: the average residence time τ, the single-sided extrusion velocity v, and the average separation d (48). The first two define the processivity as λ = 2τv, which is the average size of a loop extruded by an unobstructed LEF. Simulation parameters are given in Table 1. For WT, we use λ/d = 1/3; cells thus operate in the dilute regime (λ/d < 1), where LEFs rarely bump into each other. CTCF-enriched boundaries of TADs are modeled by barriers that block extrusion of LEFs with probability 90% in WT (16). Having a finite permeability is consistent with the turnover time of CTCF being considerably shorter than that of cohesin [≈1–2 min (49, 50) vs. >5–30 min (12, 50⇓⇓–53)], although the exact value of the permeability may vary across the genome and may depend on the number and occupancy of CTCF sites, cofactors, and details of interactions between CTCF and cohesin. Values of these and other parameters are chosen to reproduce TAD patterns observed in Hi-C data and are systematically varied to examine their effects on chromatin organization. Positions of the TAD boundaries are randomly generated based on the above characteristics. They are not intended to reproduce specific genomic regions, since our goal is to demonstrate that a single model can reproduce genome-wide quantities from three different phenotypes (removal of cohesin, CTCF, and WAPL) observed in different organisms (mouse and human). In our simulations, loop extrusion is effective in both compartment types, consistent with the presence of TADs in experimental Hi-C in both A and B regions. Unless otherwise mentioned, we allow for some passing of two parts of the chromatin fiber through each other by imposing a finite repulsive core on the monomer interaction potential (SI Appendix, Fig. S1). This represents the effect of topoisomerase II and is discussed further below.

Compartmental Segregation by Phase Separation.

Compartment organization is modeled by a block copolymer composed of A and B blocks that have the same local properties (monomer size and fiber flexibility) but interact differently. Positions of A and B blocks are randomly generated with sizes of blocks chosen to yield an autocorrelation length of the compartment profile inferred from experimental Hi-C data (SI Appendix). The spatial segregation of A- and B-type chromatin is induced by a weak B–B attraction, which we refer to as compartmental interaction. It is parametrized by E_attr, the minimum value of the monomer interaction (SI Appendix, Fig. S1A), but can also be modeled differently (SI Appendix, Fig. S1E). This is sufficient to induce compartmental segregation in the absence of anchoring to the lamina (54). We choose the interaction parameter E_attr = 0.12 k_BT to achieve a similar degree of compartmentalization (see below) in experiments and simulations. We point out that this attraction is far too weak to turn B regions into a collapsed polymer state: The densities in the A-rich and B-rich phase differ by only about 10% (SI Appendix, Fig. S1D). Taken together, within our model, heterochromatin is phase separated from euchromatin, but not collapsed (Discussion).

To quantify the degree of phase separation, we examine the local densities of A and B monomers in small boxes and compute the normalized difference of A and B particles per box: (n_A − n_B)/(n_A + n_B) (histograms in Fig. 1C). As we increase E_attr, the histograms become bimodal, which demonstrates the emergence of an A-rich and a B-rich phase. As an order parameter, we compute the mean absolute value of the normalized number difference, N = <|(n_A − n_B)/(n_A + n_B)|> (SI Appendix, Fig. S1C), which shows the microphase separation characteristic of block copolymers. The phase separation is reduced by the presence of loop extrusion, which we will explore in detail throughout the paper.

The degree of compartmentalization can also be computed from contact frequencies for both simulated and experimental Hi-C maps as the normalized contact frequency difference between same-type contacts, AA and BB, and different-type contacts AB, namely COMP = (AA + BB − AB)/(AA + BB + AB). We point out that our compartmentalization score measures the checkerboard contrast of a contact map and is by construction independent of the contact probability scaling P(s) (SI Appendix). For simulated data, the compartmental identities of all loci are known, while for experimental data they need to be inferred from the Hi-C maps. To do so, we compute compartment profiles from eigenvector decomposition of the Hi-C maps (2, 3) and assign compartmental segments of type A/B to intervals with positive/negative compartment profile. Note that a locus of a given type may not be able colocalize with other loci of the same type. Compartmental segments assigned from Hi-C maps may thus differ from the underlying A/B types of the loci (SI Appendix).

