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Bacterial chromosome organization by collective dynamics of SMC condensins

A prominent organizational feature of bacterial chromosomes was revealed by Hi-C experiments, indicating anomalously high contacts between the left and right chromosomal arms. These long-range contacts have been attributed to various nucleoid-associated proteins, including the ATPase SMC condensin. Although the molecular structure of these ATPases has been mapped in detail, it still remains unclear by which physical mechanisms they collectively generate long-range chromosomal contacts. Here, we develop a computational model that captures the subtle interplay between molecular-scale activity of slip-links and large-scale chromosome organization. We first consider a scenario in which the ATPase activity of slip-links regulates their DNA-recruitment near the origin of replication, while the slip-link dynamics is assumed to be diffusive. We find that such diffusive slip-links can collectively organize the entire chromosome into a state with aligned arms, but not within physiological constraints. However, slip-links that include motor activity are far more effective at organizing the entire chromosome over all length-scales. The persistence of motor slip-links at physiological densities can generate large, nested loops and drive them into the bulk of the DNA. Finally, our model with motor slip-links can quantitatively account for the rapid arm--arm alignment of chromosomal arms observed in vivo.


Introduction
The bacterial chromosome is highly structured over a wide range of length-scales [1,2,3,4]. Indeed, microscopy experiments have revealed a remarkable degree of sub-cellular organization of the chromosome across many species, including Bacillus subtilis, Caulobacter crescentus and Escherichia coli [5,6,7,8,9]. The structural organization of chromosomes was further exposed by recent Hi-C experiments measuring the contact probability between pairs of loci on the chromosome [10,11,12,6,13]. On small length-scales, the chromosome appears to be organized in so-called Chromosome Interaction Domains: genomic domains with above-average contact probabilities between pairs of loci. On large length-scales, the most prominent feature is the emergence of a cross-diagonal in Hi-C maps spanning the whole length of the genome, indicating anomalously high contact probabilities between opposing pairs of DNA-loci positioned on the left and right chromosomal arms. Such a cross-diagonal is observed in B. Subtilis and C. crescentus [11,12,10,14], but not in E. Coli [15].
The measured cross-diagonal indicates a robust juxtaposition of the left and right arms of the chromosome [10,12,11,14], an organizational feature that is important for faithful chromosome segregation [16,6,17,9,18]. Indeed, this juxtaposed organization depends on the ParABS partitioning system, and is largely controlled by the highly conserved ATPase SMC condensin [11,12,6]. While much is known about condensin at the molecular level [19,20,21,22,23,24], it is unclear how small numbers of condensins (3-30 per chromosome [25]) are capable of collectively organizing the chromosome over such a range of length-scales. Thus, the physical principles underlying the juxtaposed organization of the chromosome remain elusive.
The functional capability of SMC condensins to organize the chromosome into a juxtaposed state crucially depends on two factors: (i) the presence of a specific loading site on the chromosome close to the origin of replication (ori ) [11,26], and (ii) the ability of condensin to bind and hydrolyze ATP [26,25]. In B. subtilis, the loading site near ori is established by a large nucleoprotein complex composed of ParB proteins bound around parS [27,28,29,30]. ChIP-seq data shows that condensins are recruited to this ParBS region, and from there propagate deep into the bulk of the DNA polymer [26,25]. Suppressing condensin recruitment to ori in ParB or parS deletion mutants (∆ParB/S ) results in a loss of the cross-diagonal in Hi-C maps [11,12]. Moreover, the translocation of condensins away from the loading site near ori depends on their ATPase activity; condensin mutants that cannot bind ATP only weakly associate with DNA, and mutants that do not hydrolyze ATP do not efficiently propagate away from the ParBS loading site (SI 2.2 and [26,25]).
The SMC condensin complex consists of two main subunits: Smc coiled coils that emanate from a hinge domain and end in two head-domains, which are connected to establish a ring-like topology [19,25,21,22,31,20]. Interestingly, condensin is rod-shaped in solution [23], but undergoes a conformational transition into an open configuration upon ATP-induced DNA-binding [19,21]. With a typical size of 25 − 50 nm, the ATP-bound condensin ring is large enough to trap a DNA loop by threading a DNA duplex through it [23,31,19]. It has been proposed that the possibility of condensins to trap DNA-loops would enable them to align the chromosomal arms by progressively extruding DNA-loops from the origin to the terminus region [11,12,6].
Important clues on the role of ATPase activity in SMC condensin come from in vitro single-molecule experiments. These experiments revealed that Saccharomyces cerevisiae condensin is a molecular motor, and performs active translocation over DNA duplexes [32]. Kymographs showed that the direction of movement of yeast condensin is a priori random, and switches direction after a typical time-scale; in other words, yeast condensin performs persistent random motion over DNA. In fact, recent single-molecule experiments have revealed that such yeast condensin performs active loop extrusion [33]. Similar experiments have not yet observed such motor activity for bacterial condensin [34]. Nevertheless, it has been widely speculated that the ATPase activity of bacterial condensin is also directed towards motor activity [35,31,25,22]. In this picture, condensin would actively extrude DNAloops, possibly by feeding DNA duplexes through its ring-like structure [36,37,38,39]. In contrast, other models have been proposed in which the ATPase activity of condensin is directed towards regulating its association with DNA [25,26,38,40]. Thus, it remains an open question whether bacterial condensin also acts as a loop extruding enzyme, or whether the ATPase activity primarily regulates the DNA-recruitment of condensins.
To elucidate the role of condensin ATPase activity on bacterial chromosome organization, we develop two minimal models for condensin-DNA interactions, where we describe SMC condensin as a slip-link that non-topologically traps a DNA-loop (Fig. 1). In the most basic model, condensin activity is assumed to be directed towards regulating its DNA-recruitment, while the dynamics of condensin slip-links on the DNA is diffusive. Interestingly, we find that such diffusive slip-links can organize the chromosome into a juxtaposed state, but not within physiological constraints. Next, we expand the model to include motor slip-links that perform persistent random motion on DNA. We find that these motor slip-links are much more effective in organizing the entire chromosome. In particular, our motor slip-link model requires at least 2-3 orders of magnitude fewer condensins to organize the chromosome than in the diffusive slip-link model. In addition, the development of the juxtaposed state exhibits sub-diffusive dynamics in the diffusive slip-link model, in contradiction with the rapid re-organization observed in vivo [12]. We show that such a fast re-organization of the chromosome is achieved by motor activity in the form of active loop extrusion. More generally, the computational model we develop here offers a framework to study how the ATPase-activity of nucleoid-associated proteins impacts chromosome organization.

