Comparison of loop extrusion and diffusion capture as mitotic chromosome formation pathways in fission yeast

Abstract Underlying higher order chromatin organization are Structural Maintenance of Chromosomes (SMC) complexes, large protein rings that entrap DNA. The molecular mechanism by which SMC complexes organize chromatin is as yet incompletely understood. Two prominent models posit that SMC complexes actively extrude DNA loops (loop extrusion), or that they sequentially entrap two DNAs that come into proximity by Brownian motion (diffusion capture). To explore the implications of these two mechanisms, we perform biophysical simulations of a 3.76 Mb-long chromatin chain, the size of the long Schizosaccharomyces pombe chromosome I left arm. On it, the SMC complex condensin is modeled to perform loop extrusion or diffusion capture. We then compare computational to experimental observations of mitotic chromosome formation. Both loop extrusion and diffusion capture can result in native-like contact probability distributions. In addition, the diffusion capture model more readily recapitulates mitotic chromosome axis shortening and chromatin compaction. Diffusion capture can also explain why mitotic chromatin shows reduced, as well as more anisotropic, movements, features that lack support from loop extrusion. The condensin distribution within mitotic chromosomes, visualized by stochastic optical reconstruction microscopy (STORM), shows clustering predicted from diffusion capture. Our results inform the evaluation of current models of mitotic chromosome formation.


INTRODUCTION
Dynamic chromatin organization during interphase is crucial for gene regulation and other nuclear processes. In mitosis, chromatin compacts to give rise to well-defined Xshaped chromosomes, a prerequisite for their faithful segregation. At the basis of higher order chromatin organization lie Structural Maintenance of Chromosomes (SMC) complexes, large protein rings that have the ability to topologically entrap DNA (1)(2)(3). SMC rings include an ATPase, suggesting that energy is expended to organize chromatin or to regulate the process. During interphase, the major chromosomal SMC complex is the cohesin complex that establishes cohesion between the newly replicated sister chromatids. It does so by topologically entrapping the two sister DNAs. Cohesin also participates in organizing interphase chromatin into topologically associating domains (TADs). As cells progress towards mitosis, a second SMC complex, condensin, rises in importance. Condensin is enriched, or activated, on mitotic chromosomes to promote chromosome compaction. Without condensin, chromosomes fail to reach their mitotic shape and are unable to segregate, leaving behind anaphase bridges. The molecular mechanism by which SMC complexes organize chromatin has remained a matter of debate. Two prominent models postulate that SMC complexes actively extrude DNA loops (loop extrusion), or that they sequentially entrap two DNAs that come into proximity by Brownian motion (diffusion capture).
Both condensin and cohesin have, under certain conditions, been observed to extrude DNA loops in vitro (4)(5)(6)(7). Once bound to DNA, these SMC complexes asymmetri-cally or symmetrically reel in DNA, thereby forming a DNA loop. However, the same experiments suggest that little ATP is hydrolyzed to rapidly move these molecules over long distances. Applied to chromatin, condensin has been proposed to similarly reel in chromatin until it reaches a neighboring condensin complex that is itself engaged in loop extrusion. This would lead to formation of a central protein scaffold from which DNA loops emerge, reflecting chromosome models based on cytological and early biochemical analyses (8)(9)(10). Simulations of this process taking place on human chromosomes have shown agreement with experimentally observed chromosome formation, chromosome axis establishment and sister chromatid resolution (11). A feature of the loop extrusion model is that condensin-mediated DNA contacts will always lie within the same chromatin chain. Whether condensin can indeed extrude loops on a chromatin substrate densely decorated by histones and other DNA binding proteins, as well as overcome the barriers that higher level chromatin organization is likely to impose (12,13), remains unclear.
An alternative mechanism by which condensin can contribute to chromosome formation is by stabilizing stochastic pairwise interactions between condensin binding sites (14). We refer to this mechanism as 'diffusion capture'. A condensin complex that has topologically loaded onto DNA might be able to embrace a second DNA that comes into proximity by Brownian motion. This mechanism could be akin to cohesin's ability to capture a second DNA, following its loading onto a first DNA (15). Alternatively, two condensin complexes that each embrace one DNA might engage with each other. A tendency of SMC complexes to form clusters on DNA in vitro (16)(17)(18) is consistent with the latter possibility. In the diffusion capture scenario, condensin establishes contacts both within chromosomes and between chromosomes, consistent with experimental observations in yeasts (19)(20)(21). Computational simulation of diffusion capture taking place on a small budding yeast chromosome has generated chromosome properties with a good fit to experimentally observed chromosome behavior (14). Whether diffusion capture suffices to govern the formation of larger chromosomes is not known.
In this study, we developed a coarse-grained Brownian dynamics simulation of a chromatin chain, the size of the long left arm of fission yeast chromosome I. We use these simulations to explore the consequences of loop extrusion and diffusion capture on chromosome formation. We compare predictions from both models to experimental observations in fission yeast. Both loop extrusion and diffusion capture result in chromosome formation and chromosome contact distributions similar to those observed in vivo. In addition, diffusion capture provides an efficient means to recapitulate condensin-dependent chromosome axis shortening and volume compaction, as well as experimentally observed chromatin mobility changes inside mitotic chromosomes. Finally, the localization of condensin within mitotic chromosomes using STORM imaging reveals condensin clusters that are predicted to arise from diffusion capture. We conclude that diffusion capture represents an appealing mechanism that we propose contributes to chromosome formation in fission yeast.