Loop Extrusion Overrides Compartmentalization on Small Scales.

Our central finding is that the active process of loop extrusion counteracts compartmental segregation. We determine this from three different experimental datasets where the loop extrusion machinery was altered in different ways, and from our corresponding polymer simulations. First, we test whether our integrated model can explain the effects of depleting chromatin-associated cohesin (11), namely, disappearance of TADs and simultaneous changes in compartmentalization such as (i) compartmental segments that span several megabases to several tens of megabases appear more crisp in Hi-C, and (ii) they become fragmented into smaller segments (Fig. 2A, Left). Strikingly, loss of loop extrusion in our model reproduces both phenomena (Fig. 2A, Right): While TADs disappear, compartmentalization, in particular of small segments, is enhanced, leading to fragmentation of large compartmental segments. Our simulations thus show that loop extrusion suppresses the inherent compartmentalization by counteracting segregation of small segments, which emerges when loop extrusion is removed.

We quantify changes in simulated chromatin upon loss of loop extrusion and compare them to changes in experimental data from ref. 11 in three ways (Fig. 2A, Lower graphs). (i) The removal of loop extrusion is detected by changes in the contact frequency as a function of genomic distance, P(s): With loop extrusion, the P(s) curve shows a characteristic hump on the length scale of TADs. This hump disappears upon removal of loop extrusion both in experiments and simulations. (ii) The strengthening of short compartmental segments (“fragmentation” of compartments) upon loss of loop extrusion is quantified by the steeper decay of the autocorrelation of the compartment profile. This steepening is evident in simulations and experiments alike. (iii) The greater contrast in Hi-C maps upon removal of loop extrusion is measured by changes in the degree of compartmentalization (see above and SI Appendix). Its increase in simulations is slightly stronger than in experiments, which could indicate that some compartment mixing remains present in experiments, either by residual cohesin (SI Appendix, Fig. S2) or some other processes in the nucleus not considered here (note in SI Appendix).

Most importantly, our simulations show that loop extrusion suppresses small compartmental segments more than large ones. We study this in detail with simulations of uniformly sized compartmental segments (Fig. 3). For small segment lengths (<320 kb, i.e., <128 monomers), we observe little compartmental segregation even in the absence of loop extrusion, while larger segments show a clear phase separation, with transition at about 500 kb. The phenomenon of a length-dependent transition is known from the physics of block copolymers: It occurs when the product of segment length and compartmental interaction parameter exceeds a critical value (40). Here, we find that adding loop extrusion shifts this phase transition to larger segments of ≈1 Mb (Fig. 3D, Middle). For example, 800-kb segments are segregated in the absence of loop extrusion but get largely mixed in its presence (Fig. 3 A–C, second panels from Left). For even larger segment lengths, loop extrusion has diminishing effects. For example, for 2-MB segments segregation is only slightly reduced by loop extrusion (Fig. 3 A–C, third panels from Left). We attribute this size-dependent impact of loop extrusion on compartmentalization to two effects: (i) Segregation of shorter compartmental segments are weaker to begin with, and can be easily perturbed by active mixing due to loop extrusion; (ii) LEFs cannot mix segments that considerably exceed their average processivity (here, λ = 250 kb). To summarize, the impact of loop extrusion on segregation, computed as the ratio of compartmentalization measures without and with LEFs (Fig. 3D, Right), is most pronounced for compartments of 500 kb to 2 Mb since very small compartmental segments (<320 kb) do not segregate even without loop extrusion, while large compartments (>2 Mb) remain largely unaffected.

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Fig. 3.