Model
Our model of condensin-DNA interactions ( Fig. 1) contains two ingredients: (i) a circular DNA polymer, and (ii) multiple SMC condensins that interact with the DNA. We employ a lattice polymer with a lattice constant set by the persistence length of DNA (SI 1). Condensins are modeled as slip-links: elastic rings that trap a DNA-loop by encircling two DNA duplexes. Importantly, we consider both diffusive and motor slip-links. Diffusive slip-links move randomly (stepping rate k 0 ) over the DNA (Fig. 1, inset); motor slip-links perform persistent random motion (translocation rate k motor ) with a persistence time-scale τ switch = k −1 switch . These persistent dynamics enable motor The bacterial chromosome has a circular topology. SMC condensins are loaded onto the DNA by the ParBS complex located near the origin of replication (ori, green). Right: Microscopic reactions in our model with associated rate constants. Condensins are represented as slip-links, which either perform diffusive or more persistent, motorized movement (inset).
slip-links to actively extrude DNA loops.
To simulate the dynamics of both the DNA polymer and the slip-links, we developed a Kinetic Monte-Carlo (KMC) algorithm (SI 1). Our KMC algorithm simulates the Rouse dynamics of DNA [41,42], the associated stochastic motion of slip-links on the DNA, as well as the microscopic reactions in which slip-links bind to (rate k + ) or unbind from (rate k − ) the DNA. For simplicity, we assume instantaneous slip-link binding k + → ∞, justified by the relatively fast cytosolic diffusion of condensin [43]. Additionally, we fix a maximum number of slip-links N p that can bind to the DNA.

Diffusive slip-links with a specific loading site can organize the chromosome
In the simplest implementation of our model, there is no specific loading site: diffusive slip-links can bind and unbind anywhere on the DNA (Fig. 2a, ∆ParB/S ). This assumption results in a homogeneous binding probability p p (i) (Fig. 2a, bottom). Importantly, in the ∆ParB/S scenario, all microscopic reactions are fully reversible, implying that the system relaxes into thermodynamic equilibrium. In equilibrium, small loops are strongly favored, owing to the increasing entropic cost of loop formation with larger loop size. This tendency to form small loops is reflected in the loop diagrams (Fig. 2a, top). Indeed, the loop sizes trapped by the slip-links are consistent with the equilibrium loop-size distribution (SI 9.1). Furthermore, in this scenario the contact maps are structureless and only exhibit a single main diagonal (Fig. 2a, middle), as for a random polymer. These results demonstrate that our KMC model with reversible microscopic reactions evolves towards thermodynamic equilibrium. Thus, although the slip-links can easily bind over the full extent of the polymer, they do not organize the chromosome. We next investigate how the presence of a slip-link loading site [26,25,11,12,16] impacts steady-state chromosome organization. Note, while recruitment of slip-links in our model is exclusive to ori, unbinding can occur anywhere on the DNA. This implies that the reactions involving slip-link binding/unbinding are partially irreversible. Hence, detailed balance is broken [44], and the system may no longer evolve towards thermodynamic equilibrium. Nevertheless, for slow dissociation kinetics (small k − ), we observe an unstructured contact map ( Fig. 2b; SI 9.1), similar to the equilibrium system lacking a specific loading site ( Fig. 2a; SI 9.1). Moreover, the loop diagrams again show that the slip-links mostly encircle small loops (Fig. 2b, top). In sum, we see that, although detailed balance is broken on the level of slip-link binding/unbinding, the diffusive slip-links with slow dissociation kinetics do not appear to organize the DNA polymer.
Interestingly, increasing the dissociation kinetics of slip-links results in dramatically different contact map. We observe the emergence of a prominent cross-diagonal, which either disperses away from the loading site (Fig. 2c) or remains clearly resolved over the whole polymer (Fig. 2d), depending on the density of slip-links. Interestingly, the loop diagrams for these systems exhibit a topology distinct from the equilibrium configuration; slip-links trap DNAloops in a cooperative, nested manner. Movies of the loop diagrams and contact maps clearly demonstrate that these nested loops propagate away from the loading site, dynamically driving a juxtaposition of the two polymer arms (SI movies 3a-b; SI 6). This dynamical arm-arm alignment is a distinct out-of-equilibrium phenomenon, and thus requires the exclusive binding of slip-links to the loading site. Thus, our observations show that diffusive sliplinks with fast dissociation kinetics in conjunction with a loading site can, in principle, generate a non-equilibrium  polymer organization similar to that found in living cells [11,12,6,10].
To investigate the role of slip-link kinetics on loop topology more quantitatively, we employ a metric that captures essential topological differences between the loop network of random and juxtaposed polymers (compare Fig. 2b to Figs. 2c-d). To this end, we define the order parameter θ ∈ [0, 1] as the fraction of nested loops ( Fig. 2h and SI 3.1). We observe a sigmoidal relation between θ and the dissociation rate k − for diffusive slip-links, with θ transitioning from low to high values with increasing k − (Fig. 3, blue; SI 9.2). The increase in θ at higher k − stems from the increased lifetime of nested loops, and can be estimated using a simple mean-field model (solid curve in Fig. 3, SI 9.2). Furthermore, we find that the characteristic dissociation rate at the inflection point of θ coincides with the transition from a random polymer to a juxtaposed organization with a cross-diagonal in contact maps (d 1 vs. d 2 in Fig. 3). In addition, we observe that the equilibrium implementation of diffusive slip-links that bind non-specifically to the DNA (∆ParB/S ) yields θ ≈ 0 (d 3 in Fig. 3), confirming our previous observation that the loading site is necessary for creating large, nested loops (SI 9). Thus, in the presence of a loading site, diffusive slip-links appear to drive a dynamical transition between phases of weak and strong nesting of DNA-loops, and this transition is crucial to establish the juxtaposed state of the chromosome.
Importantly, in our model with diffusive slip-links, we find that having many nested loops is a necessary, but not sufficient condition to organize the polymer into a juxtaposed state (SI 8). These loops also need to propagate into the bulk of the polymer. Indeed, we find that the propagation of diffusive slip-links is a density-driven process: the nested loops only propagate over the full length of the polymer for very high slip-link densities (SI 4). This can be clearly seen in the binding profiles p p (i); at low slip-link densities, the binding profile is sharply peaked around the loading site (Fig. 2c, bottom), whereas this peak broadens as we increase the slip-link density (Fig. 2d, bottom).

