Chromosome organization by one-sided and two-sided loop extrusion

1Structural Maintenance of Chromosomes (SMC) complexes are ring-like protein complexes that are integral to chromosome organization in organisms ranging from bacteria to humans. SMC complexes linearly compact mitotic chromosomes in metazoan cells, maintain topologically associated domains (TADs) in interphase vertebrate cells, and juxtapose the arms of circular chromosomes in bacteria. In each of these processes, SMC complexes form chromatin loops. These diverse chromosome phenomena are hypothesized to be driven by a common underlying physical mechanism by which SMC complexes processively extrude chromatin or DNA loops. However, it is not known what molecular-level requirements loop extrusion must satisfy in order to robustly reproduce the 3D chromosome structures observed in these in vivo phenomena.

2The loop extrusion model posits that a loop-extruding factor (LEF), such as condensin, cohesin, or a bacterial SMC complex (bSMC) is in part comprised of two connected motor subunits that bind to chromatin and form a small chromatin loop by bridging two proximal chromatin segments. The SMC complex progressively enlarges the loop by reeling chromatin from outside the loop into the growing loop. To reel in chromatin from both sides of the complex, each motor subunit of the LEF translocates in opposite directions, away from the initial binding site. This ‘two-sided’ extrusion model recapitulates experimental observations of mitotic chromosome compaction and resolution, interphase TAD and loop formation, and juxtaposition of bacterial chromosome arms. However, until recently, loop extrusion by SMC complexes had not been directly observed.

3Recent in vitro single-molecule experiments have imaged loop extrusion of DNA by individual SMC condensin and cohesin complexes, demonstrating that yeast, human, and Xenopus condensin and Xenopus cohesin complexes extrude DNA loops in an ATP-dependent, directed manner at speeds on the order of 1 kb/s. Strikingly, however, yeast condensins and a significant fraction of both human and Xenopus SMC complexes reel in DNA from only one side, while the other side remains anchored to its DNA loading site. This contrasts with prior observations in bacteria demonstrating the direct involvement of SMC complexes in two-sided loop extrusion in vivo. One-sided extrusion also conflicts with existing versions of the loop extrusion model, which generally assume that extrusion is two-sided. Furthermore, recent theoretical work shows that purely ‘one-sided’ loop extrusion, as it has been observed in vitro so far, is intrinsically far less effective in linearly compacting DNA than two-sided extrusion. Thus, we investigated the extent to which one-sided loop extrusion might impact the 3D structure of chromosomes and whether variants of one-sided loop extrusion can recapitulate in vivo observations. In particular, we focus on three chromosome organization phenomena that are driven by SMC complexes: (1) mitotic chromosome compaction and resolution, (2) interphase chromosome domain formation, and (3) juxtaposition of bacterial chromosome arms. These three phenomena encompass the major physical processes associated with chromosome organization by SMC complexes: compaction and segregation, cis loop formation and linear scanning, and progressive juxtaposition of DNA flanking a loading site.

4The SMC condensin complex in metazoan cells plays a central role in mitotic chromosome compaction and segregation. In mitotic chromosomes, electron microscopy reveals that chromatin is arranged in arrays of loops. This results in dramatic linear compaction of the chromatin fiber into a polymer brush with a > 100 fold shorter backbone. Fluorescence imaging and Hi-C show that these loops maintain the linear ordering of the genome. Together, these features may facilitate the packaging, resolution, and segregation of chromosomes during mitosis by effectively shortening and disentangling chromatids. Each of these experimental observations is reproduced by the two-sided loop extrusion model, in which dynamic loop-extruding condensins collectively form arrays of reinforced loops by locally extruding chromatin until encountering another condensin. The simplest one-sided loop extrusion process, in contrast, can only linearly compact chromosomes 10-fold because it leaves unlooped (and thus, uncompacted) polymer gaps between loop extruders (; it is unclear whether 10-fold compaction is sufficient for robust chromosome segregation. Nonetheless, variants of one-sided loop extrusion in which loop extruders are effectively two-sided may robustly compact mitotic chromosomes. This raises the question of what abilities an individual one-sided loop extruder must possess to compact and spatially resolve chromosomes.

5In interphase in vertebrate cells, Hi-C reveals that the SMC cohesin complex is responsible for frequent but transient loop formation, which results in regions of high intra-chromatin contact frequency referred to as TADs. These regions are bordered by convergently oriented CTCF protein binding sites, which act as obstacles to loop extrusion and translocation of cohesin. The two-sided loop extrusion model explains the emergence of TADs and their ‘corner peaks’ (or ‘dots’) and ‘stripes’ (sometimes called ‘lines’, ‘tracks’ or ‘flames’) in Hi-C maps as an average collective effect of multiple cohesins dynamically extruding chromatin loops and stopping at the CTCF boundaries (;; reviewed in ). Existing models for loop extrusion during interphase have assumed LEFs with two mobile subunits, whether they be active or inactive. While it is clear that a one-sided LEF will necessarily leave an unlooped gap between its initial loading site and one of the CTCF boundary elements, the extent to which one-sided loop extrusion can recapitulate the experimental observations remains entirely unexplored.

6In bacteria, SMC complexes and homologs play an important role in the maintenance of proper chromosome organization and efficient chromosomal segregation (;;; and others). In Bacillus subtilis and Caulobacter crescentus, the circular chromosome exhibits enhanced contact frequency between its two chromosomal arms (often called ‘replichores’), as shown by Hi-C. This signal is dependent on the bacterial SMC complex (bSMC). Experiments show that bSMC is loaded at a bacterial parS site near the origin of replication, and then, while bridging the two arms, actively and processively moves along the chromosome, thus juxtaposing or ‘zipping’ the arms together. The symmetry of the juxtaposed chromosome arms implies that bSMC should be a two-sided LEF. Indeed, previous modeling has shown that pure one-sided loop extrusion produces contact maps that differ from experimental observations. However, it is unknown whether variations of one-sided extrusion can properly juxtapose the arms of a circular bacterial chromosome.

7Two-sided loop extrusion models can account for the various chromosome organization phenomena described above, but in vitro single-molecule experiments suggest that at least some SMC complexes are one-sided LEFs. We therefore investigate whether a mechanism of one-sided loop extrusion can account for in vivo observations of 3D chromatin organization, as listed above, namely metazoan mitotic chromosome compaction and resolution, interphase chromatin organization in vertebrate cells, and juxtaposition of bacterial chromosome arms. To study these processes, we construct a model for one-sided loop extrusion and simulate the collective dynamics of SMC complexes and chromatin in these three distinct scenarios. We also explore several one-sided extrusion variants. By comparing our results to experimental data, we find that pure one-sided loop extrusion fails to capture most of the in vivo phenomenology. However, simple variants of the one-sided model that make loop extrusion effectively two-sided or otherwise suppress the formation of unlooped chromatin gaps can restore the emergent features of chromatin organization observed in experiments.

8In our model, loop extrusion is performed by loop-extruding factors (LEFs), which may be a single SMC complex, a dimer of SMC complexes, or any other oligomer of SMC complexes. A LEF is comprised of two subunits, which can either be active or inactive. Each active subunit can processively translocate along the chromatin fiber, thus creating and enlarging the chromatin (or DNA) loop between the subunits (Figure 1a). An inactive subunit can either be anchored or passively slide/diffuse along the fiber, depending on the specific model (see below).

9In existing simulation models of loop extrusion, LEFs are ‘two-sided,’ that is they have two active subunits that on average grow a chromatin loop by translocating in opposing directions (Figure 1b). Here, we consider ‘one-sided’ LEFs that have one active subunit and one inactive (passive) subunit.

10LEFs in our one-sided extrusion model have binding and translocation dynamics that mimic turnover and translocation of SMC complexes, as has been observed in experiments. In our model, LEFs bind to chromatin with association rate kbind and unbind from chromatin with dissociation rate kunbind (mean residence time τ = 1/kunbind). A LEF’s active subunit translocates at speed v along the chromosome, away from its passive subunit, thus growing the chromatin loop. Furthermore, LEF subunits cannot translocate through other LEF subunits unless otherwise stated; extrusion by an active LEF subunit halts when it encounters another LEF subunit. Extrusion may continue if the obstacle is removed (for example, by unbinding). This constraint is relaxed for one model variant, as described in the Results section.

11The pure one-sided and two-sided loop-extrusion models are primarily controlled by two length scales, λ and d. The LEF processivity λ is given by λ = qv/kunbind, where q = 1 or q = 2 for one- and two-sided, respectively; thus, one-sided LEFs with extrusion velocity v grow loops at half the speed of two-sided LEFs with the same v (see arch diagrams in Figure 1b and c, bottom). d = L/Nb, is the mean distance between the Nb LEFs bound to the fiber of length L (where Nb = N kbind/(kbind+kunbind)). For λ <d, LEFs are sparse and on average do not meet. For λ >d, LEFs are densely loaded on the chromatin, and a translocating LEF typically encounters other LEFs.

12While there are many possible variants of the one-sided loop extrusion model, we mainly focus on three general variants of one-sided loop extrusion that differ by LEF subunit translocation dynamics.

13In pure one-sided loop extrusion, the passive subunit of the bound LEF remains stationary on the chromatin fiber for the entire residence time of the LEF, while the active subunit translocates at speed v away from the passive subunit. LEFs bind with a random orientation. Individual LEFs asymmetrically extrude loops, as observed in. Figure 1c shows a typical trajectory and corresponding arch diagram for LEF subunits in the pure one-sided extrusion model.

