In the model, LEFs representing SMC complexes perform loop extrusion on a polymer fiber representing the chromosome (Alipour and Marko, 2012; Goloborodko et al., 2016b; Fudenberg et al., 2016; Banigan et al., 2020). Each LEF is composed of two subunits or ‘sides’, which may have different translocation abilities, so the entire LEF may be either symmetric or asymmetric. A subunit may be either active or inactive. An active subunit processively translocates at speed v along the polymer fiber, thus creating and enlarging the polymer loop between the subunits. In this work, inactive subunits are immobile. We refer to LEFs with two active subunits as ‘two-sided’. LEFs with one active subunit and one inactive subunit are referred to as ‘one-sided’. Each LEF subunit is assumed to act as a barrier to the translocation of other LEFs so that an active subunit cannot pass through another LEF subunit. Thus, pseudoknots or ‘Z-loops’ (Kim et al., 2020) are prohibited; the scenario in which one-sided LEFs may traverse each other has been considered previously (Banigan et al., 2020) (moreover, compaction by mixtures of one- and two-sided LEFs in that model would simply rescale the mean processivity compared to the pure one-sided LEF model). LEFs bind with equal probability to any location on the polymer fiber, following a previous analysis of condensin localization that found condensin loading to be largely sequence-independent (Gibcus et al., 2018). Furthermore, each one-sided LEF has two possible binding orientations (← or →), which determines the direction in which extrusion proceeds along the polymer. Following in vitro experiments (Ganji et al., 2018; Golfier et al., 2020) and lacking molecular evidence that binding to chromatin in a particular orientation could be favored, the extrusion orientation for each LEF is selected randomly. Each LEF stochastically unbinds at rate k, which releases the corresponding polymer loop. Altogether, these LEF dynamics lead to a dynamic statistical steady state in which loops formed by LEFs stochastically appear, grow, and vanish. The steady-state fold linear compaction, ℱ⁢𝒞, is calculated from the fraction, f, of the fiber that is extruded into loops as ℱ⁢𝒞=(1-f)-1 (Banigan and Mirny, 2019). This quantity primarily depends on the ratio, λ/d, of the processivity (λ=p⁢v/k, where p=1 or 2 for one- or two-sided LEFs, respectively, and in mixtures, λ denotes the population-averaged processivity) to the mean separation (d) between LEFs (Goloborodko et al., 2016b; Banigan and Mirny, 2019). Further details and a public link to the simulation code are provided in the Materials and methods section.

Previous computational models of loop extrusion generally assumed that all LEFs have the same average unbinding rate and that all active LEF subunits translocate at the same average speed (Alipour and Marko, 2012; Sanborn et al., 2015; Goloborodko et al., 2016b; Fudenberg et al., 2016; Miermans and Broedersz, 2018; Banigan and Mirny, 2019; Banigan et al., 2020). Since experiments observed SMC complex dynamics that are contrary to these assumptions, we consider models in which these assumptions are relaxed. In the Results section, we first present results for systems with LEFs that perform asymmetric two-sided loop extrusion. Second, we investigate mixtures of one- and two-sided LEFs with different extrusion velocities, mixtures of one- and two-sided LEFs with different residence times, and mixtures of only one-sided LEFs with different residence times. Third, we specifically consider mixtures of LEFs with the properties measured for condensins I and II in experiments. Additionally, the simulation results for mixtures of one- and two-sided LEFs are explained by theoretical arguments, which are presented in detail in Appendix 1.

To determine the ability of asymmetric two-sided loop-extruding condensins (Golfier et al., 2020) to compact mitotic chromosomes, we simulated an asymmetric variant of the two-sided loop extrusion model. In these simulations, each LEF has two active subunits. One of these is a fast subunit that extrudes at speed vfast, while the other is a slow subunit that extrudes at speed vslow. LEFs unbind from the chromatin polymer fiber at rate k=1/τ, where τ is the mean residence time. Thus, the mean processivity (i.e. how large a loop a LEF can extrude before unbinding) is λ=(vfast+vslow)⁢τ. A prototypical trajectory, a schematic illustration, and an arch diagram are shown in Figure 1a.