Motor slip-links are highly effective in organizing the chromosome, even at low densities
Motivated by recent observations of motor activity of yeast condensin in single-molecule experiments [32,33], we next explore how such activity impacts the ability of slip-links to organize the chromosome. In our model, motor slip-links are assumed to perform persistent random motion ( Fig. 1, inset). Such persistent slip-links actively extrude loops [33,36,37,38,39]. The active dynamics of motor slip-links is characterized by the switching rate k switch . For k switch → 0, the motor slip-links never reverse direction, whereas for k switch k motor they behave as diffusive slip-links. We find that motor slip-links with small k switch organize a system-spanning cross-diagonal ( Fig. 2f), whereas the cross-diagonal retracts for increasing k switch (Fig. 2g).
Overall, we observe that the persistence of such motor slip-links renders them much more effective at producing the cross-diagonal (Figs. 2f-g). Even for low slip-link densities, a system-spanning cross-diagonal is formed together with an extended binding profile (Fig. 2f). Indeed, motor slip-links can readily drive the chromosome into a state with a high degree of loop nesting. Even in the absence of a loading site (m 3 in Fig. 3), motor slip-links efficiently create nested loops. However, the degree of loop nesting θ is sensitive to k switch (Fig. 3, red). We observe a decline of θ with increasing k switch , although θ remains above 50% even if the motor slip-link on average switches direction with each step (k switch = k motor ). Since a high degree of loop nesting is necessary for establishing the juxtaposed state (SI 8), the rate of directional switching must remain sufficiently small for the motor slip-links to organize the chromosome.
Importantly, in the absence of a specific loading site, motor slip-links still extrude large loops and efficiently propagate along the polymer, as reflected in the higher contact probability away from the main diagonal (Fig. 2e). However, there is no breaking of translational symmetry by a loading site, resulting in a leveled time-averaged contact map. Interestingly, we observe that systems without a loading site can organize transiently into a juxtaposed state (SI 10 and SI movies 7a-b), but the location of the fold diffuses randomly over the polymer.