S. pombe strains and culture
All the S. pombe strains used in this study are listed in Supplementary Table S1. To construct the Cut14-SNAP strain, the SNAP coding sequence (New England Biolabs) was cloned into a pFA-based fission yeast C-terminal tagging vector, then the C-terminus of the endogenous cut14 + locus was fused to SNAP by PCR-based gene targeting (22). Strains were cultured in Edinburgh minimal medium (EMM) supplemented with 2% glucose and 3.75 g/l of Lglutamic acid as a nitrogen source. To arrest cells in mitosis, 5 g/ml of thiamine was added to the EMM culture to repress Slp1 expression and incubated for 3 h at 25 • C. For Cut14 depletion, cells were incubated for 90 min after the addition of 5 g/ml thiamine at 25 • C to repress both Slp1 and Cut14 expression, and then 0.5 mM of the auxin 3indoleacetic acid (IAA) was added to the culture to degrade Cut14 and incubated for another 90 min at 25 • C before cells were collected.

Measurement of DNA volume and chromatin loci distance
Cells were fixed with 70% ethanol and then stained with 4 ,6-diamidino-2-phenylindole (DAPI). Images were acquired as serial sections along the z axis on a DeltaVision microscope system (Applied Precision). To measure the DNA volume, all the images were deconvolved in Soft-WoRx and then the voxels over an arbitrary DAPI signal intensity threshold were counted using the 3D objects counter in Fiji (23,24). Distance distribution data between chromatin loci was adopted from (25).

Chromatin mobility tracking and mean square displacement (MSD) calculation
For chromatin mobility tracking, a single focal plane of live cells was imaged at 20 ms intervals using a custom-built spinning-disc confocal microscope system (Intelligent Imaging Innovations) (25,26). The movement of a fluorescent dot was automatically traced using Virus Tracker (https://github.com/djpbarry/CALM/wiki/ Virus-Tracker). The weighted mean of the MSD was calculated using the @msdanalyzer Matlab class (27). Further details are described in (24).

Anisotropy of motion determination from trajectories at short times
From a trajectory in 2D, we determined anisotropic motion where the diffusion constant is not the same in all directions and/or there are different constraints in one direction or another. In both cases, the MSD along each direction will be different. In the case of a polymer like chromatin, there is in general no good frame of reference, since the local environment rearranges over time. Over long times the MSDs along two axes will therefore be the same. However, over short times the local environment will be relatively constant, and the MSDs along two axes will show a difference if there are anisotropic constraints. For this reason, we define timedependent anisotropy η in the following way: is the MSD in the y-direction and r 2 ( t) = x 2 ( t) + y 2 ( t) the total MSD in 2D. Note that η is an average over a number of trajectories, where for each trajectory the MSD is calculated by an average over all displacements with delays t. η is roughly the squared normalized average difference between the diffusion constants in x and y directions, D x and D y , and so we can roughly relate the ratio of these diffusion constants to η in the following way where without loss of generality we assume that D x ≤ D y , by assuming the x direction is that corresponding to the smaller diffusion constant.