Eliminating or avoiding unlooped gaps between LEFs is critical to achieving a high degree of linear compaction (Banigan and Mirny, 2019; Banigan et al., 2020). One-sided LEFs cannot close all gaps because one quarter of all pairs of neighboring LEFs are in a divergent orientation (←→); thus, they extrude loops by collecting chromatin from opposite directions, while leaving the chromatin between the LEFs unlooped (Banigan and Mirny, 2019). In contrast, at sufficiently high processivities (λ) and linear densities (1/d), symmetric two-sided LEFs eliminate unlooped gaps (Goloborodko et al., 2016b; Banigan et al., 2020). Based on the idea of closing gaps between LEFs, we previously argued that asymmetric two-sided extrusion could fully linearly compact mitotic chromosomes provided that the residence time is sufficiently long; in particular, we require λslow/d≫1(Banigan and Mirny, 2019; Figure 1b).

In simulations, we controlled asymmetry by varying the relative extrusion speeds of the active subunits, quantified by the dimensionless ratio vslow/vfast. For all cases of asymmetric two-sided extrusion (i.e. vslow>0) over the simulated range (10-4≤vslow/vfast<1), chromosomes can be linearly compacted 1000-fold, provided that λfast/d (and thus, λslow/d) is sufficiently large (Figure 1c). Specifically, fold compaction, ℱ⁢𝒞, grows rapidly for λslow/d>1 (Figure 1—figure supplement 1a), as predicted. This occurs because gaps between LEFs are eliminated even if the gap is bordered by two slow LEF subunits (Figure 1b and Figure 1—figure supplement 1b). As shown in Figure 1c, 1000-fold compaction can be achieved for all asymmetries of two-sided LEFs, notably including the asymmetries and λ/d in the range of expected physiological values (see below and Materials and methods).

We next determined whether 1000-fold linear compaction can be achieved with the asymmetries (vslow/vfast) observed in single-molecule experiments (Golfier et al., 2020) and previously estimated values of λ/d for condensin (Goloborodko et al., 2016b; Banigan et al., 2020). We performed three sets of simulations with asymmetric two-sided LEFs (see Materials and methods): (1) simulations with a 50/50 mixture of asymmetric two-sided LEFs and one-sided LEFs, which has the same distribution of ‘symmetry scores’ (S=(vfast-vslow)/(vfast+vslow)) as measured for Xenopus condensins by Golfier et al., 2020, (2) simulations with only asymmetric two-sided LEFs that also reproduce the symmetry score distributions measured by Golfier et al., 2020, and (3) simulations with asymmetric two-sided LEFs that can both grow and shrink loops according to our reanalysis of the experiments of Golfier et al., 2020.

These three scenarios lead to two qualitatively different outcomes. The mixture of asymmetric two-sided LEFs and one-sided LEFs (scenario 1) can achieve only ~60-fold linear compaction (blue circles in Figure 1d). Because half of the LEFs are one-sided in this scenario, a relatively large number of unlooped gaps remain in steady state, which limits linear compaction. In contrast, simulations with different distributions of asymmetric two-sided extrusion (scenarios 2 and 3) do not have this limitation, which results in >1000-fold linear compaction for plausible values of λ/d (<1000) (red squares and purple triangles in Figure 1d). Although a significant fraction of LEFs are highly asymmetric (≥20% of LEFs with vslow<0.1⁢vfast), they are typically able to close gaps within their residence times. Thus, we conclude that even highly asymmetric two-sided LEFs can close gaps and compact chromosomes, while a modest amount of strictly one-sided LEFs significantly inhibits compaction.

It is unclear how a condensin complex could perform asymmetric two-sided extrusion, while still compacting chromosomes in vivo. For instance, diffusive sliding of one side combined with active translocation by the other is unable to form large DNA loops in vitro (Ganji et al., 2018), and it is insufficient to consistently eliminate chromatin gaps and achieve 1000-fold compaction in simulations (Banigan et al., 2020). Instead, complete compaction requires directed translocation to collect chromatin from both sides of the condensin complex. Directed translocation could be governed by one or more of several mechanisms.

One possibility is that asymmetric two-sided extrusion could occur due to asymmetry that is intrinsic to the complex. The degree of asymmetry of a condensin could be controlled by the species of the kleisin or Hawks (HEAT proteins associated with kleisins) associated with the complex. Consistent with this hypothesis, it has been observed that the yeast kleisin Brn1 and Hawk Ycg1 can act as a ‘safety belt’ that anchors condensin to DNA (Kschonsak et al., 2017). However, anchoring by the safety belt combined with condensin translocation (Terakawa et al., 2017) generates pure one-sided loop extrusion (Ganji et al., 2018), which is incompatible with the high degree of mitotic chromosome compaction observed in metazoan cells (Banigan and Mirny, 2019). Conceivably, a looser safety belt combined with the ability of the complex to perform two-sided extrusion could generate asymmetric two-sided extrusion in higher eukaryotes.