Condensins need motorized movement to efficiently propagate into the bulk of the chromosome
In vivo experiments have demonstrated that condensins propagate far from their loading site [26,25]. To quantify the distribution of condensins on the chromosome in our model, we measure the extentX p of slip-link propagation as the standard deviation of the binding profile p p (i) (Fig. 2h). We eliminate the system-size dependence by considering the scaled propagation lengthx p =X p / 1 2 N m (SI 4) as a function of slip-link density φ p = 2N p /N m . The scaled propagation length of diffusive slip-links only approaches the in vivo value,x p ≈ 0.5 ( [26,25] and   Figure 4: Propagation length of slip-links and associated arm-arm juxtaposition (a) Scaled propagation lengthx p ≡X p / 1 2 N m as a function of slip-link density φ p for diffusive (blue) and motor (red) slip-links. Indicated are also experimental measurements from [26] for wildtype cells ("WT", red star) and mutants whose SMC condensins have suppressed ATPase activity ("ATP-mutant", blue star) (SI 2.2). (b): State diagram of scaled crossdiagonal lengthx c =X c / 1 2 N m for diffusive slip-links as a function of dissociation rate k − (in units of k 0 ) and slip-link density φ p . (c): State diagram of scaled crossdiagonal lengthx c for motor slip-links as a function of the switching rate k switch (in units of k motor ) and sliplink density φ p . Both (b-c) have N m = 100. Marker "exp.": is an estimate of WT behavior, using the values φ p = 10 −4 . . . 10 −3 as in B. Subtilis [25], k −1 switch = 222 min (SI 2.1) and k −1 motor = 0.12 s (SI 1.3.3). Markers m 1 (SI movie 10a) and m 2 (SI movie 10b) are respectively at low and high switching rates. Hatched area indicates values of φ p that we did not reach computationally. For both panels (b) and (c), we definex c as the 75 th percentile of the cross-diagonal contacts (SI 3.2). SI 2.2), when we use slip-link densities 10% in our simulations (Fig. 4a, blue). Importantly, this slip-link density would correspond to thousands of condensins on the chromosome, 2-3 orders of magnitude more than reported in vivo [25]. This further illustrates that diffusive slip-links are not efficient at forcing the nested loops into the bulk of the polymer at low densities. In contrast, motor slip-links propagate over the full length of the polymer at all slip-link densities we considered (red data in Fig. 4), as observed experimentally in cells ( [26,25] and SI 2.2).
Our results are summarized in a "state diagram" (Figs. 4b-c), indicating the scaled extentx c =X c / 1 2 N m of the cross-diagonal in contact maps. Diffusive slip-links require both fast dissociation kinetics as well as a high slip-link density to bring the polymer into the juxtaposed state (Fig. 4b). For high densities of motor slip-links φ p 10% and low k switch , the slip-links readily juxtapose the DNA polymer (e.g. m 1 in Fig. 4c). Contrarily, for increasing k switch , motor slip-links antagonize each other's translocation (e.g. m 2 in Fig. 4c), impeding the collective propagation of slip-links away from the loading site, thereby resulting in a reducedx c (compare m 1 , m 2 in Fig. 4c; SI movies 10a-b). In the limit k switch k motor , motor slip-links effectively behave as diffusive slip-links with enhanced unbinding kinetics, placing them in the fast dissociation regime (data for k switch 10k motor in Fig. 4c). In contrast, for φ p 10%, motor slip-links only require k switch to be sufficiently low. We estimate that wild-type condensin is indeed in this slow switching regime ("exp." in Fig. 4c). In sum, our simulations indicate that condensins at physiological densities can drive nested loops into the bulk of the polymer, crucial for establishing arm-arm alignment, only if they perform motorized, persistent motion.
2.4 SMC condensin requires motor activity to rapidly re-organize the chromosome SMC induction experiments revealed that condensin can propagate from the loading site into the bulk of the DNA, thereby organizing an entire bacterial chromosome in a timespan of only T WT ≈ 24 min (SI 2.3 and [12]). Based on these experiments, we estimate that condensins translocate away from their loading site with a velocity of ≈ 300 nm/s. To understand the origin of these remarkably fast dynamics, we compute time-traces X p (t) of the width of the slip-link binding profile for our minimal models. From these traces, we extract a typical propagation time T for slip-links to establish a steady-state binding profile on a DNA polymer of physical length L = aN m (Fig. 5, inset), where a ≈ 50 nm is the size of one monomer (SI 1.3).
The propagation time of diffusive slip-links scales strongly with DNA length: T ∼ L x , with x ≈ 2.5 (Fig. 5, blue). This scaling differs from simple diffusive motion (x = 2, black in Fig. 5), which we attribute to the loopentropic forces that impede slip-link movement away from their loading site. Based on this observed scaling of T , we estimate that diffusive slip-links propagate several orders of magnitude slower over the DNA than observed in live cells (Fig. 5, "WT"). In contrast, the propagation time of motor slip-links exhibits ballistic scaling, T = L/v (Fig. 5, red), where the translocation velocity v depends on the motor activity and DNA relaxation dynamics, but not on the system size employed (SI 5). Interestingly, our model prediction of motor slip-links, T = vL, is remarkably close to the observed propagation time in vivo (Fig. 5, "WT"). Indeed, recent single-molecule experiments have revealed that yeast condensin can extrude DNA loops with a velocity of up to 425 nm/s [33]. These data combined with our simulations, strongly indicate that rapid re-organization of the chromosome by SMC condensin requires fast and active loop extrusion.  For diffusive sliplinks (blue), we quantify the propagation time-scale T by an exponential fit to the propagation length X p (t) (legend, top). For motor slip-links (red), we computed the average translocation velocity v and define T = L/v (legend, bottom). Diffusive slip-links exhibit sub-diffusive scaling T ∼ L 2.5 (blue, dashed); diffusive scaling T ∼ L 2 is shown for comparison (black, dashed). The extrapolation of the simulated data for motor slip-links (red) (SI 5) is in accord with in vivo data ("WT", see SI 2). Simulation units were converted to real units as described in SI 2.