14We also considered a model in which the active LEF subunit translocates at speed v, while the inactive LEF subunit stochastically diffuses (slides) along the fiber. This model is primarily motivated by the experimental observation of the yeast condensin ‘safety belt’. This condensin component is thought to anchor the LEF in place as it extrudes loops in a one-sided manner, but the safety belt can be released via protein alterations, allowing the passive subunit of the SMC complex to diffuse along DNA. In addition, we note that several in vitro experiments have imaged cohesins and condensins diffusively translocating along naked DNA with diffusion coefficients of D = 0.001–4 μm2/s (or D = 0.01–35 kb2/s).

15In the model, the inactive subunit stochastically translocates by taking diffusive steps in either direction. The stepping rate in each direction is modulated by the entropic penalty for polymer loop formation (see Materials and methods). As a result of this effect, the sliding tends to shrink small loops, while having little effect on large loops. A typical trajectory and arch diagram for the subunits of a semi-diffusive LEF are shown in Figure 1d.

16To evaluate the importance of passive extrusion as compared to active extrusion, we study loop extrusion as a function of the scaled diffusive stepping rate. This quantity is the ratio, vdiff/v, of the characteristic diffusive stepping rate, vdiff, to the active loop extrusion speed, v. vdiff/v < 1 indicates that diffusive stepping is slow as compared to active stepping, while vdiff/v > 1 indicates that diffusive stepping is relatively rapid. The scaled diffusive stepping rate may be converted to a diffusion coefficient by D = a v, where a is the length of a lattice site.

17As another alternative model, we consider a scenario in which LEFs are instantaneously one-sided (i.e., one subunit is active and the other is inactive and stationary), but stochastically switch which subunit actively translocates. This model captures the dynamics of a proposed mechanism dubbed ‘asymmetric strand switching’ (see Figure 2d in ). As described in, switching could be achieved through a stochastic segment/loop-capture mechanism. In our model, switches occur at rate kswitch; by switching, inactive subunits become active and vice versa. Thus, LEF subunits have trajectories similar to the one shown in Figure 1e, top panel, and loops grow as shown in the arch diagram at the bottom of Figure 1e. Although not yet observed experimentally, we hypothesize that switching activity of SMC complexes could potentially be induced by exchange of subunits within the SMC complex, different solution conditions, or post-translational or genetic modifications, all of which can alter SMC complex behavior in experiments.

18We explore the switching model by varying the switching rate scaled by either the dissociation rate kunbind (for the eukaryotic chromosome models) or the chromosome traversal rate v/L (for the bacterial chromosome model). For the eukaryotic models, the dimensionless ratio kswitch/kunbind determines the mean number of switches before a LEF unbinds from the chromatin fiber. For kswitch/kunbind <1, switches rarely occur and LEF trajectories typically appear to be pure one-sided. In contrast, for kswitch/kunbind >1, the active and inactive LEF subunits may frequently switch before unbinding chromatin, and trajectories appear as in Figure 1e, top panel. For bacteria, the dimensionless quantity kswitchL/v is a dimensionless measure of the switching rate, chosen because chromosome-traversing bacterial SMC complexes (like B. subtilis SMC complexes) do not have a well defined unbinding rate. When this ratio is large, switching occurs many times during chromosome traversal; when it is small, switching is rare.

19We investigated the degree to which the above models reproduce physiological chromosome structures via 3D polymer simulations. To do this, we coupled each of the 1D loop-extrusion models in Figure 1 to a 3D model of a polymer chain and performed molecular dynamics simulations using OpenMM (see Materials and methods for details). In this coupled model, LEFs act as a bond between the two sites (monomers) to which the LEF subunits are bound; these bonds have the dynamics described for LEFs above. We simulated each of the three models, as well as several other variants, for various values of λ, d, vdiff/v, and either kswitch/kunbind or kswitchL/v. From these simulations, we obtain 3D polymer structures, images of compacted chromosomes and/or contact frequency (Hi-C-like) maps. By analyzing these data, we compare the models to experiments.

20In addition to 3D polymer simulations, we generated contact maps semi-analytically from the 1D models of the underlying SMC dynamics. This method allowed us to explore a broad range of parameter values and assess the resulting Hi-C-like maps in a computationally inexpensive manner. The semi-analytical method is compared to the 3D polymer simulation method in Appendix 3. The semi-analytical method is not used for modeling the eukaryotic systems because the Gaussian approximation used is not appropriate for highly compacted mitotic and ‘vermicelli’ (i.e., Wapl depletion) interphase chromosomes, which have linearly dense arrays of chromatin loops. However, as shown in Appendix 3, this method can be used to study bacterial chromosome conformations.

21We analyze these models for three chromosome phenomena that depend on SMC complexes. Each of the following results sections briefly describes the scenario, explains the relevant model observables, and subsequently, explores each model variant.

22We determined whether variants of the one-sided loop extrusion model can explain mitotic chromosome compaction and the spatial resolution of connected sister chromatids. Experimentally, it has been shown that these phenomena are driven by the condensin complex. During mitosis, mammalian chromosomes are linearly compacted ~1000 fold, leading to the formation of rod-like chromatids. Such compaction is thought to facilitate the spatial resolution of sister chromatids, which are connected at their centromeres.

23Previous work suggests that the two-sided loop extrusion model can rapidly achieve 1000-fold linear compaction in the regime in which LEFs are densely loaded on the chromosome (λ/d≳10), which is expected for mitotic chromosomes in metazoan cells. With a loop extrusion speed of v≈1 kb/s, two-sided extrusion can achieve full linear compaction within one residence time (1/kunbind ~ 2–10 min) and full 3D compaction and loop maturation occurs over a few (<10) residence times, consistent with the duration of prophase and prometaphase and in vivo observations of mitotic chromosome compaction and resolution.

24In contrast, theoretical work has demonstrated that pure one-sided loop extrusion cannot linearly compact a chromatin fiber by more than ~10 fold. Linear compaction in these models depends only on the dimensionless ratio of length scales λ/d. However, the 3D structures of such chromosomes have not yet been studied, and compaction by the semi-diffusive model, switching model, and other model variants has not been comprehensively investigated. Furthermore, sister chromatid resolution by variations of the one-sided loop extrusion model has not been investigated.

25We therefore performed simulations to measure linear compaction and characteristics of 3D chromosome organization of individual, compacted chromosomes. To measure linear compaction, we define the compacted fraction, f, as the fraction of chromosome length that is contained within looped regions and the resulting linear fold compaction as FC = 1/(1-f). We measure the resulting 3D compaction by computing chromosome volume, V, which is expected to decrease by >2 fold during mitotic compaction. We thus look for scenarios in which chromosomes are linearly compacted ~1000 fold and form the spatially compact rod-like arrays of chromatin loops observed in experiments.

26We also characterize the ability of one-sided loop extrusion models to resolve sister chromatids connected at their centromeres. We quantify chromatid resolution by measuring the median inter-chromatid backbone distance, ΔR, scaled by the polymer backbone length, Rb. As a supplementary metric, we also compute the inter-chromatid overlap volume, Vo, compared to the overlap volume without loop extrusion, Vo(0) = 3.6 μm3. Larger distances, ΔR/Rb >1, indicates that typical inter-chromatid distances are sufficient to prevent contacts between backbones. Larger median distance and smaller overlap are expected to contribute to the disentanglement of chromatids, which facilitates chromosome segregation by preventing anaphase bridge formation. Models are thus evaluated on the basis of whether compacted chromatids are fully spatially resolved.

27Mean-field theory predicts that pure one-sided loop extrusion can achieve at most ≈10-fold linear compaction, 100-fold less than expected for mammalian mitotic chromosomes. Figure 2c (i) shows linear fold compaction, FC, as a function of λ/d in the simulations, and results for λ/d>>1 are consistent with the theoretical predictions. The compaction limit is due to the unavoidable presence of ‘gaps’ of uncompacted (unlooped) chromatin between some adjacent loops (Figure 2c (ii)); of the four possible orientations of adjacent translocating LEFs, →→, ←←, →←, and ←→, the last one necessarily leaves an unlooped gap; the mechanistic connection between gaps and deficient compaction is illustrated by simulations broadly spanning λ/d (Figure 2c (ii)).

28We find that the presence of unlooped gaps along the chromatin fiber additionally has severe consequences for the 3D conformations of simulated mitotic chromosomes. As shown in Figure 2b (left), chromosomes compacted by one-sided LEFs are more spherical, and compacted regions are interspersed with uncompacted (unlooped) chromatin fibers. Moreover, compaction by one-sided LEFs only reduces the volume, V, by up to 2-fold from the uncompacted volume of V(0) = 3.6 μm3 (Figure 2c (iii)). This contrasts with the structures observed and >2.5 fold 3D compaction in the two-sided loop extrusion model (Figure 2a, left). Moreover, adding a small number of two-sided LEFs does not close a sufficient number of gaps to achieve 1000-fold linear compaction (Figure 2—figure supplement 1a; ) or 2.5-fold volumetric compaction (Figure 2—figure supplement 1c) because even a small number of gaps prevents full compaction (Figure 2—figure supplement 1b). A fraction of >80% of two-sided LEFs is necessary for sufficient compaction and resolution. One-sided extrusion thus leads to loosely compacted chromosomes that are qualitatively different from mitotic chromosomes observed in both the two-sided loop extrusion model and in vivo.