Alternatively, complexes performing asymmetric two-sided extrusion could be dimers of condensins, for which asymmetries could arise by several mechanisms. The extruding complex could contain condensins with two different protein compositions. Each side of the resulting complex might have different extrusion dynamics. However, this possibility is diminished by the strong tendency for particular kleisins and HAWKs to associate together (i.e., form either condensin I or II) (Ono et al., 2003) and the different patterns of spatial localization of these proteins (Ono et al., 2003; Shintomi et al., 2017; Walther et al., 2018). A more likely possibility is that the compositions of the condensins within a dimer are identical, but asymmetry arises if the components are differentially regulated by post-translational modifications. This would be consistent with the observation of asymmetric two-sided extrusion by Xenopus condensins in their native context (Golfier et al., 2020). Yet another possibility is that dimerization of condensins generates an asymmetric complex. This could arise due to chirality within the joined molecules or through differential conformational changes that are required to form the dimer. These scenarios are not mutually exclusive; several of these mechanisms could act together to generate asymmetric two-sided extrusion.

What could give rise to the predicted stability of two-sided SMC complexes? One hypothesis is that condensin I performs one-sided extrusion, while condensin II performs two-sided extrusion (Banigan and Mirny, 2020). The mean residence time of dynamic condensin II is about three times longer than that of condensin I, and there is an immobile subpopulation of condensin II with a much longer residence time (Gerlich et al., 2006a; Walther et al., 2018). Thus, two-sided complexes (condensin II in this scenario) would be long-lived compared to one-sided complexes (condensin I). Such a mixture of condensins I and II could generate 1000-fold chromosome compaction (Figures 2 and 4c). Further supporting this hypothesis, yeast condensin, which is evolutionarily conserved as condensin I (Hirano, 2012), performs one-sided loop extrusion in vitro (Ganji et al., 2018). Thus, it is appealing to think that condensin II is a two-sided counterpart to one-sided condensin I in metazoan cells.

However, single-molecule experiments with condensins I and II suggest otherwise, at least for human cells (Kong et al., 2020). It has been observed that human condensins I and II can both perform either one-sided or two-sided extrusion in vitro. Moreover, two-sided extrusion events are observed more frequently for condensin I as compared to condensin II. Interestingly, photobleaching experiments show that two-sided loop-extruding condensin I complexes are dimers of condensin I molecules (Kong et al., 2020). This observation suggests that perhaps individual condensins are one-sided extruders, but they can dimerize to form two-sided loop-extruding complexes.

Taking the in vitro experiments (Kong et al., 2020) together with our simulation results (Figure 2e and Figure 3b), we propose that condensin II complexes dimerize to form two-sided loop-extruding complexes with very long residence times. Dimerization could facilitate two-sided extrusion by combining two one-sided complexes, while also stabilizing binding of condensin II to DNA by increasing (or otherwise modifying) the condensin-DNA binding surface. This combination of effects would naturally generate a tight coupling between two-sided extrusion and long residence times. Thus, dimerization could simultaneously allow two-sided extrusion and increase the residence time. This combination would generate the necessary conditions for chromosome compaction by mixtures of one- and two-sided condensins.

Metazoan chromosomes are composed of large (~400 kb) chromatin loops with condensin II at their bases, with smaller (~80 kb) loops mediated by condensin I nested within (Gibcus et al., 2018; Walther et al., 2018). Consequently, condensin II is tightly localized to the central axis of the rod-like chromosome, while condensin I is localized along the axis in a broader pattern (Ono et al., 2003; Shintomi et al., 2017; Walther et al., 2018).

This hierarchy of loop nesting naturally emerges from the loop extrusion model for mixtures of LEFs with two different residence times. LEFs with longer residence times, τ, have a larger processivities, λ=v⁢τ, and thus tend to form larger loops. In 3D, the bases of these loops localize along a central axis (Goloborodko et al., 2016a; Gibcus et al., 2018). This hierarchy emerges in mixtures of LEFs with the properties of condensins I and II (Figure 3—figure supplement 1), mixtures of one- and two-sided LEFs (Figure 2b, bottom left), mixtures of two populations of one-sided LEFs (Figure 2—figure supplement 4e), and mixtures of two populations of two-sided LEFs (Gibcus et al., 2018). While not all combinations of LEF symmetries and dynamics can fully compact mitotic chromosomes (two-sided extrusion is required; Figure 4), differences in residence times can generate the nesting structure required for the patterns of spatial localization of condensins I and II observed in vivo. Consistent with this idea, the residence time of condensin II in vivo is longer than that of condensin I (Gerlich et al., 2006a; Walther et al., 2018).