Discussion
Our computational framework reveals the basic physical requirements for condensins to collectively organize the bacterial chromosome as observed in live cells [11,12,6,10]. In the presence of a specific loading site and with physiologically relevant numbers of condensins, we find that motor activity is required to robustly and rapidly generate a system-size spanning juxtaposition of the chromosomal arms. In contrast, purely diffusive condensins would require more kinetic fine-tuning and unphysiologically high copy numbers to organize the chromosome.
Our minimal model for the action of SMC condensin as a motor slip-link accounts for several key observations, including the rapid development of the juxtaposed state [12] and the crucial role of the ParBS nucleoprotein complex as a specific loading site [11,12,16,6]. Without a well-defined loading site (∆ParB/S ), motor slip-links still transiently organize the polymer into a juxtaposed state. However, the chromosomal fold diffuses along the chromosome, resulting in a structureless time-averaged contact map with enhanced long-range contacts ( Fig. 2e and SI 10). This may account for observations in E. coli, where the action of the SMC complex MukBEF does not seem to involve an exclusive loading site [15]. Indeed, Hi-C maps of E. coli do not display a cross-diagonal, but rather an elevated contact probability at large length-scales [15], in line with our simulations (Fig. 2e and SI 10).
Experiments of SMC condensin propagation in B. subtilis suggest that two condensin complexes might link together in a hand-cuff topology, with each of the two condensins in the dimer actively extruding a separate DNA duplex [12,13]. In our model, motor slip-links extrude DNA in a symmetrical fashion, as expected for condensins in a hand-cuff configuration: both sides of a slip-link move over a separate DNA duplex with the same translocation rate. However, in recent in vitro assays, single yeast condensin complexes actively extrude DNA loops asymmetrically: one end of the complex appears anchored at a DNA locus, while the opposite end actively translocates over DNA [33]. Interestingly, contact maps of such asymmetric motor slip-links contain a star-shaped pattern around the loading site (SI 11), a feature that is also visible in Hi-C maps of B. subtilis [11,6]. This suggests that there is at least some fraction of condensins performing asymmetric translocation. An equally tantalizing explanation is that the movement of one of the condensins in a dimer is impeded by other DNA-bound factors, thereby forcing a condensindimer to propagate asymmetrically. Indeed, there is growing evidence that the movement of SMC complexes can be antagonized by oncoming transcription factors [11,13,12].
It has also been suggested that the cylindrical geometry of many bacteria, combined with DNA-cell-pole tethering, facilitates chromosome organization [45,46]. However, our simulations indicate that the effect of confinement alone on chromosome organization is weak, and hence cannot be solely responsible for the juxtaposed organization observed in live cells (SI 7). This is in line with the observation that mutants lacking SMC condensin, but with the ori -proximal loci still tethered to a cell-pole, also lack the cross-diagonal [12].
Our simulations further indicate that even purely diffusive slip-links with fast dissociation kinetics can induce arm-arm alignment, but at low slip-link densities this organized state remains localized near the loading site (SI 6). For a given number of condensins (3-30 per chromosome [25]), however, the slip-link density depends on DNA length. Therefore, we expect that a physiological number of non-motorized condensins can organize a mini-chromosome or plasmid of tens of microns in length. Indeed, plasmids are known to bind SMCs [47] and contain ParBS nucleoprotein complexes that could act as a condensin loading site [48].

*Supplementary Information 1 Kinetic Monte-Carlo algorithm
Our lattice Kinetic Monte-Carlo (KMC) framework employs a Gillespie-type algorithm [52]. This KMC algorithm is schematically illustrated in Fig. S1, and consists of the following steps: (a) Randomly select one of the transitions T j in Ω. The probability to perform transition T i is k i / i k i , which we implemented using tower sampling [53].
(b) Update the KMC time t → t + ∆t, ∆t = − log r/K, where r is a uniformly sampled random number r ∈ 0, 1] and K = i k i is the total rate of the system [52].
(c) Perform transition T j that was selected in (a), which can affect the DNA polymer or the particles interacting with the polymer.
(d) Update the entire rate catalog of possible transitions Ω based on the transition T j that was just performed. For more details, see section 1.1.
(e) Return to step (a).   [42]. Right columns: Slip-link moves include association to the DNA polymer and movement along the polymer.

Updating the KMC rate catalog
Every iteration of the KMC loop (see Fig. S1) requires an update of the rate catalog Ω. For computational efficiency we only update the part of the rate catalog δΩ j ∈ Ω that is possibly affected by the previous transition T j . Let Θ j be the set of coordinates of monomers that are displaced by the transition T j . We update all coordinates q in the rate catalog that satisfy |p − q| ≤ m, p ∈ Θ j , since our move set displaces particles by at most m = 2 lattice points (Fig. S2, "crankshaft").

Lattice KMC moves
The lattice KMC moves that we employ are illustrated in Fig. S2. Importantly, this move-set with local moves for the polymer dynamics is guaranteed to give rise to Rouse dynamics [42].

Justification of simulation units
Because cytosolic diffusion of condensin is fast compared to diffusion of condensin over dsDNA [43], we assume a regime in which the condensin binding rate is much higher than the slip-link movement attempt rate, k + k 0 . We implemented loading to the ParBS region [26] by binding slip-links only to a predetermined loading site (ori ) on the polymer. Slip-links can, however, unbind anywhere on the DNA, as long as the slip-link encircles a loop of only one lattice width (Fig. S2).

Length of B. subtilis genome in simulation units
Each monomer size (i.e. the lattice constant 0 ) in our lattice simulation is set equal to the persistence length of dsDNA, approximately 50 nm [54]. This is a convenient choice since the size of SMC condensin is also ∼ 25 − 50 nm [23,31], which we thus also take to be one lattice size. In these units, the B. subtilis genome of 4 Mbp (1.4 mm) [26] corresponds to 28 × 10 3 monomers.

Monomer relaxation time
The monomer relaxation time τ monomer for a cylindrically shaped polymer bond, according to the Rouse model is [55,56] where a ≈ 50 nm is the length of one monomer (SI 1.3.1), b ≈ 2 nm is the radius of dsDNA [57], k B T ≈ 4.2 pN nm [57] and η ≈ 1 mPa/s is the cytosolic viscosity [58, 59, 60].