29We therefore investigated whether the inability of one-sided LEFs to compact chromosomes also impacted their ability to resolve sister chromatids. We find that one-sided LEFs can spatially resolve chromosomes that are physically linked at their centromeres, but far less effectively than two-sided LEFs. With one-sided extrusion, there is a small relative separation between chromatid backbones (ΔR/Rb <1, Figure 2c (iv)) and large overlap of chromatids (Vo/Vo(0)≈0.3; Figure 2—figure supplement 2c). In contrast, with two-sided extrusion, there is a larger distance between chromatid backbones (ΔR/Rb >10), and consequently, less overlap of chromatids (Vo/Vo(0) ≈ 0.1). The resulting linked chromatids are reminiscent of microscopy images of mitotic chromosomes (Figure 2a, right panel, and e.g.,), as has been observed in previous simulations. Thus, we find that chromatin gaps left by pure one-sided extrusion inhibit the spatial resolution of linked chromosomes; moreover, determining the presence or lack of unlooped chromatin gaps in 1D is sufficient to predict the effects on 3D compaction. Together, these results indicate that while the two-sided loop extrusion model can explain condensin-mediated metazoan mitotic chromosome resolution, the pure one-sided loop extrusion model cannot.

30We next investigated the semi-diffusive one-sided extrusion model, in which the inactive LEF subunit may passively diffuse. We find that semi-diffusive LEFs can compact chromatin to a greater extent than pure one-sided LEFs in some scenarios, but are unable to achieve 1000-fold linear compaction for a plausible values of λ/d (i.e., λ/d < 1000, which is expected from experimental measurements (;;;;;;;; Figure 2d (i)). The enhanced compaction by semi-diffusive one-sided LEFs arises from their ability to close some unlooped gaps (Figure 2d (ii)). LEFs may suppress gaps in two ways: 1) inactive but diffusive LEF subunits may stochastically slide toward each other and 2) diffusion of an inactive subunit of a ‘parent’ LEF may be rectified if a ‘child’ LEF is loaded within the loop so that the active subunit of the child LEF moves toward the inactive subunit of the parent LEF, leading to Brownian ratcheting (Figure 2—figure supplement 3a). The first mechanism is ineffective in eliminating gaps because it is opposed by the conformational entropy of the extruded loop, and the LEFs may also diffuse apart, causing the unlooped gap to reappear. The second mechanism can be enhanced by the active subunit of the child LEF actively ‘pushing’ the parent’s inactive subunit (Figure 2—figure supplement 4 and Appendix 1). These active processes are more effective at closing gaps. Nonetheless, Brownian ratcheting by nested LEFs does not sufficiently linearly compact chromosomes for all λ/d < 1000, while active pushing can only achieve a high degree of compaction if the active subunit can simultaneously reel chromatin through multiple inactive subunits and λ/d≈1000.

31To understand how semi-diffusive LEFs enhance linear compaction in some particular scenarios, we investigated how compaction depends on the scaled diffusion speed, vdiff/v. For reference, with v = 1 kb/s as in vitro, vdiff/v = 1 corresponds to D = 0.5 kb2/s or D = 0.06 μm2/s on naked DNA, which is in the range of measured in vitro measured diffusion coefficients (D = 0.01–35 kb2/s or 0.001–4 μm2/s) for SMC complexes on DNA. For vdiff/v<<1, the inactive subunit diffuses very slowly, so the LEFs behave similarly to pure one-sided LEFs; moreover, thermal ratcheting by nested LEFs is very slow since the translocation speed of the active subunit of the child LEF is effectively limited by the diffusion of the inactive subunit of the parent LEF. Interestingly, in the case with rapid diffusion, vdiff/v > 1, semi-diffusive LEFs linearly compact chromosomes even less effectively than pure one-sided LEFs. Because conformational entropy favors shrinkage of parent loops, the diffusive subunit shrinks loops more rapidly than the active subunit grows loops. Since loops remain small, nesting of loops (i.e., LEFs extruding loops within loops) becomes less likely (Figure 2—figure supplement 3). Thus, gaps remain because they are not closed by Brownian ratcheting. Intriguingly, our simulations reveal that vdiff/v≈1 is an optimal case in which diffusion is sufficiently slow to permit loops to grow large enough to allow loop nesting, but fast enough to promote loop growth by thermal ratcheting. However, even this ‘optimal’ case leaves a large number of gaps. Thus, we find that for all vdiff/v unlooped gaps remain (Figure 2d (ii)) and 1000-fold compaction cannot be achieved with λ/d < 1000 (Figure 2d (i)).

32In the semi-diffusive model, as in the pure one-sided model, the limited ability to linearly compact chromosomes impairs 3D compaction. Simulated chromosomes are generally not rod-like (Figure 2d (iii), inset), and the loop architecture remains gapped and weakly reinforced. Consequently, for optimal scaled diffusion speeds, vdiff/v≈1, the volume, V, is reduced by less than in the case of two-sided extrusion (≤2 fold vs. >2.5 fold, Figure 2d (iii)). Similarly, modest linear compaction of chromatids leads to only a slight increase in inter-chromatid distance (Figure 2d (iv)) and moderate overlap volume (Vo/Vo(0)≈0.2). Thus, 3D compaction and sister chromatid resolution in the semi-diffusive model can exceed that of the pure-one sided model, but they still fall short of the far more dramatic compaction and distinct spatial resolution expected for mitotic chromosomes in vivo and reproduced by the two-sided loop extrusion model. The failure of this one-sided loop extrusion variant is again due to the inability to robustly eliminate unlooped gaps.

33The results of the previous sections suggest that robust mitotic chromosome compaction and chromatid resolution requires LEFs that consistently and irreversibly eliminate unlooped gaps. We therefore consider a variation of the one-sided extrusion model in which only one LEF subunit translocates at a time, but the LEFs stochastically switch which subunit is active at rate kswitch. In principle, in this scenario, LEFs may be ‘effectively two-sided,’ which allows LEFs initially in a divergent orientation (←→) to eliminate the initially unlooped gap.

34To study mitotic chromosome compaction within the switching model, we vary both λ/d and the scaled switching rate, kswitch/kunbind. The scaled switching rate determines the number of times that a LEF will switch before unbinding; each switch allows a LEF the chance to close a gap. Accordingly, we observe that the ability of LEFs to linearly compact chromatin increases with kswitch/kunbind. For very slow switching rates (kswitch/kunbind<<1, or roughly kswitch<<1 min−1 for experimentally observed kunbind), loop extrusion is effectively one-sided because switches rarely occur and gaps are not closed, so linear compaction is limited to ~10 fold (Figure 2e (i), (ii), blue). For faster scaled switching rates (0.1 < kswitch/kunbind ≤1), switches are more likely to occur during each LEF’s residence time, so greater numbers of LEFs are effectively two-sided and more gaps can be closed (Figure 2e (i), (ii), gray). In these cases, LEFs linearly compact chromosomes 10- to 100-fold. For very fast switching (kswitch/kunbind >1 or kswitch >1 min−1), many switches occur per residence time. Thus, all LEFs are effectively two-sided so that all unlooped gaps are eliminated for large λ/d, and 1000-fold linear compaction can be achieved (Figure 2e (i), (ii), red).

35Concordant with observations for linear compaction, we find that 3D chromosome compaction and resolution varies from the one-sided to two-sided phenotypes with increasing scaled switching rate, kswitch/kunbind. Chromosomes with rapidly switching LEFs can undergo a large reduction in volume, V (>2.5 fold, Figure 2e (iii)), comparable to what is observed for two-sided extrusion. Similarly, sister chromatid resolution can be achieved in the switching model for kswitch/kunbind > 1. The distance between chromatid backbones increases (ΔR/Rb >8, Figure 2e (iv)), and overlap is greatly reduced (Vo/Vo(0)≈0.1), comparable to what is achieved in the two-sided model. We thus conclude that the switching model with fast switching rates, kswitch ~1 min−1, can reproduce the experimentally observed 3D compaction and resolution of mammalian mitotic chromosomes.

36Of the three main variants of one-sided loop extrusion that we tested, only the switching model can reproduce mammalian mitotic chromosome compaction and resolution. In each of these models, the ability of LEFs to eliminate unlooped gaps governs compaction and resolution. Chromatin segments that are not linearly compacted into loops are longer, and thus have a larger 3D size. Therefore, the average number of unlooped gaps that remain, a 1D quantity, determines the 3D structure and organization of simulated mitotic chromosomes. Effectively two-sided extrusion is required to eliminate these gaps, and of the models considered here, this physical mechanism is reliably present in only the switching model.

37As an alternative to the models above, which are dominated by the effects of extrusion-driven linear compaction, we performed polymer simulations to determine whether gaps created by one-sided loop extrusion could be eliminated by 3D attractive interactions between LEFs or between different polymer segments (e.g., poor solvent). Moreover, we explored whether such interactions could volumetrically compact chromosomes and generate rod-like mitotic chromosomes, as previously suggested. We find that 3D attractions can volumetrically compact polymers (Figure 2—figure supplement 5a), but the resulting structures do not resemble mitotic chromosomes. When LEFs attract each other, compacted chromosomes form extended, clumpy structures (Figure 2—figure supplement 5b, top), and chromatin gaps remain visible. Moreover, sister chromatids do not spatially segregate (Figure 2—figure supplement 5b, bottom). When the simulated chromosomes are instead treated as polymers in poor solvent, chromosomes are compacted into spherical structures and sister chromatids cannot be spatially resolved (Figure 2—figure supplement 5c). Attractive interactions have little effect on chromosome structure when the interaction strength, ε, is low, but when ε is large, the chromosome is compacted into a spherical globule. These findings are consistent with previous theoretical and computational work on polymer combs, showing that 3D attractive interactions lead to a coil-globule transition.

38We also considered the possibility that interactions between one-sided LEFs might alter their residence times. We hypothesized that such interactions could stabilize LEFs that had closed gaps. However, we found that linear compaction in this model is still limited to 10-fold because gaps are still created by divergently extruding LEFs (Figure 2—figure supplement 6). Altogether, we find that attractive interactions between LEFs or between different polymer segments cannot be the mechanism of gap closure for mitotic chromosomes.