Furthermore, our results suggest specific properties of condensins that could regulate mitotic chromosome morphology. Previously, it was shown that mitotic chromosome morphology can be controlled by the relative ratio of condensin I to condensin II (Shintomi and Hirano, 2011). Generally, condensin II is responsible for lengthwise ‘axial’ compaction along the central axis of a chromatid, while condensin I drives ‘lateral’ compaction, reducing the width of a chromatid (Ono et al., 2003; Shintomi and Hirano, 2011; Green et al., 2012; Bakhrebah et al., 2015; Shintomi et al., 2017; Hirano, 2016; Kalitsis et al., 2017). We find that morphology and the degree of linear compaction could alternatively be controlled by the relative residence times of condensins I and II and the fraction of condensins that perform one-sided (instead of two-sided extrusion). For example, a perturbation (such as a post-translational modification) that increases the residence time of condensin II could lead to greater lengthwise compaction due to the larger loops that would be formed by condensin II. Similarly, perturbations that promote two-sided extrusion, especially by condensin II, could also increase the degree of lengthwise compaction. Moreover, molecular perturbations affecting the coupling between symmetry and residence time could have a significant effect on compaction and morphology. These types of biomolecular perturbations would enable cells to regulate chromosome compaction without requiring global changes to levels of condensin expression.

In the loop extrusion model, LEFs can linearly compact chromosomes by extruding chromatin into loops within approximately one LEF residence time. Chromosome morphology, as quantified by loop sizes, equilibrates over longer timescales of approximately 5–10 residence times (Goloborodko et al., 2016a; Goloborodko et al., 2016b; Banigan et al., 2020). In mixtures of LEFs with different dynamics, there are multiple timescales that could govern compaction kinetics. Linear compaction into a series of loops may still occur within ~1–2 residence times of the short-lived LEFs, but equilibration of large loops formed by the long-lived LEFs will occur over several residence times of the long-lived LEFs.

With the experimentally measured residence times of condensins I and II (Gerlich et al., 2006a; Walther et al., 2018), loop extrusion could compact chromosomes into loop arrays within a few minutes during prophase (Gibcus et al., 2018). However, loops should continue to merge and grow as mitosis progresses since the residence time for stably bound condensin II is comparable to the duration of mitosis (Gerlich et al., 2006a; Walther et al., 2018). Consequently, chromosomes should axially shorten and undergo 3D compaction as mitosis progresses, as observed in vivo (Nagasaka et al., 2016; Gibcus et al., 2018). Simultaneously, chromosomes should be further compacted by progressive loading of condensins, which increases throughout mitosis (Walther et al., 2018) and hyper-compacts chromosomes when mitosis is stalled (Sun et al., 2018). Thus, even with mixtures of condensins with long residence and equilibration times, 1000-fold linear compaction is achievable during mitosis.

The principles for mitotic chromosome compaction developed here are generalizable to other chromosome organization scenarios. In particular, we previously showed that the physical principles underlying mitotic chromosome formation by loop extrusion are relevant for topologically associated domain (TAD) formation in interphase (Banigan et al., 2020), where extrusion by the SMC complex cohesin can be paused by barriers such as CTCF (Sanborn et al., 2015; de Wit et al., 2015; Fudenberg et al., 2016; Busslinger et al., 2017; Nora et al., 2017; Wutz et al., 2017). Thus, we expect that asymmetric two-sided LEFs and mixtures of LEFs might be able to form TADs. There are similar requirements for juxtaposition of bacterial chromosome arms by bacterial SMC complexes, but there also are additional constraints due to the specific loading site near the origin of replication (Banigan et al., 2020). Accordingly, asymmetric two-sided LEFs and mixtures of LEFs with different residence times might be able to form TADs but not be able to juxtapose chromosome arms.