Diffusion and translocation times of SMC complexes
The one-dimensional diffusion coefficient of dsDNA-bound cohesin (an SMC complex related to condensin [61]) at physiologically relevant salinity is D ≈ 1 µm 2 /s [43], so that the diffusion time over one polymer bond of a ≈ 50 nm [54] is approximately The average velocity of persistent motion of yeast condensin without the application of force is approximately v ≈ 1.25 kbp/s ≈ 425 nm/s [33]. This means that the average time to traverse one polymer bond by active translocation is τ motor.slip−link ≈ a v ≈ 0.12 s.
2 Measurement of metrics from empirical data

Motor direction switching time in yeast condensin
In vitro DNA curtain experiments have shown that a fraction of condensins can reverse their active motion within a time τ r [32]. In these experiments, the condensins can only be monitored over a maximum distance L assay = 16.49 µm. The time a condensin can be observed to move with a constant velocity v ≈ 20 nm/s is, therefore t = L assay /v ≈ 825 s [32]. In [32], the probability for the motor to be reversed at least once within this time t has been measured to be p r (t ) ≈ 6% . Assuming Poissonian statistics, the probability of the condensin having reversed within a time t is p r (t ) = 1 − exp(−t /τ r ), from which we estimate a typical switching time for yeast condensin τ r ≈ 222 min. Based on a typical velocity of v ≈ 20 nm/s [33], we estimate that a yeast condensin travels on average a distance L max = vτ r ≈ 266 µm ≈ 785 kpb before switching direction. Crowding on the DNA by other proteins in vivo could affect these estimates.

Measuring propagation length of condensin in B. subtilis
We analyzed ChIP-seq data from [26] of the strains "WT" (wild-type), "ATP − " (mutant with strongly suppressed ATP hydrolysis) and ∆ParB (mutant without ParB). As a baseline for the SMC signal, we used the ∆ParB data for both the WT and ATP − strains and computed the difference in their SMC ChIP-Seq signals c SMC (i) (Fig. S3), We then smoothed the data with a Savitzky-Golay-filter of window 21 and order 2 and extracted typical widthŝ X SMC , defined as the standard deviation of the smoothed ChIP profiles. We computed the typical widthsX SMC,WT ≈ 7000 0 ,X SMC,ATP− ≈ 2000 0 -or, when scaled by the system sizex SMC,WT ≈ 13%,x SMC,ATP− ≈ 45%.

Measuring condensin propagation timescale
Recent in vivo experiments have been performed in which ChIP-seq data was measured at various time-points after induction of SMC condensin [12]. From these ChIP-seq data, the typical widthX p (t) of condensin propagation was measured as a function of time. Since the curve ofX p (t) was well approximated by an exponential curve, we extracted a typical timescale of ≈ 24 min from these curves (Fig. S4).
3 Measurement of metrics from simulation data

Measuring the loop network topology
We quantify the loop network topology using an order parameter θ ∈ [0, 1] that measures the fraction of nested loops. This order parameter is defined for a slip-link-polymer network, so that we can measure θ for each time-point t, i.e. θ(t) is well-defined. Variances (varθ) were computed as the variance of the time-sequence {θ(t 1 ), θ(t 2 ), . . .}.
Whenever we refer to θ in the main manuscript without explicit reference to the time-dependence, we imply θ = θ(t) where we used a time average for · . Below we have reproduced a pseudocode to measure θ:

Determination of typical length of cross-diagonal
We first extract an 'unprocessed' cross-diagonal probability p c,unprocessed (k) ∈ [0, 1] as illustrated in Fig. S5. Neither the moments k n nor the p−th percentile of the 'unprocessed' distribution p c,unprocessed (k) correlated well with the  Fig. S4. The ChIP-seq data of the SMC condensin propagation width has a typical timescale of ≈ 24 min. Fit parameters τ ≈ 24 min, amplitude ≈ 1368 kbp were found by fitting the data from [12] with an exponential.  [12,10]). (d): Average the data from step (c) over the 10 rows, returning a cross-diagonal probability p c (k) where k = − 1 2 N m , . . . , + 1 2 N m is the distance from ori. In order to isolate the cross-diagonal from the rest of the contact map, we select the 10 most central rows from the contact map (i.e. |k| ≤ 5).
cross-diagonal width based on visual inspection. The reason for this is that p c,unprocessed (k) contains large flanks at |k| ≈ N m /2 1 (Fig. S5d) that were found to significantly impact estimates ofX c . The flanks in p c,unprocessed (k) (Fig. S5d) arise due to the circular topology of the polymer, with an increasing contact probability for |k| ≈ N m /2 1. A naive estimate of the flanks is a power-law with Flory-scaling p c,naive ∼ ( 1 2 N m − |k|) dν , d = 3, ν ≈ 3/5 [62]. Subtracting this naive estimate from p c,unprocessed (k), was found to sometimes lead to negative probabilities. To avoid this, we instead subtracted an underestimate of the flanks, namely a power-Input : slip-link sites {(i, j)n}, n = 1, . . . , Np (with Np the number of slip-links). output: loop network topology order parameter θ ∈ [0, 1] n nested ← 0; for n ← 1 to N do get slip-link sites (in, jn) of slip-link n; compute self-loop-size of slip-link n: ∆ self ← |i − j|; get sites (i, j)n−1,n+1 of neighboring slip-links n − 1, n + 1; compute nested-loop-size between slip-link n and n − 1, n + 1: law with 50% stronger scaling than the naive estimate: (Fig. S6, compare black and blue histograms). By visual inspection on a variety of representative test-cases, we found that this method indeed suppressed the flanks of p c,unprocessed (k), but left the cross-diagonal itself intact (Fig. S6, blue  histogram).
After subtracting the flanks from p c (k) (Fig. S6, blue histogram), we calculate the typical length X c,p = 1 2 m p with parameter p as the p−th percentile of the distribution p c (k). We computed various percentiles {m 50% , m 75% , m 90% , m 95% } on a representative collection of contact maps and empirically found by visual inspection that m 75% was a good measure for the typical length of the cross diagonal in contact maps.

Universal scaling of slip-link propagation length for diffusive sliplinks
The propagation lengthX p and the number of slip-linksN p depend on N m (Fig. S7, left). For diffusive slip-links in the regime of fast dissociation kinetics, the relation between their scaled intensive counterpartsx p =X p /N m ,φ p = 2N p /N m appear to be described by a single master curve (Fig. S7, right).