STORM microscopy
Cells were fixed with 3.7% formaldehyde in PEM buffer (0. STORM images were collected on a Bruker Vutara 352 commercial 3D biplane single molecule localization microscope using a 60× silicone objective (Olympus) with a numerical aperture of 1.2 (28). We used a 640 nm laser with 50% laser power for illuminating Alexa Fluor 647 and a 405 nm laser with 0.5% laser power for photo-activation. Fluorescent signals were captured on an ORCA-Flash4.0 CMOS camera (Hamamatsu) using 20 ms exposure. We collected 30 000 frames and eliminated the first 10 000 frames for data processing.
To determine precise particle localization, we followed a previously described data processing method (28) with slight modifications. Briefly, we removed localizations with lower quality score (<0.8, the value ranging from 0 to 1) according to the goodness-of-fit metric of each localization event. We then removed localizations that did not blink for longer than three frames. Finally, we eliminated all localiza-tions with a lower axial precision (>100 nm). Filtering was performed using Bruker's SRX software.

Simulation of a coarse-grained chromatin chain
A virtual chromatin chain was constructed to study the expected behavior of the long left arm of Schizosaccharomyces pombe chromosome I. The chain comprises 1880 consecutively connected beads with a radius of 25 nm, each reflecting a string of ∼10 nucleosomes covering a genomic size of ∼2 kb. This chain thus corresponds to a genomic length of ∼3.76 Mb, equivalent to the long S. pombe chromosome I left arm. Any two connected beads elastically interact and any two beads that overlap mutually repel. In the absence of introduced condensation mechanisms, each chromatin bead undergoes Brownian motion, constrained by attractive and volume exclusion forces. Effectively, the relaxed chain behaves as a self-avoiding Rouse polymer ( Figure 1A).

Chromatin bead unit.
A linear array of 10 nucleosomes with 10 nm diameter including linker DNA reaches just over 100 nm (radius 50 nm). Tight hexagonal packing of 10 nucleosomes in turn results in an assembly with radius 15 nm. This gives us upper and lower bounds for the size of a 10 nucleosome unit. Based on fine-grained simulations of a histone chain (14), we observe that 10 nucleosomes in a chromatin chain typically occupy a volume with a radius of ∼25 nm. This volume is only partially filled with nucleosomes and is accordingly modeled as a soft sphere without a rigid boundary.
Special sites on the chromatin chain. While the virtual chromatin chain is a homopolymer in a physical sense, a few beads are marked as special sites corresponding to their biological roles. The first and last beads of the chain represent the telomere and centromere, respectively. A group of beads with 0.1, 0.7, 1.2, 1.8 and 2.2 Mb genomic distance from a locus close to the centromere are labeled to correspond to fluorophore-tagged sites, allowing inter-fluorophore distances to be monitored akin to experimental observations (25,29). 158 beads are selected to be 'condensin binding sites'. Their distribution is based on a condensin ChIP experiment in fission yeast (25). The mean distance between neighboring condensin binding sites is 11.7 beads (23.4 kb), the median distance is six beads (12 kb) ( Figure 1B). These beads are either the 'host' sites of condensin to mediate diffusion capture or the starting positions of condensin to initiate loop extrusion.
Boundary condition and initial configuration. To resemble conditions in the interphase S. pombe nucleus, the chromatin chain was placed in a spherical volume of 14.14 m 3 (1.5 m radius) with a rigid boundary to represent the S. pombe nucleus. S. pombe interphase chromatin contains few defined structural domains, such as TADs (25,30). For this reason, a set of 'random' conformations was created within a cylindrical subsection (3.84 m 3 ) of our virtual nucleus, corresponding to the fraction that the chromosome I left arm represents of the total fission yeast genome. The cylindrical constraint was removed and evolution of the ini-Nucleic Acids Research, 2021, Vol. 49, No. 3 1297 tialized chromatin chain was then subject to the rules and physics-based forces introduced below.
Forces employed. In the absence of active processes governing condensation, a bead i in the chromatin chain is subject to a stochastic force F sto i sourced from collision with molecules in the nucleoplasm, a tension force F ten i imposed by connected beads, and a volume-exclusion repulsive force F rep i exerted by spatially overlapping beads. Additionally, a damping force F dam i = − λ v i acts on the moving bead, representing the viscous effect of the nucleoplasm, the magnitude of which is assumed to be proportional to instantaneous speed.