Impact of loop extrusion on compartments of different size. (A) Contact frequency maps without/with loop extrusion (upper/lower triangles). The lengths of A/B segments are indicated above the maps. (B and C) Example conformations of 50-Mb fibers without/with loop extrusion, in periodic boundary conditions (see SI Appendix, Fig. S10 for a comparison with spherical confinement). The approximate segment length where phase separation (PS) occurs is indicated in gray. (D) The degree of phase separation as a function of segment length is measured from contact frequency maps (COMP) and from spatial configurations (N; see text for details). The impact of loop extrusion on compartmentalization (Right) is measured by dividing each of the above order parameters in the absence of loop extrusion by their value with loop extrusion. The impact is maximal for segment lengths that exceed the segregation transition but not the mixing length, which is of the order of several LEF processivities (∼1 Mb).

These results are in very good agreement with recent cohesin depletion experiments that reveal finer-scale compartmentalization: Small compartmental segments are suppressed by loop extrusion in WT cells and emerge in the mutant without loop extrusion; large compartments are present in both cases but may be diminished by loop extrusion. Our simulations suggest that the emergent fine structure is the intrinsic compartmentalization that is overridden in WT cells by loop extrusion by cohesin. This is in line with the observation that epigenetic marks correlate better with finer emergent than with the coarser WT compartmentalization (11). Taken together, our results suggest that loop extrusion suppresses the inherent compartmental segregation on the length scale of several cohesin processivities and leaves only larger-scale compartmentalization visible. When loop extrusion is removed by depletion of chromatin-associated cohesin, the intrinsic compartmental segregation emerges.

Removing Loop Extrusion Barriers Suppresses TADs but Not Compartments.

Next, we asked whether our model of loop extrusion and compartmental segregation is compatible with depletion experiments of the TAD boundary element CTCF (37). Namely, CTCF depletion leads to a loss of TAD boundaries while having little effect on compartmentalization (Fig. 2B, Left). We simulated CTCF depletion by removing extrusion barriers, which led to a 1.2-fold increase in loop size (from 173 to 216 kb). In agreement with experiments, we observe a loss of TADs, while compartmentalization is mostly unaffected (Fig. 2B, Right; see SI Appendix, Fig. S3 for a parameter sweep). Unlike in cohesin depletion, no fine compartmentalization emerges. The distinction from cohesin depletion arises because upon CTCF removal loop extrusion is still present, but not restricted to specific domains.

Although TADs are diminished upon both CTCF removal and cohesin loss, these two perturbations have vastly different effects on chromatin organization. The lack of changes in P(s) curves upon CTCF removal suggest that local chromatin compaction by loop extrusion is unaffected, as evident from the hump for s < 1 Mb in the P(s) curve. Our simulations reproduce this phenomenon: The loss of extrusion barriers while maintaining loop extrusion removes TADs but preserves P(s). Loss of chromatin-associated cohesin in experiments, on the contrary, leads to the reduced compaction as evident by the loss of the hump in the P(s) curve. Simulations with diminished loop extrusion activity reproduced these changes (see above and Fig. 2A). Corresponding changes in compartmentalization upon cohesin loss and the lack of such changes upon CTCF removal suggest that it is the loop extrusion activity of cohesin that led to coarsening of compartmentalization in the WT.

Increased Loop Extrusion Activity Suppresses Compartments and Enhances TADs.