Universal scaling of the translocation rate of motor slip-links
What is the relationship between the slip-link translocation attempt rate k motor , the monomer diffusion attempt rate k 0 , the polymer size N m , and the effective motorized slip-link translocation velocity v motor ? To answer this question, we distinguish two regimes: a fast relaxation regime k motor k 0 and a slow relaxation regime k motor k 0 . In both regimes, for a stiff slip-link to make a step, the two polymer bonds in the direction of movement need to be parallel (Fig. S8a). If we denote the prior probability of observing these two polymer bonds to be parallel by C ≈ 3/16 (Fig. S8b), then the effective rate of bond-bond alignment is k eff 0 ≈ Ck 0 . Hence, the characteristic rate k motor that sets the transition from the fast relaxation to the slow relaxation regime occurs at k motor = k eff 0 . Consistent with the prediction that k motor only depends on local kinetics, our data shows that k motor is independent of the system size (Fig. S9a).
In the fast relaxation regime, the rate-limiting factor is k motor . In this case, the velocity of slip-links is approximated by v motor ≈ C k motor . As already stated, the fast relaxation regime transitions into the slow relaxation regime at k motor = k eff 0 , where the movement of motor slip-links becomes limited by polymer relaxation so that v max = C k motor = C 2 k 0 .
This heuristic argument suggests that the scaling form of dimensionless variablesṽ ≡ v/( k 0 ),k ≡ k motor /k 0 will collapse the data of k motor , v, k 0 onto a universal curveṽ = Ck fork < C andṽ = C 2 fork ≥ C. Indeed, our numerical data is well-described by this scaling (Fig. S9b-c). By combining in vitro with in vivo empirical data, we estimate that SMC condensin in B. subtilis is well within the fast relaxation regime k motor < 10 −5 k 0 k 0 (Fig. S9c, "WT").

Stability of juxtaposed organization by diffusive slip-links
To investigate the stability of the juxtaposed state with diffusive slip-links, we prepared a polymer in a juxtaposed organization. This was achieved by positioning immobilized slip-links at regular intervals along the polymer but nevertheless allowing for polymer relaxation (Fig. S10a). After we turned on slip-link movement, we recorded kymographs of the cross-diagonal.
Kymographs clearly reveal that the cross-diagonal in the regime of slow dissociation propagates away from the loading site (Fig. S10b). The region around ori in the contact maps slowly loses the anomalously high contact probabilities over time, meaning that the cross-diagonal disappears for diffusive slip-links in the slow dissociation regime. For slip-links in the fast dissociation regime, however, the cross-diagonal persists and remains stable over time for N p above a critical value (Fig. S10c). Ensemble averaged movies of contact maps for N p = 1, 2, 4, 8, 16, 32 confirm that the cross-diagonal only persists in the fast dissociation regime (SI movies 4[a-f] for fast dissociation and 5[a-f] for slow dissociation). Fig. S6. Computation of the length of the cross-diagonal using the p−th percentile. We first measured the 'unprocessed' cross-diagonal contact probability p c,unprocessed (k) from contact maps (black, see Fig. S5 for the procedure). Then, we subtract an estimate for the left and right flanks (blue, see section 3.2) and computed the p−th percentile m p of p c (k). The half-width of the p−th percentile 1 2 m p is shown for p = 50, 75, 90, 95% (blue vertical lines, from left to right), and the typical width of the curve is defined as X c ≡ 1 2 m p .

Cell anisotropy and confinement does not explain juxtaposed organization
To investigate the role of an anisotropic confinement on the cross-diagonal, we use an ellipsoidal harmonic potential, where the k x,y,z specify the 'spring-constants' of the confinement for monomers with positions x, y, z. The ellipsoidal potential simulates the combined action of the cell membrane and crowding agents. We assume cylindrical symmetry [6], x 2 = y 2 = r 2 where r is the radius of the cylindrical confinement. We use a length-to-width ratio of 4, slightly larger than the typical length-to-width ratio ≈ 3 of many bacteria [57]. We additionally include a spring-like tether capturing the effect of proteins, such as DivIVA in B. subtilis and PopZ in C. crescentus, that localize ori to one of the cell poles [1]. We use equilibrium Monte-Carlo simulations [27] to investigate the organization of DNA in the presence of a confinement potential and tether. We find that the contact maps do not exhibit a cross-diagonal. The confinement potential does result in a slight asymmetry in the contact maps that increases weakly with the number of slip-links N p (Fig. S11a-b). This slight asymmetry resembles a cross-diagonal under conditions of large N p and small polymer-length N m , but only becomes visually apparent in   Fig. S8) separates the "fast relaxation" and "slow relaxation" regimes. Propagation of wild-type condensins ("WT" marker, inset) is within the "fast relaxation" regime. Data for "WT" was calculated using k motor = 425 nm/s/ [33] where = 50 nm [54], k 0 = 1.25 µs −1 (section 1.3.2) andṽ = v/ k 0 with v = 46 kbp/min [12] so thatk = 6.8 × 10 −6 ,ṽ = 4 × 10 −6 . our contact maps when the density of slip-links is 10%, ¿100 times higher than the SMC condensin density in vivo [25]. Even so, even at this unphysiologically high concentration of slip-links, the asymmetry in the contact maps is very diffuse compared to that of the in vivo Hi-C maps ( Fig. S11a-b). The width of the off-diagonal structure broadens as we increase N m . Indeed, the cross-diagonal for N m = 200 is only clearly resolved when we increase the asymmetry of the confinement potential to unrealistically high values z 2 /r 2 > 30 ( Fig. S11c-d). In sum, our simulations indicate that ellipsoidal confinement and tethering alone cannot be responsible for a cross-diagonal. 8 Relationship between θ,X p andX c Our simulations reveal that the cross-diagonal lengthx c is well approximated by an 'and-gate' of θ andx p (Fig.  S12). This shows that one needs cooperative loops (high θ) that also propagate deep into the bulk of the polymer (highx p ). 9 Loop topology of polymers with diffusive slip-links 9.1 Non-specific binding (∆ParB/S ) and exclusive binding in the slow dissociation regime We implemented a system with diffusive slip-links that can bind and unbind anywhere (∆ParBS), so that this system obeys detailed balance and hence relaxes into thermodynamic equilibrium. The scaling we observe for the ∆ParB/S scenario p ∼ ∆ 1.7 (Fig. S13a, left) is nearly identical to that of a random polymer p ∼ ∆ dν , dν ≈ 1.8 [62]. The distribution of θ is highly peaked at θ = 0, showing that the diffusive slip-links almost always enclose self-loops and almost never enclose nested loops (Fig. S13a, right). The system with diffusive slip-links that can only bind to the loading site breaks detailed-balance, but we find that for slow dissociation kinetics, the resulting distribution of loop-sizes p ∼ ∆ 1.8 is nevertheless consistent with that of a random polymer (Fig. S13b).