39Recent single-molecule experiments report the first observations of effectively two-sided loop extrusion that results from the coordinated activity of two one-sided loop extruders. Single-molecule experiments have shown that yeast condensins can form ‘Z-loops’ that act as an effectively two-sided extruder. In this scenario, condensins can pass each other as they translocate along DNA, thus forming structures that reel in DNA from two directions. To analyze this possibility, we simulated chromosomes compacted by LEFs that can freely traverse each other. In this model, linear chromosome compaction, as quantified by loop coverage, increases exponentially with λ/d, as expected from theory (Figure 2f and Appendix 2). Correspondingly, we observe that chromosomes in this model form compact, rod-like structures (Figure 2g). We find that ~ 1000 fold linear compaction is achieved for λ/d ~ 7, which can be satisfied with reasonable physiological values of loop sizes, ℓ=λ~140 kb and densities of one LEF per d ~ 20 kb. In addition, LEFs in this model can spatially resolve sister chromatids (Figure 2h). Thus, one-sided LEFs that can freely traverse each other may be sufficient to compact and resolve mitotic chromosomes.

40Next, we determined whether one-sided extrusion can recapitulate prototypical features in Hi-C and micro-C maps of vertebrate cells during interphase, such as TADs, ‘stripes’ (also called ‘lines,’ ‘tracks,’ or ‘flames’), and particularly, the ‘dots’ (or ‘corner peaks’) found at the boundaries of TADs (Figure 3a). Dots are foci on Hi-C maps that reflect enriched contact frequency between specific loci, often found at the corners of TADs and/or between proximal (<1–2 Mb) CTCF sites. TADs, stripes, and dots are cohesin-mediated, and they can be modulated by changes to cohesin and/or CTCF. Thus, we evaluate extrusion models based on whether they can generate these hallmarks of interphase chromosome organization.

41We perform polymer simulations for each model, sweeping λ and d, as well as model-specific parameters. CTCF barriers are modeled as partially permeable loop-extrusion barriers. In Figure 3 we use experimental values for λ and d for wild-type (WT) conditions (Materials and methods and Figure 3—figure supplement 1); other values for λ and d are explored in the figure supplements. We compute and visualize contact maps from these simulations and quantify the dot strength by the enhancement of dot contact frequency over background, as in Figure 3—figure supplement 2; ).

42In models of two-sided loop extrusion in interphase, a TAD arises due to the formation of extruded loops within a particular region, usually bounded by convergently oriented CTCF sites. A stripe emerges if one extruding subunit of a LEF is stalled by CTCF while the other subunit continues extruding (Figure 3—figure supplement 3). A dot arises when two barriers to extrusion (e.g., convergently oriented CTCF sites) are brought together by one or a few LEFs that close a gap between two barriers (Figure 3—figure supplement 3;; ).

43While two-sided extrusion can reproduce TADs, stripes, and dots, we found that the simplest model of one-sided extrusion can recapitulate only some of these features. When LEFs are uniformly loaded onto chromatin, pure one-sided extrusion can form the bodies of TADs and stripes, but does not form dots (Figure 3b, right panel). For one-sided extrusion, stripes are an average effect of LEFs loading at different loci and extruding up to a barrier (Figure 3—figure supplement 3), while dots are not formed because only one-sided LEFs loaded at a barrier can pair two barriers (Figure 3—figure supplement 3). This problem cannot be resolved by increasing the processivity, λ, or decreasing the separation between LEFs, d (Figure 3—figure supplement 4). In contrast, two-sided extrusion with increased processivity generates the strong dots seen in wild-type data as well as the ‘extended dots’ (Figure 3b and Figure 3—figure supplement 5) seen in Wapl depletion data. This failure to form dots is due to inevitable gaps that one-sided extrusion leaves between LEFs and between LEFs and CTCF barriers (Figure 3c).

44The semi-diffusive model creates a phenotype that is similar to that of pure one-sided extrusion for simulations of WT conditions (Figure 3d); it can generate TAD bodies and stripes, but neither dots nor extended dots (Figure 3—figure supplement 2). We conclude that the semi-diffusive one-sided model works similarly to the pure one-sided model, and it is also limited by its inability to close gaps between LEFs and between LEFs and barriers.

45Next, we considered variations of the model in which one-sided LEFs are loaded nonuniformly, with increased probability of loading at barriers (;; Figure 3d). Each barrier has two loading sites and one-sided LEFs are loaded directionally so that they translocate away from the boundary. Loading of LEFs at CTCF sites increases both the primary and extended dot strengths, qualitatively reproducing both wild-type conditions (λ = 200 kb, d = 200 kb) (Figure 3d) and Wapl depletion (λ = 2 Mb, d = 200 kb) conditions (Figure 3—figure supplement 6). To clearly observe dots, however, LEFs must have a strong loading bias, i.e., >100 fold preference to bind barrier sites as compared to body sites. While contacts within the TAD body are reduced for this large bias (Figure 3d), it is possible to find a loading bias and LEF density such that both dots and the TAD body are clearly visible (Figure 3—figure supplement 6). Although current experimental evidence does not support preferential loading of cohesin at CTCF sites in mammals, such a mechanism of TAD, stripe, and dot formation is feasible and may be operational under some conditions, in some cell types, or in other species.

46We hypothesized that mechanisms other than loading at CTCF could enable one-sided extrusion to reproduce interphase Hi-C features. We considered the switching model because a LEF, when switching frequently enough, might bring two barriers together, even if it is not loaded at a barrier. Moreover, switching could eliminate gaps between nearby LEFs.

47The switching model for slow switching rates approximates the pure one-sided model; primary and extended dots are not present (Figure 3d, third column) and they do not appear with increased λ (Figure 3—figure supplement 2). For faster switching rates, primary and extended dots appear (and loop strengths increase with λ, Figure 3—figure supplement 2), as they do in the two-sided model (Figure 3d, third column). The switching model approaches the two-sided extrusion model, as quantified by primary and extended dot strengths for kswitch/kunbind≈10 (Figure 3—figure supplement 2). Thus, the model suggests that cohesin must undergo a switch once per minute for characteristic residence times of ~10–20 min. In addition to dots, switching generates a high frequency of intra-TAD contacts and stripes (Figure 3d, third column). Thus, one-sided LEFs that switch sufficiently fast can account for features of interphase chromosome organization.

48A mix of one- and two-sided LEFs approaches either the one-sided or the two-sided phenotype depending on the percentage of two-sided LEFs (Figure 3d, right column). Dots are visible, but weak for a mix with 20% two-sided LEFs, while a mix with 60% two-sided LEFs approaches the two-sided dot strength and generates stripes and intra-TAD contacts (Figure 3d, right column). A lower percentage of two-sided extruders, however, is needed to reproduce interphase organization (~50%) as compared to the percentage needed for strong mitotic compaction (>80%). While even a small fraction of gaps can be detrimental to mitotic compaction, gaps between LEFs are less damaging for the interphase, in which LEFs are more sparse along the chromosome (Figure 3c).

49Next, we considered one-sided LEFs that may traverse each other upon encountering each other as a model for ‘Z-loops,’ which have been observed for yeast condensins on DNA. We find that under WT conditions (d=λ=200 kb), such LEFs do not form noticeable dots (Figure 3—figure supplement 7). While the ability of LEFs to traverse each other can eliminate both gaps between LEFs and gaps between LEFs and boundaries, one-sided extruders with LEF traversal are still less efficient in pairing CTCF sites than two-sided LEFs. Dots become stronger when the separation between LEFs is reduced (d ≤ 50 kb) while maintaining the WT processivity or the processivity is increased (λ >2 Mb) while maintaining WT LEF densities for the simulated TAD sizes. Nonetheless, dots remain weaker than those of two-sided LEFs with the same separation and processivity.

50Our simulations show that features of interphase chromosome organization can be reproduced by variants of one-sided extrusion where (a) extruders can switch their directionality approximately every minute; (b) one-sided extruders are mixed with two-sided extruders; (c) extruders have a > 100 fold preference for loading at CTCF sites; or (d) extruders may traverse each other and have a small average separation (d ≤ 50 kb) or large processivity (λ >2 Mb).

51The bacterial SMC complex (bSMC) plays a direct role in juxtaposing the arms of the circular bacterial chromosome. In bacteria such as B. subtilis, the strong site-specific loading of bSMC followed by loop extrusion forms a distinctive pattern different from the case of uniform loading (assumed for eukaryotic systems). The bSMC loading sites (i.e., parS sites) are typically located near the origin of replication (<100 kb away). A secondary diagonal is visible emanating from the parS site in the bacterial Hi-C maps; it indicates long-ranged, high frequency contacts between chromosomal loci on opposite sides of the replichore (Figure 4a;;; ). This secondary diagonal arises due to the high processivity of bSMCs (λ >4 Mb), which brings together DNA segments approximately equidistant from the origin-proximal parS loading sites. Recent modeling studies show that the shape and trajectory of the secondary diagonal can be theoretically predicted by a stochastic model of bSMC two-sided loop extrusion. In light of these recent models and data, we explore the extent to which variations of one-sided extrusion might recapitulate these results.

52We compare the models for one-sided extrusion as follows. We perform 1D simulations of LEF dynamics, and then use our semi-analytical approach (see Materials and methods and Appendix 3) to produce Hi-C-like contact maps. In contrast to the previous sections, we only consider the limit of large λ/d > 1 as suggested by experiments (i.e., d < 4 Mb < λ; see Appendix 3;;; ). We evaluate the model by qualitatively comparing the width, intensity, and length of the experimental secondary diagonals to what is produced by our models.

53It was recently shown by 3D polymer simulations that the pure one-sided loop extrusion model cannot reproduce the secondary diagonals visible by Hi-C. In contrast, two-sided loop extrusion qualitatively reproduced the experimentally observed secondary diagonal, with an intensity that depends on the number of LEFs (Figure 4—figure supplement 1, left column).