Finally, we consider how increased processivity and amount of cohesin due to depletion of the cohesin unloading factor WAPL can affect compartmentalization. Hi-C data for WAPL-depleted cells show weaker compartmentalization and a strengthening of large TADs and corner peaks (Fig. 2C, Left) (12, 14). To determine simulation parameters for WAPL depletion, we note that in experiments the amount of chromatin-associated cohesin in WAPL-depleted cells increases moderately (≈1.5- to 2-fold), while the residence time increased considerably (>5-fold) (12, 14). We thus increased the LEF density 1.5-fold (reducing the average separation from 750 to 500 kb) and the residence time 10-fold, which results in larger processivity (2.5 Mb instead of 250 kb). The average loop size increased only 2.6-fold (from 173 to 449 kb), as expected theoretically (48), indicating that extrusion becomes limited by LEFs bumping into each other. In agreement with experiments, this leads to TADs with more pronounced corner peaks (Fig. 2C, Right). Corner peaks between nonadjacent TAD boundaries are particularly enhanced. We point out that such secondary corner peaks do not per se imply that extruded loops extend beyond TAD boundaries. In simulations, such secondary peaks can emerge either by LEFs crossing a permeable barrier at the TAD boundary, or from nonadjacent extrusion barriers being brought into spatial proximity when LEFs extrude most of the intervening fiber in each TAD (Fig. 2C and SI Appendix, Fig. S3). To what extent actual loop extrusion enzymes cross TAD boundaries will be an interesting question for future experiments. The change in the contact probability P(s) in our simulations is also consistent with changes in experimental P(s) curves (Fig. 2C, Lower), which show an extension of the characteristic hump to larger genomic separations, reflecting larger extruded loops.

Also in agreement with experiments, our simulations of WAPL depletion show reduced compartmentalization (Fig. 2C). We attribute this to increased compartment mixing by the increased number of LEFs and increased loop length. Further suppression of compartments in WAPL-depleted cells might be due to formation of axially compressed and stiff “vermicelli” chromosomes (52), which can limit far-cis contacts and interactions with the lamina, thus affecting compartmentalization.

The Nonequilibrium Nature of Loop Extrusion Is Central to Compartment Mixing and TAD Strength.

We have shown above that compartment mixing by loop extrusion explains the changes of TADs and compartmentalization for all considered experimental perturbations. We thus aim at understanding physical mechanisms behind this mixing effect.

The active process of loop extrusion can bring loci into contact irrespective of their compartmental identity and thereby counteract the phase separation structure from compartmental interaction. We thus asked whether the reduced compartmentalization due to loop extrusion can be simply understood by an effective reduction of the compartmental interaction. To test this, we run simulations without loops, but instead we lowered the B–B attraction until the degree of compartmentalization fell below the value achieved by adding loop extrusion (SI Appendix, Fig. S4). We find, however, that Hi-C maps (SI Appendix, Fig. S4A) and the compartment profile autocorrelation (SI Appendix, Fig. S4E) behave differently. Indeed, for reduced B–B attraction, we see little evidence of compartment coarsening, that is, loss of shorter compartment regions, as the autocorrelation barely changed. We thus conclude that the impact loop extrusion on compartmentalization cannot be described by a reduced compartmental interaction.

Next, we asked whether the active, nonequilibrium nature of loop extrusion is essential for its interference with compartmentalization. The process of loop extrusion (i.e., loops are born, grow, and then are released when the LEF dissociates) can interfere with compartmentalization in two distinct ways: (i) by the mere presence of loops that compact chromatin and connect loci irrespective of their compartmental identity, and (ii) by the active nature of loop extrusion that can increase compartment mixing because loci need some time to resegregate after being brought into contact by active loop extrusion.