Exclusive loading site, arbitrary dissociation kinetics
When increasing the dissociation kinetics, we observe a sigmoidal increase of θ to a θ max at k − → ∞. θ max tends to 1 as we increase the system size (Fig. S14).
Fig. S14. There is a sigmoidal relation between θ and k − and θ max → 1 as N m → ∞. We define θ max as the the saturating value of θ as k − → ∞ (inset). As the system size N m → ∞, we observe θ max → 1 (main plot).
We can understand the relationship between θ and k − by constructing a simple mean-field model. In particular, we model the loop topology of the slip-links as N p uncorrelated two-state systems: (i) self-loops (S) with a lifetime τ S ; (ii) nested loops (NS) with a lifetime τ NS . The fraction of nested loops θ is simply the weighted lifetime of a nested loop: where the p 0,S , p 0,NS are the probability for a diffusive slip-link to enclose a self, nested loop after loading to ori. The lifetime of a self-loop is τ S = k −1 − . We estimate the lifetime of a nested loop by where v is the mean velocity of a slip-link moving through the bulk of the polymer. Since the nested loops propagate ballistically in the high density phase (Fig. S15, inset; SI movie 3b), there exists a well-defined velocity, but this velocity v does depend on the system size. In particular, we empirically find for where k 0 is the slip-link movement attempt rate and c ≈ 9 (Fig. S15). This 1/N 2 m scaling is distinct from the 1/N m scaling observed in the Simple Symmetric Exclusion Process [63], likely due to the presence of polymer loop entropy that impedes the movement of slip-links away from the loading site. . The data in this graph were computed in the high-density and fast-dissociation regime φ p = 0.4, k − = k 0 . Inset: Individual trajectories of diffusive slip-links l(t) were collected (thin, gray), where l(t) is the distance traversed by a tracer slip-link at time t after it was loaded onto the polymer. The ensemble-averaged trajectory l(t) (thick, blue; shaded region is the standard deviation) was computed by binning the trajectories. The average velocity is defined as v = l(t max ) /t max (blue, dashed).
In sum, our estimate for θ is Given an estimate of c from a measurement of v (Fig. S15), this form for θ does not contain any free fitparameters, and collapses data for various N m , k − onto a single master curve (Fig. S16).
10 Diffusing cross-diagonal for motor slip-links without a loading site Time-averaged contact maps of a polymer with motor slip-links but no loading site (∆ParB/S ) are, apart from the main diagonal, homogeneous. Upon closer inspection, however, movies of the contact maps (SI movie 7a) suggest  N m = 100). A meanfield-theoretical estimate (equation (S1)) describes these data well (solid curves). Inset: The mean-field-theoretical estimate for θ (equation (S1)) collapses the data onto a single master-curve x/(1 + x). The reduced parameters in this graph are:θ = (θ − θ min )/(θ max − θ min ), θ max = 1, θ min = 1 a diffusing cross-diagonal. Additionally, movies of the loop diagrams show that the motor slip-links extrude large loops until they collide and block further motion (SI movie 7b). To investigate this quantitatively, we perform Hough transforms of the contact maps. The Hough transform maps Cartesian space (x, y) into Hough-space (r, θ). Lines in Cartesian can be represented as r = x cos θ + y sin θ, and the Hough transformed image is simply the image of these (r, θ). Thus, we can easily identify the lines in our contact maps.
Intuitively, Hough transformed contact maps of this system with one motorized slip-linkN p = 1 only contain a locus that corresponds to the main diagonal (Fig. S17a). Interestingly, however, when we have two motor slip-linkŝ N p = 2, there appears a very clear additional locus in the Hough transformed contact maps whose position diffuses randomly over time ( Fig. S17b and SI movie 8).

The impact of asymmetrical translocation on chromosome organization
We implemented asymmetric active translocation or loop extrusion as detailed below (see SI movie 9 for the dynamics of a loop diagram and the contact map in Fig. S18): 1. A slip-link is added to the loading site 2. One of the two sides of the slip-link is immobilized 3. The side of the slip-link that is not immobilized is assigned a random translocation direction (i.e. positive or negative). For these data, we bound one asymmetric highly persistent motorized slip-link to the loading site. Note, in this simulation, none of the slip-links perform symmetric loop extrusion, which is the type of motor activity discussed in the main text.