54Using our semi-analytical approach, we recapitulate these previous results (Figure 4b) and explore a broader range of parameter values. As seen in Figure 4b (right panel), with bSMC loading only at a predetermined site (with up to 30 bSMCs per origin of replication), one-sided extrusion fails to yield the secondary diagonal that is characteristic of the chromosome contact maps of B. subtilis (Figure 4a) and other bacteria. Instead, pure one-sided extrusion exhibits a ‘+”-shaped pattern overlaid on the main diagonal, which indicates contacts of the parS loading site with all other chromosomal loci. This results from the fact that in pure one-sided loop extrusion, one LEF subunit is fixed at the parS loading site, while the other subunit translocates away from it. Thus, we conclude that pure one-sided loop extrusion fails to reproduce the symmetric chromosome arm juxtaposition that is characteristic of many bacterial Hi-C maps.

55We next considered the semi-diffusive case in which one subunit of the LEF actively translocates, while the other diffuses. Despite the increased mobility of the inactive subunit, the qualitative patterns of the contact map remained largely unchanged from the pure one-sided model (Figure 4c). Increasing the scaled subunit diffusion rate, vdiff/v, broadened the ‘+”-shaped pattern and did not produce the secondary diagonal (Figure 4c and Figure 4—figure supplement 2). Interestingly, for high enough values of vdiff/v (Figure 4c, right panel), the ‘+”-shaped pattern is replaced by a square TAD-like structure, reminiscent of two large macrodomains separating each of the sister replichores from each other. No secondary diagonal was observed even when the number of LEFs that is present on the chromosome is changed (Figure 4—figure supplement 3). Thus, for all values of vdiff/v, the semi-diffusive loop-extrusion model does not explain the available Hi-C data for B. subtilis and C. crescentus (and other bacteria with a secondary diagonal).

56We also tested whether one-sided loop extrusion with traversal could explain the experimental data. Similarly to the semi-diffusive case in which vdiff/v is large (Figure 4c, right panel), we found that LEF traversal generated a square, TAD-like structure between the left and right replichores (Figure 4—figure supplement 4), rather than a secondary diagonal characteristic of prokaryotes with an SMC/parABS system.

57We next tested whether one-sided LEFs that stochastically switch which subunit is active can recapitulate the available data. We performed a parameter sweep over a range of numbers of bSMCs and scaled switching rates, kswitchL/v, and we generated Hi-C contact maps (Figure 4d and Figure 4—figure supplement 5). The width of the experimentally observed secondary diagonal constrains the possible values of kswitchL/v in our model. In experiments, the secondary diagonal is narrow, with a width of ~100 kb across the entire map. This suggests that there is very little variance in the extrusion speeds along each chromosome arm. With more frequent switches (larger kswitchL/v), the progression of each extruding subunit along each arm varies less relative to the mean extrusion trajectory (Figure 4d). We found that fast enough switching rates (kswitchL/v > 200) can produce the secondary diagonal (Figure 4d), irrespective of the number of bSMCs (Figure 4—figure supplement 5). For B. subtilis and C. crescentus, we calculate that the upper bound on the mean time between switches is approximately 2–10 s and 10–20 s, respectively, with v = 50 kb/min in B. subtilis and v = 25 kb/min in C. crescentus as measured experimentally (Figure 4d, right panel).

58Thus, in contrast to other models that we considered, one-sided extrusion with switching can juxtapose chromosomal arms, as demonstrated by the presence of the Hi-C secondary diagonal that is prominent in many bacterial maps. In our model, this requires a relatively fast switching rate, which effectively makes a one-sided LEF behave like a two-sided LEF at the physiologically relevant time scales of a few minutes. Other variants of one-sided mechanism cannot achieve juxtaposition of bacterial arms due to tethers that remain between distal chromosome loci and the LEF loading site, indicating that bSMC is an effectively two-sided extruder.

59SMC complexes are ubiquitously found in all domains of life, and strong evidence is emerging that SMC protein complexes function by DNA loop extrusion, which appears to be central to their function. By forming loops, SMC complexes promote chromosome contacts spanning tens of kilobases to megabases in bacteria and hundreds of kilobases in metazoan cells (e.g.,). Proper function of the SMC machinery is vital to chromosome organization and compaction. Improper chromosome compaction and segregation can lead to anaphase bridges in metazoan cells and mispositioning of origins of replication in prokaryotes, all of which might cause aneuploidy (or anucleate cells in bacteria) and DNA damage (e.g.,). Additionally, the loss of interphase chromosome structure in vertebrates by loss of cohesin SMC complexes can affect gene expression (e.g.,). Similarly, mutations that perturb cohesin or condensin can lead to human developmental disorders, such as Cornelia de Lange syndrome and microcephaly.

60Recent in vitro imaging studies showed that loop extrusion by Saccharomyces cerevisiae condensin SMC complexes is purely one-sided. To determine the biophysical implications and to test the generality of this striking molecular observation, we explored whether one-sided loop extrusion could explain SMC-dependent phenomena observed in vivo for a range of organisms beyond S. cerevisiae. These phenomena included mitotic chromosome compaction in metazoans, formation of TADs and dots (corner peaks) in vertebrate interphase Hi-C maps, and juxtaposition of chromosome arms in rapidly growing bacteria. Together, these three systems exhibit the main features of chromosome organization that are attributed to loop extrusion: linear and 3D compaction, spatial segregation, cis loop/domain formation, linear scanning in cis, and progressive juxtaposition of chromatin flanking a loading site.

61Our work, along with recent theoretical modelling, indicates that pure one-sided loop extrusion does not generically reproduce these three phenomena, except under specific conditions. Therefore, biophysical capabilities beyond the one-sided loop extrusion observed for yeast condensins in vitro should be present for other organisms. Indeed, recent experimental evidence suggests that pairs of yeast condensins may be able to cooperatively grow loops bidirectionally, while human and Xenopus condensins and cohesins can perform either one- or two-sided loop extrusion. Thus, we explored simple variations of the pure one-sided loop extrusion model and identified a class of one-sided extrusion models that can reproduce in vivo experimental observations (Table 1). Our results suggest modes of loop extrusion that might be observed in future experiments.

62We focused on several variations of the one-sided loop extrusion model and investigated the consequences for 3D chromosome organization (Table 1). Our aim was not to exhaustively enumerate all possible model variations of one-sided extrusion. Instead, we sought to obtain and evaluate a set of minimalistic requirements to explain experimental data. We modeled SMC complexes as LEFs with two subunits with distinct dynamics; subunits could be either active (i.e., moving processively), inactive and anchored, or inactive but diffusive. Within this framework of varying the dynamics of the subunits, we primarily focused on the following models for LEFs: 1) one subunit active, the other subunit inactive and anchored (‘pure one-sided’), 2) one subunit active, the other subunit inactive but diffusive (‘semi-diffusive’), 3) one subunit active, the other subunit anchored, with kinetic interchange of active and anchored subunits (‘switching’). We also considered several related variants for each chromosome organization scenario, such as preferential loading at CTCF by one-sided cohesins during interphase. As a point for comparison, we quantitatively compared all results with those of two-sided extrusion, which previous works have shown to recapitulate key experimental observations.

63Our modeling demonstrates that the ability to robustly eliminate unlooped gaps is essential to the chromosome-organizing role of LEFs. As a result, models in which gaps persist in steady state, such as the pure one-sided model, fail to reproduce hallmarks of chromosome organization found in several physiological scenarios. One-sided extrusion generally does not reproduce mitotic chromosome compaction and chromatid segregation or hallmarks of interphase Hi-C maps, without further assumptions beyond what has been observed experimentally. Importantly, even dynamic LEF turnover (i.e., allowing dynamic chromatin unbinding with uniform rebinding) does not eliminate gaps because LEF unbinding (and even LEF binding) can introduce new gaps. Instead, chromosome compaction, resolution, and interphase organization can readily be explained by physical mechanisms that either eliminate gaps by turning one-sided extrusion into effectively two-sided extrusion (e.g., as in the switching model) or suppress the creation of gaps (e.g., by biased loading at boundaries).

64In the case of mitotic chromosome compaction, linear compaction by pure one-sided loop extrusion is limited to ~10 fold because it unavoidably leaves gaps between SMC complexes (Figure 2c (i), (ii) and). By simulations, we showed that 10-fold linear compaction is not sufficient to reproduce the classical 3D shapes of mitotic chromatids and chromosomes are volumetrically compacted at most twofold in 3D (Figure 2 b,c (iii)). This defect in 3D compaction leads to defects in mitotic chromosome resolution (Figure 2 b,c (iv)). Allowing the SMC complexes’ anchor points to diffuse (i.e., slide) along chromosomes also does not close gaps because loop formation is opposed by the conformational entropy of the formed loop (Figure 2d (ii) and Figure 2—figure supplement 3). Therefore, the LEFs cannot generate a sufficient increase in linear compaction for any diffusive stepping rate, vdiff (or diffusion coefficient, D) (Figure 2d (i)); in vitro experiments also show that one-sided condensins with diffusing safety belts do not grow large DNA loops. More generally, with one-sided LEFs, uncompacted gaps are pervasive, so simply adding a small fraction of two-sided LEFs is unable to sufficiently compact chromosomes; in vivo levels of compaction requires >80% two-sided LEFs (Figure 2—figure supplement 1; ). Similarly, a model in which LEFs are effectively two-sided, such as the switching model in which the active and inactive subunits dynamically switch, can generate greater than twofold 3D compaction and clear resolution of sister chromatids (Figure 2e (iii), (iv)), as observed in vivo. Such a switching mechanism could be achieved in vivo by a stochastic strand switching mechanism in which both upstream and downstream DNA can be captured by the loop extruder.