Stochastic force.
A stochastic force is applied to each bead, both chromatin beads and condensins (see below), at each simulation step as: Any component of the stochastic force ξ (t) has a timeaverage of zero and is uncorrelated in space and time. Namely, In practice, at each step, the instantaneous value of any component of the force ξ (t) is calculated as √ c 1 multiplied by a random number drawn from a Gaussian distribution with a zero mean and a standard deviation of 1. The constant c 1 is set as: with Stokes-Einstein relation: where D represents the diffusion coefficient, η dynamic viscosity and r is the bead radius. Since the coarse-grained bead does not represent a rigid sphere but rather a flexible chain of ∼10 nucleosomes, the relation λ = 6πηr does not apply. For simplicity, we introduced a plausible damping constant λ. The value of c 1 allows the bead to have an average movement on a scale consistent with experimental observations (24,31). Coarse-grained bead movement is principally regulated by the entropic force and the spring constant of the chromatin bead linker. This parameter pair was chosen such that the bead displacement distribution over short (20 ms) time intervals was compatible with that observed in the S. pombe interphase nucleus (25).

Tension force.
A linear elastic force (e.g. Hookean spring) is applied to describe the interaction between two consecutively connected coarse-grained beads: where K ten is the spring constant of the linker, d i, i +1 is the distance between the centers of two consecutively connected beads i and i +1; c 2 is a constant describing the equilibrium (non-stretched or non-compressed) length of the bead linker;û i, i +1 are unit vectors determining the direction of the force.
Repulsion force. In order to limit overlaps between any two beads, a constant volume exclusion force between two beads within d rep0 < 50 nm of each other is applied. Unless stated otherwise,û a, b denotes a unit vector from object a to object b.
where c 3 is a constant equal to 0.5; j is a bead different from i.

Diffusion capture simulations
The diffusion capture model describes the crosslinking of distal genomic sites mediated by condensin. To implement this model an additional pair of beads, representing a condensin molecule, is bound to condensin binding sites. When two condensins bound to different binding sites stochastically become spatially adjacent, they have a probabilistic propensity of association ( Figure 1C). In some simulations, we titrated the condensin concentrations such that we removed randomly 85, 75, 50 or 25% of condensin molecules from their binding sites.
Forces employed. Three additional forces are employed in order to implement the diffusion capture model. A condensin structure force describes the interaction between the two condensin 'feet' that reflects the structural integrity of a condensin molecule; a condensin attachment force describes the interaction between both condensin feet and a chromatin bead that maintains condensin attachment to the chromatin chain; a condensin capture force describes the interaction between condensins on different beads that mediates diffusion capture.
Condensin structure force. While condensin is modelled as two beads, a front and a rear 'foot', only the front foot participates in diffusion capture. A linear elastic force is applied between the two feet to maintain their spatial proximity, which becomes important later in the loop extrusion model.
where c 4 , representing the equilibrium distance between the two feet, equals 0 nm. The radius of both feet is 25 nm.
Condensin attachment force. The interaction between each condensin foot and its binding site is described as: F attach f oot = K ten d f oot, j − c 4 û f oot, j , where c 4 is equal to 0 nm; j refers to the chromatin bead that condensin is attached to.
1298 Nucleic Acids Research, 2021, Vol. 49, No. 3 Condensin capture force. The condensin capture force, F condensin i , is applied between condensin front feet on different binding sites as an elastic spring following Hooke's law: This force is exerted when the center of front foot ii and the center of another front foot j j are within a cut-off distance d ii, j j = 77 nm. This equates to a distance of 27 nm between the bead surfaces, a conservative estimate for a distance that might be bridged by a condensin molecule. p is the dissociation probability which represents turnover of diffusion capture pairs. Algorithmically, it is implemented through a random number generated at each time step : if a random number r p is less than a threshold p then F → (d ii, j j , p) condensin i becomes zero. c 5 is the equilibrium distance between two interacting condensins. Here, we define c 5 as 52 nm, meaning that two condensins lie adjacent. The valence of diffusion capture sites, F → (d i, j , p) condensin i , representing the number of interacting partners, is naturally regulated by the forces in the system and steric constraints.