To examine the relative contributions of these factors, we compare dynamically growing loops and static loops. We choose an ensemble of static loops from simulations with loop extrusion, but now loops remain static while the chromatin fiber is subject to thermal motion (see SI Appendix for details). We find that TADs are still visible in the corresponding Hi-C maps (Fig. 4A) and TAD averages (Fig. 4B), albeit weaker. Surprisingly, the degree of compartmentalization for static loops is almost as strong as when loops are completely absent (Fig. 4C). Also, the compartment profile autocorrelation for static loops resembles that without loops (SI Appendix, Fig. S5E). To generalize the dichotomy of static vs. dynamic loops, we varied the loop extrusion speed while keeping the thermal motion of the fiber constant (Fig. 4 A–C and SI Appendix, Fig. S5). We found that compartmentalization progressively decreases for faster LEFs (Fig. 4C). This suggests that the nonequilibrium nature of loop extrusion is central to its role in compartment mixing: After being zipped together by the passage of an LEF, the fiber has less time to resegregate before passage of the next LEF when LEFs dynamics are fast. Furthermore, we found that the speed of extrusion also affects TAD strength: Fast LEFs lead to stronger TADs, while static loops lead to weak TADs (Fig. 4 A–C; see SI Appendix for definition of TAD strength). The impact of LEF speed on TAD strength is particularly apparent when averaging contact maps over many TADs (Fig. 4B; see SI Appendix, Fig. S5 for TAD size dependence). This, again, reflects the nonequilibrium nature of loop extrusion: TADs are more compact and thus more pronounced in Hi-C data if LEF dynamics are accelerated relative to polymer diffusion dynamics (see SI Appendix, Fig. S6 for a cartoon explanation). Taken together, our results suggest that static loops contribute little to the observed compartment mixing and to TAD strength, indicating that the nonequilibrium nature of active loop extrusion is central both to interference with compartmentalization and establishment of pronounced TADs.

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Fig. 4.

The nonequilibrium nature of loop extrusion. (A–C) Effects of the speed of loop extrusion relative to thermal polymer dynamics. (A) Contact frequency maps. (B) Averages of TADs of sizes 675–825 kb rescaled to fixed size (see SI Appendix, Fig. S5 for other TAD sizes). (C) Strength of compartmentalization and of TADs score as a function of LEF speed: Compartmentalization decreases while TAD strength increases from no loops over static loops to extruding loops of increasing speed (the polymer dynamics due to thermal motion are kept constant). (D) Importance of chain passing: The impact of loop extrusion on compartmentalization, measured by the ratio of compartmentalization strength without/with loop extrusion, increases for reduced topoisomerase II activity, that is, reduced chain passing (implemented by increasing E_rep, the repulsive part of the monomer interaction potential; SI Appendix, Fig. S1). (E) Length scales relevant for equilibration of a loop: radius of gyration of an extruded loop R_g and diffusional displacement during loop growth. (F) R_g follows equilibrium theory (gray) for static loops, while with increasing LEF speed loops are more compact. R_g is measured in units of one monomer diameter of ≈50 nm. (G) The root-mean-square displacement of chromatin with/without loop extrusion differs on the LEF residence timescale, but not globally, indicating that loop extrusion cannot be described as an elevated effective temperature.

It is important to emphasize that when changing LEF speed in simulations we made sure to maintain other macroscopic characteristics such as processivity, loop sizes, and the distribution of genomic LEF positions: We only altered LEF dynamics relative to thermal polymer diffusion (this applies in particular to our static loops, which are found at all positions within TADs and should not be confused with hypothetical loops connecting only TAD borders). The important effect of increasing LEF speed is that the fiber has less time to equilibrate by thermal motion between passages of LEFs and is thereby kept further from thermodynamic equilibrium. Our finding that TAD strength and compartment mixing depend on LEF speed is thus a direct consequence of the nonequilibrium nature loop extrusion.

The nonequilibrium effect of active loop extrusion can be further strengthened by topological effects such as entrapment of the fiber in the dense network of chromatin surrounding it (55, 56). It is well known that the amount of chain passing, which is enabled by topoisomerase II activity in the cell nucleus, has a great influence on relaxation times of polymer systems (39, 40). We thus alter the stringency of such topological constraints by changing the energy barrier for chain passing, that is, the repulsive core of the monomer interaction potential E_rep. We find that more stringent topological constraints reduce compartmentalization (SI Appendix, Fig. S7) and that the impact of loop extrusion on compartmentalization increases (Fig. 4D). Thus, our findings suggest that loop extrusion keeps chromatin far from equilibrium, with topological constraints reinforcing this effect.