Two ingredients are essential for mitotic chromosome compaction by condensins: sufficiently long residence times and some amount of either symmetric or asymmetric (effectively) two-sided extrusion. Strikingly, the presence of even a small (~20%) fraction of such condensins in a mixture with other, purely one-sided condensins could be sufficient to achieve compaction under physiological conditions. It remains to be determined what mechanisms are responsible for different types of extrusion dynamics and what factors might facilitate a transition from one-sided to two-sided loop extrusion in some molecules in vivo. However, our analysis suggests that two-sided extrusion should be tightly coupled to stable chromatin binding. In particular, two-sided extrusion by stably bound condensin II is sufficient to linearly compact mitotic chromosomes in simulations. We thus hypothesize that condensins, particularly condensin II, may bind chromosomes as a dimer of condensin complexes in vivo. Such a dimer might have a longer residence time due to a larger protein-DNA interface, while also performing two-sided loop extrusion via its two protein motors.

Further single-molecule, biochemical, and structural studies could also help to understand how kleisins, Hawks, and post-translational modifications might generate diverse SMC complex symmetries and dynamics, and thus, functions. Such experiments, together with the principles established by our models, could clarify how the molecular properties of loop-extruding SMC complexes compact and organize chromosomes throughout the cell cycle.

Stochastic simulations of LEFs on a chromatin fiber are performed as previously described (Goloborodko et al., 2016b; Banigan et al., 2020) with adaptations as described in the Model section. The chromatin fiber is a one-dimensional lattice of length L=60000 sites, each of which is taken to be a=0.5 kb. LEFs bind to chromatin at rate kb, and the two LEF subunits initially occupy two adjacent lattice sites upon binding the fiber. Each active subunit of the LEF may translocate. Translocation occurs in a directed manner away from the sites originally occupied by the LEF unless otherwise noted. Simulations typically consist of N=1000-2000 LEFs (i.e. d=L/N=30-60 lattice sites or d=15-30⁢kb). Each simulation is run for a duration of ttotal=400⁢τlongest, where τlongest is the longest mean residence time. For each simulation, 100 data points are collected long after achieving steady state (which occurs after t≈10⁢τlongest) from the time interval 300⁢τlongest≤t≤400⁢τlongest. Each simulation is run at least twice. The fraction, f, of the fiber compacted into loops is thus measured for each set of parameters with a standard error that is <5% of the mean. The simulation code is publicly available at (https://github.com/mirnylab/one_sided_extrusion/tree/master/mitotic/; Banigan, 2020; copy archived at swh:1:rev:b27012e95d354e8deaac5bcfdcb3c36b375626ce).

The physiological range of the ratio of the processivity to the mean separation is estimated as 10<λ/d<1000, as calculated previously (Banigan et al., 2020). Processivity, λ, was estimated from experimental measurements of condensin’s extrusion speed of ∼1⁢kb/s in vitro (Ganji et al., 2018; Golfier et al., 2020; Kong et al., 2020) and measured and estimated residence times of order 1–100 min in vivo and in vitro minutes (Gerlich et al., 2006a; Terakawa et al., 2017; Walther et al., 2018; Gibcus et al., 2018). Mean separation, d, was determined by measured linear densities of 1 per 10–100 kb in vivo (Takemoto et al., 2004; Fukui and Uchiyama, 2007; Walther et al., 2018).

For the general estimate of the physiological range of asymmetries, the expected range of asymmetries is calculated from the symmetry scores measured by Golfier et al., 2020. The symmetry score is given by S=(vfast-vslow)/(vfast+vslow), where S=0 indicates perfectly symmetric two-sided extrusion and S=1 indicates (completely asymmetric) one-sided extrusion. Using experimentally measured symmetry scores, we calculate that vslow>0.025⁢vfast for two thirds of condensin loop extrusion events (green box in Figure 1c).

We also considered three scenarios of asymmetric extrusion to model experiments by Golfier et al., 2020. In the first scenario, all LEFs perform asymmetric two-sided extrusion, but there is a distribution of asymmetries; half of the LEFs have symmetry scores, S, uniformly randomly selected from the interval [0,0.8] and half of LEFs have S randomly selected from [0.8,1]. In the second scenario, only half of the LEFs perform asymmetric two-sided extrusion with asymmetries in the interval [0,0.8]; the other half of the population performs one-sided extrusion (i.e., S=1).

In the third scenario, we calculated loop-growing and loop-shrinking velocities for condensins from single-molecule experiments by Golfier et al., 2020. Ten trajectories from the experiments were smoothed with a Savitsky-Golay filter with a second order polynomial and a window of 63 frames as in the previous analysis by Golfier et al., 2020. For each condensin trajectory, we then calculated the mean size of loop-growing and loop-shrinking steps and computed mean loop-growing and loop-shrinking speeds for each of the two sides. We simulated two-sided asymmetric extrusion with these ten sets of velocities, with each set of velocities assigned to one tenth of the LEFs.