65For interphase organization in vertebrate cells, the ability of one-sided loop extrusion to reproduce major features of Hi-C maps is more complicated. We found that one-sided extrusion with uniform association and dissociation of LEFs can generate TADs (Figure 3b, right) and ‘stripes’ (or ‘flames,’ ‘tracks,’ or ‘lines’) on Hi-C maps (Figure 3a). However, one-sided extrusion cannot reliably bring CTCF barriers together, and thus, cannot generate the dots (corner peaks) that are prominent features of Hi-C and micro-C maps and are reproduced by two-sided extrusion (Figure 3b, right and Figure 3—figure supplement 2). The presence of unavoidable gaps between LEFs and between LEFs and barriers is the reason for this deficiency. This can be remedied by introducing a comparable number of two-sided LEFs to close gaps (Figure 3d, right). One-sided extrusion alone, however, can reproduce dots when undergoing frequent stochastic switches in translocation direction, turning one-sided into effectively two-sided extrusion. Additional mechanisms to generate two-sided or effectively two-sided extrusion have also been proposed, and gap closure may be achieved by several other mechanisms, as we discuss below in the subsection ‘Molecular evidence and plausibility of different modes of SMC function.’ Another strategy to eliminate gaps between boundaries and generate dots is to have strongly (>100 fold) biased loading of LEFs at barriers. Loading of cohesin at CTCF sites has been proposed since the two were found to colocalize. Available experimental evidence, however, argues against loading at CTCF sites; it was previously shown that CTCF is dispensable for cohesin loading, and more recently, CTCF-degradation experiments appear to have little effect on the levels of chromatin-associated cohesin and the extent of loop extrusion.

66In many bacteria, bSMCs loaded near the origin of replication (by the parABS system) generate contacts centered about the ori-ter axis, which is visible in Hi-C maps as a secondary diagonal. The challenge for one-sided loop extrusion models in bacteria is to explain how one-sided (i.e., asymmetric) LEF translocation might generate symmetrically aligned contacts between chromosome arms. Pure one-sided extrusion does not work because it creates a ‘+”-shape on the contact map instead of a secondary diagonal (Figure 4c and). Furthermore, we find that allowing diffusion of the anchor point does not help because this type of asymmetric extrusion cannot promote symmetric juxtaposition of the chromosome arms.

67The switching model, however, with a switching time on the order of seconds (<10 s for B. subtilis and <20 s for C. crescentus, i.e., rates kswitch≳0.1 s−1; Figure 4d) exhibits the desired effectively two-sided property and naturally creates the desired symmetry of contacts between left and right chromosome arms. Interestingly, if bSMCs function by one-sided extrusion with switching, this constraint suggests that bSMCs can switch their direction of extrusion within a few ATPase cycles (the B. subtilis SMC complex has an ATPase rate of 0.7 ATP/s). Switching, however, has not been observed in single-molecule experiments with yeast condensin SMC complexes, and such fast switching may appear as two-sided extrusion in vitro. We note that it was recently suggested that B. subtilis SMCs have two independent motor activities for extrusion; this observation is consistent with either two-sided extrusion or one-sided extrusion with rapid switching. Thus, our model suggests that microscopically one-sided extrusion can explain juxtaposition of chromosome arms, provided that bSMCs act as effectively two-sided extruders.

68One-sided loop extrusion was first imaged for budding yeast (S. cerevisiae) condensins. Yeast chromosomes are organized differently from chromosomes of higher eukaryotes. In budding yeast, cohesin is responsible for moderate compaction of mitotic chromosomes, while condensin compacts rDNA and proximal regions into insulated domains and, in quiescent cells, forms 10–60 kb chromatin domains that silence transcription. In fission yeast (S. pombe), cohesin forms small (<100 kb) domains, while during mitosis, condensin compacts chromatin by forming larger (100’s of kb) domains.

69The ~10 fold linear compaction achievable by pure one-sided loop extrusion is consistent with fluorescence in situ hybridization imaging of yeast mitotic chromosomes. Moreover, previous modeling of budding yeast mitotic chromosomes indicated that just ~30–40% coverage by cohesin-extruded loops (i.e.,~2 fold linear compaction, Figure 2—figure supplement 7) produces chromosome contact maps consistent with those obtained from Hi-C experiments. This lesser degree of compaction generally leads to poorly resolved sister chromatids in our model (Figure 2b,c (iii), and c (iv)), but chromatid resolution in yeast could be facilitated by spindle tension and the shorter length of yeast chromosomes. These observations could be consistent with compaction by cohesins performing one-sided loop extrusion.

70In contrast, one-sided extrusion could account for some, but not all, of the observations of chromatin domains in yeast Hi-C, micro-C, and ChIA-PET experiments. Yeast condensins compact pre- and post-rDNA genomic regions (in S. cerevisiae) and mitotic chromosomes (S. pombe) into insulated domains that do not exhibit the dots that are indicative of bringing boundaries together. In a similar manner, fission yeast cohesins organize small chromatin domains without dots. As shown in Figure 3b, pure one-sided loop extrusion can generate domains without dots.

71Nonetheless, recent observations of chromatin domains with dots under certain conditions in budding yeast challenge the viability of one-sided extrusion by both condensin and cohesin. In quiescent cells, condensins generate dots at the corners of small (10–60 kb), transcription-silencing domains in micro-C maps. In exponentially growing cells arrested during mitosis, cohesins can also generate dots in S phase. This observation suggests that budding yeast condensins and/or cohesins are either effectively two-sided loop extruders or loaded at specific sites because one-sided extrusion alone cannot generate dots (Figure 3b). However, a mix of two-sided cohesins and one-sided condensins (e.g., similar to Figure 3d, right panels) could generate dots as in micro-C/Hi-C experiments, while remaining consistent with single-molecule experiments.

72Cohesin-dependent dots have also been observed at sites of convergent transcription in Hi-C maps when cohesin is overexpressed in G1. While such dots can be explained by two-sided extrusion, we also considered the possibility that one-sided extrusion assisted by RNA polymerases that can push one side of an SMC complex. For one-sided extrusion, this effect could in principle generate effectively two-sided (but asymmetric) extrusion, where the slower extruding subunit moves at the speed of transcription (~1 kb/min). For typical cohesin residence times, this model suggests that small loops of 10–60 kb could be generated by the combined activity of loop extrusion and transcription (Figure 3—figure supplement 8).

73In summary, one-sided extrusion by condensin and cohesin can reproduce some, but not all, of the chromosome organization phenomena observed in yeast. The lower degree of mitotic chromosome compaction and formation of chromatin domains without dots is consistent with one-sided extrusion by yeast SMC complexes. However, pure one-sided extrusion alone is insufficient to form dots in Hi-C and micro-C. Consistent with single-molecule experiments, budding yeast condensins could be one-sided, but then cohesins must be two-sided or effectively two-sided in order to generate Hi-C patterns in quiescent cells. In metaphase, budding yeast cohesins may be one-sided extruders, but their interphase activity during exponential growth requires two-sided or effectively two-sided extrusion.

74Our work identifies two requirements for loop extrusion by SMC complexes to generate known chromosome structures. First, unlooped chromatin gaps between SMC complexes must be closed in order to compact mitotic chromosomes, and they occasionally must be closed between extrusion barriers during interphase to generate enrichment of CTCF-CTCF interactions. Second, particularly in prokaryotes, we find that extrusion must be two-sided or effectively two-sided in order to juxtapose bacterial chromosome arms. Although we studied the switching model in detail, we note that several molecular mechanisms can give rise to such effectively two-sided, gap-closing extrusion. Based on the available experimental evidence, we also considered several physical factors and additional models, discussed below.

75Whether loop extrusion can compact and resolve chromosomes within physiological limits is a persistent question for chromosome organization in higher eukaryotes. Previous work on two-sided loop extrusion showed that LEFs can compact and resolve metazoan chromosomes (~100 Mb in length) for physiological densities of LEFs (1 per d = 10–30 kb). Compaction and resolution are completed within a few (~5) residence times (1/kunbind ~ 2–10 min), provided that extrusion is fast, i.e., v > 0.2 kb/s. The extrusion rate of v≈1 kb/s recently observed in vitro confirms that loop extrusion is sufficiently rapid to compact metazoan chromosomes during prophase and prometaphase. Moreover, this rate is consistent with expectations from studies of the molecular dynamics of loop-extruding SMC complexes.

76Furthermore, we can estimate an upper bound on the energy required to compact human chromosomes. Conservatively estimating that condensin or cohesin require two ATP per extrusion step and several attempts to traverse each nucleosome (~150 bp), the ATP cost to extrude 6 Gb is of order 10 x (6 × 109/150)~108 (we assume only ~5 attempts because in vitro extrusion speeds are not measurably altered by nucleosomes). This upper limit estimate is still less than the ~109 ATP present in the cell and less than the ~109 ATP/s that the cell produces. Moreover, there are only ~105 cohesins and condensins in each living cell; given an ATPase rate of ~1 s−1, we estimate that the rate of actual energy consumption by loop extrusion is ~105 s−1, well within the cell’s energy budget. We conclude that genome compaction and organization by loop extrusion is energetically feasible.