The nonequilibrium nature of loop extrusion not only leads to compartmental mixing but also directly affects other characteristics of the chromatin fiber that can potentially be addressed experimentally. In particular, we consider the 3D size of an extruded loop, as measured by its radius of gyration R_g (Fig. 4E and SI Appendix). We find that actively extruded loops are more compact than static loops and that the compaction increases with LEF speed (Fig. 4F; see SI Appendix for details). This is expected, because loci that are brought into proximity by loop extrusion need time to move apart by thermal diffusion (Rouse diffusion, Fig. 4E). Finally, we ask how active loop extrusion is reflected in the overall dynamics of the chromatin fiber by measuring its mean square displacement (MSD). Specifically, we asked whether loop extrusion could be understood as an increased effective temperature, a conceivable consequence of the energy input from molecular motors. We find, however, that the MSD is elevated only on the timescale of loop extrusion without affecting the displacement on longer times (Fig. 4G). This is inconsistent with an elevated effective temperature, which would increase MSDs uniformly.

In conclusion, we found that neither (i) elevated effective temperature, nor (ii) static or very slow loops, nor (iii) reduced compartmental interaction can reproduce the effects of loop extrusion, which underlines that it is a true nonequilibrium effect that can be thought of as active mixing of the polymer system. Experimental ramifications of these findings are discussed below.

Changes in TADs and Compartmentalization Can Reveal the Underlying Mechanisms.

To consolidate our results, we consider how the strengths of TADs and compartments are connected to each other, and how they can be altered by biological perturbations at the molecular level. To this end, we measure how the strengths of TADs and compartments change as we vary (i) the characteristics of the loop extrusion machinery, namely LEF processivity (or residence time, SI Appendix, Fig. S9), LEF density (SI Appendix, Fig. S2), and LEF speed (SI Appendix, Fig. S5); (ii) topological properties, that is, the frequency of chain passing (SI Appendix, Fig. S7); (iii) the permeability of extrusion barriers (SI Appendix, Fig. S3); (iv) the strength of epigenetically encoded compartmental interaction (SI Appendix, Fig. S4); and (v) nuclear volume (SI Appendix, Fig. S8). In each case, we start from our “WT” parameters and sweep a single parameter to examine how compartmentalization and TAD strengths change. Strikingly, we find that different perturbations lead to different changes in the compartmentalization-vs.-TAD strength diagram.

We find (Fig. 5) that alterations of the loop extrusion process, namely of the residence time of LEFs, their linear density, and the speed of extrusion, result in simultaneous changes in TADs and compartmentalization: Reduced loop extrusion activity leads to weaker TADs and stronger (more segregated) compartments. Interestingly, changes in topological properties, simulating activation or inhibition of topoisomerase II (i.e., allowing more or fewer chain passings), show a similar trend. Alteration of the extrusion barrier permeability, however, shows a different pattern: It strongly affects TADs but leaves compartmentalization almost unaffected (as loop extrusion is preserved; see above). Strikingly, when nuclear volume or the compartmental interaction (i.e., B–B attraction) is changed, we observe a third type of behavior: changes in compartmentalization but not in the strength of TADs.

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Fig. 5.

Effects of different mechanisms on TAD and compartment strength. Three main classes of responses to perturbations are identified: A trade-off between compartmentalization and TADs is observed for parameter changes related to cohesin dynamics and for the frequency of chain passing (topoisomerase II activity). Compartmental interaction and nuclear volume mainly affect compartmentalization. The permeability of extrusion barriers mainly affects TADs. The black dots indicate our simulations of WT interphase cells and removal of cohesin (by Nipbl deletion), of CTCF, and of WAPL.

Our joint analysis of variations in TADs and compartmentalization provides an approach to interpreting existing and future experimental data, suggesting that coordinated changes in TADs and compartments reflect changes in the loop extruding machinery of cohesin or topoisomerase II activity; changes in TADs that leave compartments unaffected most likely come from altered extrusion barrier permeability [determined by binding of boundary proteins such as CTCF, and potentially YY1 (57) and Znf143, either globally or at specific loci]; and changes in compartments that do not affect TADs reflect changes in nuclear volume or in the epigenetic landscape of histone modifications or the molecules that mediate their interactions.