77It has previously been suggested that 3D attractive interactions between LEFs could facilitate compaction of mitotic chromosomes. For mitotic chromosomes, our results, along with previous work on polymer combs, suggests otherwise. It is possible that SMC complexes may attract each other, but such interactions must be weak enough that the chromosome does not collapse into a spherically symmetric polymer. With weak interactions, however, gaps created by one-sided extrusion cannot be closed, and mitotic chromosomes cannot be formed (Figure 2—figure supplement 5). Thus, 3D interactions cannot be the mechanism of chromatin gap closure, and thus, they cannot be essential for mitotic chromosome compaction. For interphase chromosomes, 3D attractions between TAD boundaries (CTCF proteins or their binding sites) could potentially close chromatin gaps. However, 3D attractions would not consistently pair CTCF boundaries in a convergent orientation nor would they distinguish between proximal and distal TAD boundaries. Furthermore, for both mitotic and interphase chromosomes, attractive 3D interactions would promote trans interactions, contrasting with in vivo observations of condensin-mediated spatial resolution of mitotic chromosomes and cohesin-driven formation of cis loops. All of these points suggest that one-sided loop extrusion together with random cross-bridging of chromatin/DNA segments as in several previous studies is not sufficient for compaction and domain formation.

78We considered the possibility that interactions between one-sided LEFs and other LEFs or protein factors might alter their residence times, which might facilitate chromosome organization. In simulations of mitotic chromosomes, we found that alterations to LEF residence times due to LEF-LEF interactions do not enhance linear fold compaction (Figure 2—figure supplement 6).

79Translocation along DNA by loop-extruding complexes often proceeds in the presence of RNA polymerases that actively translocate as they transcribe genes. We therefore evaluate whether active transcription can help one-sided loop extrusion become effectively two-sided extrusion, or otherwise promote the chromosome organization scenarios studied above. As discussed above, modeling of condensins and RNA polymerases on bacterial chromosomes, along with experimental evidence for other cell types, suggests that translocating RNA polymerases can push translocating SMC complexes, and thus alter chromosome organization. While transcription can occur during mitosis, inhibiting transcription does not visibly alter mitotic chromosome compaction. Furthermore, only condensin and a few other protein factors are required to form mitotic chromosomes in vitro. Therefore, pushing of condensins by RNA polymerases cannot be the primary mechanism underlying the predicted requirement for effectively two-sided loop extrusion in mitosis. In contrast, formation of cohesin-dependent dots between convergent genes in budding yeast Hi-C requires either effectively two-sided extrusion by cohesin or a hypothetical mechanism in which one-sided extrusion is assisted by transcription: one-sided cohesins could become effectively two-sided if RNA polymerase (translocating at v ~ 1 kb/min) is able to efficiently push the passive side of the cohesin complex (Figure 3—figure supplement 8). This assistance would further require specific orientations of multiple genes (Figure 3—figure supplement 8). In bacteria (B. subtilis and C. crescentus), the ability of bSMCs to juxtapose chromosome arms is largely unaffected by transcription inhibition. Additionally, pushing of bSMCs by RNA polymerases cannot drive chromosome arm juxtaposition because genes are not universally transcribed from ori to ter; as such, RNA polymerase together with one-sided extrusion would be unable to juxtapose the entire length of two chromosomal arms. Furthermore, condensin seems to be able to traverse highly transcribed genes within mere seconds. Thus, for bacteria, transcription is also not an essential driver of effectively two-sided loop extrusion. Altogether, transcription cannot be the driving force of metazoan mitotic chromosome compaction and bacterial chromosomal arm juxtaposition, but it could help drive effectively two-sided, but asymmetric, extrusion by cohesins in yeast in some specific scenarios.

80It has previously been proposed that SMC complexes with purely diffusive subunits might organize interphase TADs. In this model, cohesins with two diffusive subunits are loaded at a loading site. Osmotic pressure arising from the successive loading of multiple cohesins at the loading site biases loop growth such that boundary elements (i.e., CTCFs) may be brought together. However, targeted loading of LEFs in vertebrate cells has not been observed. Moreover, our modeling shows that even a semi-diffusive model fails to compact and resolve mitotic chromosomes (Figure 2d), generate TADs with dots (Figure 3d, left), or juxtapose bacterial chromosome arms (Figure 4c). Consistently, previous modeling demonstrated that slip links could only juxtapose bacterial chromosome arms at unphysiologically high densities. Thus, diffusive slip links are not sufficient to account for various chromosome organization phenomena.

81SMC complex oligomerization could facilitate chromosome organization by suppressing gap formation and/or promoting symmetric extrusion in various scenarios. In eukaryotes, in situ amino acid crosslinking and in vitro gel filtration suggest that condensins can oligomerize. Several experiments similarly suggest that cohesin may form oligomeric complexes in vitro or in vivo. Formation of such complexes could lead to effectively two-sided extrusion and gapless chromosome compaction. In prokaryotes, such as E. coli (which have MukBEF complexes, SMC complex homologs), experiments show that MukBEF forms dimers of complexes linked by the kleisin molecule, MukF. MukBEF complexes promote long-ranged contacts within E. coli chromosome arms, and they are proposed to function by two-sided loop extrusion. Dimerization has also been suggested for other bacterial SMC complexes, but it is still unknown whether bSMCs in well studied organisms like C. crescentus and B. subtilis dimerize in vivo. Functional dimerization of bSMCs in vivo could be directly tested by photobleaching experiments with endogenous fluorescently tagged versions of bSMC, as in. Additionally, to determine whether MukBEF dimerization is needed for DNA loop formation, we suggest a Hi-C experiment on a MukBEF mutant deficient in dimerization. If long-ranged chromosome interactions and proliferation under fast-growth conditions persist, then dimerization is not required for MukBEF function. These experiments could therefore investigate the possible functional role of SMC complex oligomerization in loop extrusion.

82Recent single-molecule experiments have reported the first observations of two-sided and effectively two-sided loop extrusion. It has been shown that ~ 80% of human condensin I and ~50% of human condensin II complexes perform two-sided DNA loop extrusion in vitro. This finding suggests that human condensins in vivo might satisfy constraints predicted by previous theory and new simulations (Figure 2—figure supplement 1), which show that ~ 85% of LEFs must be two-sided in order to achieve 1000-fold linear chromatin compaction and robust 3D compaction of mitotic chromosomes. Similarly, recent single-molecule experiments observe mostly two-sided extrusion by human and Xenopus cohesin, which is consistent with our finding that > 50% two-sided extrusion is needed to reproduce the ‘dots’ that reflect elevated CTCF-CTCF contact frequency in interphase (Figure 3d, right).

83Other single-molecule experiments have shown that yeast condensins can traverse each other, which in turn may act as effectively two-sided extruders. We simulated and analyzed a simple realization of this scenario, in which condensins, cohesins, or bSMCs can pass each other as they translocate along DNA/chromatin. For simulations of mitosis, this leads to loop coverage that increases exponentially with λ/d and compacted rod-like chromosomes (Figure 2f–g). However, our model with LEFs traversing each other generates many pseudoknots, and thus, linear spatial ordering of the mitotic chromosome is not maintained on length scales comparable to the loop size, ℓ≈λ, which may be >100 kb (as estimated from measured condensin speed and turnover rate). For interphase simulations, the ability of cohesins to pass each other increases the strength of dots (corner peaks) as compared to pure one-sided extrusion. However, dots are not as strong as they are with two-sided extrusion, and they only appear for a high cohesin densities and/or processivities (Figure 3—figure supplement 7). In contrast, LEF traversal does not facilitate juxtaposition of bacterial chromosome arms because the one-sided LEFs maintain contacts between the origin and distal regions of the chromosome (Figure 4—figure supplement 4).

84Moreover, several questions remain about the in vivo relevance of LEF traversal, and the formation of the ‘Z-loop’ structure. We assumed that each LEF may traverse any other LEF that it encounters, but it is unknown how SMCs contributing to Z-loop structures actually interact. A more restrictive set of traversal rules could severely limit linear compaction and corner peak formation. For example, if each active subunit can only traverse a single anchored subunit, then linear compaction is limited to 50-fold (following arguments for the ‘weak pushing’ model, see Appendix 1 and Figure 2—figure supplement 3). In addition, it is unknown how Z-loop formation is altered when condensins or cohesins extrude chromatin instead of DNA. Thus, while our preliminary modeling suggests that effectively two-sided extrusion by Z-loops might compact mitotic chromosomes and pair CTCF sites, a number of experimental and theoretical factors remain unexplored.

85In Table 1, we list possible mechanisms of loop extrusion and whether they are able to reproduce in vivo experimental observations; however, many of these mechanisms have not yet been observed or tested. Single-molecule experiments could assay different types of SMC complexes from a range of organisms in order to establish which loop extrusion models are applicable. We predict that SMC complexes in vivo may constitute effectively two-sided motors or exhibit biased loading in order to robustly organize and compact chromatin. However, a variety of microscopic (molecule-level) modes of extrusion may achieve the same macroscopic organization of the chromosomal DNA.

86We make several testable predictions. First, if switching of extrusion direction is observed, switching should be fast (occurring at least once per 10 s for bSMCs and at least once per minute for human SMC complexes cohesins and condensins). In addition, we predict that if a mixture of one-sided and two-sided extrusion is observed for a population of SMC complexes, then the fraction of two-sided extrusion should be at least 50% for cohesin and at least 80% for condensin (Table 1). We also predict that bSMCs from eubacteria are either two-sided monomeric complexes or a dimer of complexes that translocate in opposing directions, enlarging a loop and resulting in two-sided extrusion.

87A few other types of experiments are critical to perform at the single-molecule level in vitro; these would be difficult to test in vivo by microscopic and biochemical methods. We suggest: 1) testing how SMC complexes interact with one another when they meet on the same chromatin/DNA substrate in vivo, as we show that LEF traversal can lead to effective compaction; 2) testing whether/what fraction of SMC complexes do one-sided or two-sided extrusion under different conditions, such as at various salt concentrations and/or with molecular crowding agents; and 3) testing whether specific factors, such as chromatin conformations (e.g., supercoils or Holliday junctions) or proteins (e.g., other SMC complexes or CTCF), affect mechanisms of extrusion.

88Finally, we note that there may be differences in functionality among condensins of different species or physiological scenarios. For example, it has been hypothesized that yeast condensins could be one-sided because they do not need to linearly compact mitotic chromosomes 1000-fold. If yeast condensin is fundamentally different from human condensin in function, its use in cell-free chromosome assembly systems should result in long, poorly folded chromosomes relative to those with condensin II only. Similarly, mutations that bias condensin activity towards one-sided extrusion could lead to catastrophic under-compaction of human chromosomes, failure to decatenate chromosomes, DNA damage, aneuploidy, developmental disorders, and cancer.

89The loop extrusion model has been hypothesized to explain a variety of chromosome organization phenomena, but until recently had remained a hypothesis. Experimental work on yeast condensins has observed that loop extrusion by yeast condensins occurs in a one-sided manner. Theory and simulations of one-sided loop extrusion challenge the generality of this observation. We have shown that pure one-sided loop extrusion generally is unable to reproduce a variety of chromosome organization phenomena in different organisms and scenarios. Instead, loop extrusion should be ‘effectively two-sided’ and/or have the ability to robustly eliminate unlooped chromatin gaps to organize chromosomes; in accord with this, recent experimental data indicate that human condensins and human and Xenopus cohesins are capable of acting in a two-sided manner. Additionally, among the models we explored, the switching model is an example that meets these requirements. Nonetheless, experimental evidence suggests that different organisms are likely to achieve macroscopic chromosome organization through diverse microscopic mechanisms. While loop extrusion remains a unifying model for chromosome organization across different domains of life, various to-be-determined microscopic mechanisms could underlie these phenomena.

90Stochastic simulations of loop-extrusion dynamics are performed with N LEFs on a lattice of length L. There are several types of events. LEFs bind to the chromatin lattice at rate kbind by occupying two adjacent lattice sites and LEFs unbind at rate kunbind. When an active subunit of a LEF makes a step, it occupies the site that was immediately adjacent to it, which frees the lattice site that it previously occupied. Directional stepping by an active subunit occurs at speed v and proceeds in the direction away from the other LEF subunit. Diffusive stepping occurs in either direction at loop-size-dependent rate v±diff(ℓ). When a one-sided LEF switches its active extrusion direction, the active subunit becomes passive and vice versa. Switches occur at a rate kswitch. In interphase simulations, LEF subunits may stall upon encountering a correctly oriented CTCF site. This occurs with probability pstall. Each simulation consists of a chromatin polymer with L sites and a fixed number, Nb, of LEFs that populate the sites at low density, Nb/L ≤ 0.05. The simulation code is publicly available at https://github.com/mirnylab/one_sided_extrusion (; copy archived at https://github.com/elifesciences-publications/one_sided_extrusion).

911D stochastic simulations of loop-extrusion dynamics modeling mitotic chromosome compaction for pure one-sided, two-sided, switching, and pushing models are performed with N LEFs on a lattice of length L, with L = 60000 sites and 100 < N < 3000. Each site is taken to be a = 0.5 kb.

92We use the Gillespie algorithm to determine the time that each kinetic event -- binding, unbinding, directional stepping, and switching -- occurs. Events are executed in temporal order, and after an event occurs, we compute the lifetimes of new events that become permissible (e.g., a LEF step that becomes possible because another LEF has moved). Simulations are run for tsim = 400 max((1/kunbind+1/kbind), L/v+1/kbind), and data is recorded for the second half of the simulation, long after the onset of the steady-state, for at least three simulations per parameter combination.

93For 1D simulations of chromosome compaction in the semi-diffusive model, 1D simulations of compaction with LEF traversal, 3D polymer simulations of chromosome compaction with all models, interphase TAD formation, and 1D simulations of LEF dynamics on bacterial chromosomes, we use a fixed-time-step Monte Carlo algorithm instead of the Gillespie algorithm. This algorithm facilitates coupling of LEF kinetics to the loop architecture (for the semi-diffusive model) and/or 3D polymer conformation (for polymer simulations). Here, each event is modeled as a Poisson process; at each LEF time step dt, an event is executed with probability kidt, where ki is the rate of event i. In the semi-diffusive model, the passive diffusive stepping rate for a LEF is v±diff(ℓ)=vdiff e∓(3/2) (a / ℓ), which is updated when the size of either the loop associated with the LEF or any loop in which the LEF is nested changes in size. The expression for v±diff(ℓ) is a discretization of v±diff(ℓ)=vdiff e∓f a / kT. Here, f = -dU/dℓ = (3/2) kT ln(ℓ/a) defines the entropic force arising from loop configurational entropy (e.g., see.

94For fixed-time-step simulations of mitotic chromosomes, L = 30000, N = 750, and a = 0.5 kb, which is assumed to be 30 nm in diameter (~3 nucleosomes). At least three simulations per parameter combinations are run for >40 residence times, and linear compaction is measured after 20 residence times. Probe radius rhull = 600 nm was used to calculate concave hulls.

95For simulations of interphase, we simulate a chain with three different TAD sizes of 100, 200, and 400 monomers. This system of 700 monomers in total is repeated 6 or eight times, giving a total size of 4200 monomers (for computing dot strengths) or 5600 monomers (for computing contact maps and scalings). When LEFs encounter a CTCF site, they are stalled (i.e. they stop moving until they are unloaded), with a probability of 80%. From the scalings, we determined that one monomer corresponds to 2 kb (Figure 3—figure supplement 1).

96We used a total of 4000 conformations to compute contact maps, scalings or dot strengths. For computing the contact maps, we used a contact radius of 5 monomers. Dot strengths are computed as follows: first, we compute observed-over-expected of a contact map (we divide out the distance dependence, by dividing each diagonal by its average), then we compute the strength of a dot of a particular TAD (Figure 3—figure supplement 2) and last, we compute the average of all the dots (each of which appears six times on one map).

97In contrast to mitotic compaction, λ and d are varied separately for interphase chromosomes, because the dot strengths depend on λ and d separately, as well as the distance between two CTCF sites, dCTCF. Based on contact probability scalings (Figure 3—figure supplement 1) and experimental observations, we consider a separation between loop extruders of d = 200 kb and a processivity of λ = 200 kb in the main text, and we consider other parameter values in the figure supplements. Furthermore, we choose typical TAD sizes of 200 and 400 kb. For simulations of Wapl depletion conditions, we use d = 200 kb and λ = 2 Mb.

98We simulate loop extrusion on bacterial chromosomes using the fixed-time-step simulations for LEF dynamics described above. LEFs are allowed to randomly load on a lattice of L = 4000 sites, where each lattice site corresponds to ~1 kb of DNA. LEFs have a strong bias to bind one site at the center of the lattice to mimic the effect of a single parS site near the origin of replication in bacterial chromosomes. The relative probability of loading at the simulated parS site was ~40,000 times stronger than that of every other site, i.e., if the relative probability of loading at the simulated parS is 1, then the total relative probability to load on any other site is 0.1 L. As a result, the overall preference to bind the parS site over all other genomic loci is approximately 10-fold.

99Bacterial LEFs were simulated as deterministic extruders with a stochastic dissociation rate kunbind = 2/L to approximate the steady decrease in bSMC density away from the ori observed via ChIP-seq (i.e., bSMC density at the ter region is ~1/3 of the value at ori). In addition to a stochastic (position-independent) dissociation rate, LEFs automatically unbind if one of the subunits reached the edge of the lattice, i.e., the ter region; ter was set to lattice positions 0–3 and 3996–3999 (i.e., diametrically opposite to the parS site at lattice site 2000).

100To model the 3D dynamics of polymers loaded with LEFs, we performed polymer molecular dynamics simulations in OpenMM using a custom, publicly available library, openmm-polymer (available at https://github.com/mirnylab/openmm-polymer-legacy;, coupled with the fixed-time-step LEF simulations described above and in;.

101In the polymer simulation, a LEF crosslinks the sites that it occupies together. LEF positions are evolved as described above. After each time step of LEF dynamics, the polymer simulation is evolved via Langevin dynamics for 200 or 250 time steps (for interphase and mitosis, respectively) with dt = 80.

102Polymers are constructed of L consecutive subunits bonded via the pairwise potential:where r = ri rj is the displacement between monomers i and j, k = 2 kT / δ2 is the spring constant, δ = 0.1, and b is the diameter of a monomer. For mitotic chromosome simulations, b = 30 nm; for other scenarios, it is unnecessary to assign a value to b. Monomers crosslinked by a LEF are held together by the same potential. Weakly repulsive excluded volume interactions between monomers are modeled as:for r<σ with σ=1.05b,, =46656/823543, and =1.5 kT. For simulations of mitotic chromosomes with 3D attractive interactions, monomers interact through the potential:for σ < r < 2b and is a parameter to be varied.

103At the beginning of each simulation, the polymer is initialized as a random walk and monomers are initialized with normally distributed velocities, so that the temperature is T. The system is thermostatted by intermittent rescaling of velocities to maintain temperature T.

104To compute contact maps for bacterial chromosomes, the contact frequency was calculated from the equilibrium contact probability for a Gaussian chain. This theoretical model agrees well with polymer molecular dynamics simulations (Appendix 3 and Figure 4—figure supplements 6, 7). Briefly, contact probability between two sites on a Gaussian chain scales with s-3/2, where s is the linear distance between the sites, excluding any loops between the two sites. Sites within the same loop obey this scaling relation with an effective s, seff, substituted for s in the scaling relation; seff = s(1 s/ℓ), where ℓ is the loop size. For sites in different loops, s in the scaling relation is replaced by the sum of the effective lengths of the regions connecting the two sites (see Appendix 3 for details). These relative contact probabilities are used to compute the contact maps for bacterial chromosome simulations. Contact maps are generated using contacts from 50,000 to 100,000 different LEF conformations.