Loop extrusion simulations
In contrast to diffusion capture, where condensins attach to their binding sites and remain in position, in the loop extrusion model condensins load at empty binding sites from where they translocate. The two condensin feet symmetrically move in opposite directions along the chromatin chain by repeatedly associating with the next chromatin bead, thereby bridging distant genomic sites to form a chromosome loop ( Figure 1D). When two condensin complexes encounter each other, movement of colliding feet is stopped. Condensin feet that are not in collision continue translocation, resulting in further asymmetric loop extrusion. The rate of translocation is given by: where p sliding is a probability of translocation, n is the number of travelled beads, n bp represents the DNA length in bp per bead and t step is the simulation timestep. In our simulation, υ LE ∼ 1.2 kb/s in line with experimentally observed values (4,32). To introduce condensin turnover, each condensin can stochastically unbind from the chromatin chain and relocate. A constant condensin concentration on chromatin is achieved such that every time a condensin is unloaded, a new condensin is loaded at an empty binding site. Algorithmically, dynamical condensin exchange is implemented such that at T exchange time intervals a dissociation probability p on/of f is calculated (akin to the condensin dissociation probability p in the diffusion capture model) for each condensin to decide whether it is unloaded from its current position and relocated to an empty binding site.

Forces employed
The loop extrusion model differs from diffusion capture in that the condensin attachment force is repurposed as an extrusion force F extrusion f oot, j , acting to elastically connect the translocating condensin foot with the associated chromatin bead.
where F extrusion f oot, j is a tension force allowing a condensin foot to interact with bead j on the chromatin chain. Bead j is iteratively being updated to the following chromatin bead j + 1 (in the case of forward-moving condensing foot) or j − 1 (in the case of backward-moving condensing foot), therefore allowing translocation of condensin along the chromatin chain and re-assignment of their F extrusion f oot, j from the current chromatin bead to the adjacent one. A condensin capture force is not operational in the loop extrusion model.

Dynamics and model implementation
The overdamped Langevin equation is employed to describe the time evolution of the coarse-grained chromatin configuration. This assumes that the inertial part m .. u i is much smaller than the damping part λ . u, on the longer timescales of interest in this study. Under this assumption, the dynamical equation to describe a free chromatin chain is as follows: In the diffusion capture model, additional forces are included to describe the movement of a specific bad i: In the loop extrusion model, movement of a specific bead i during simulation is controlled by a summation of forces: At each simulation step, the Euler integration has been applied to the dynamics equation in order to describe time evolution of the system, therefore movement of each bead is described for velocity v i x (t) and tension u i x (t + t) as follows: where F f ree , F DC , F LE , as given by the sum of terms in the equations above, correspond to forces employed in free chromatin chain, the diffusion capture model, and the loop extrusion model, respectively.

List of parameters regulating bead movement
See Table 1.

Measurements and readouts
All simulations were run for 1200 s with a simulation timestep dt = 10 −4 s. Each simulation condition for diffusion capture and loop extrusion was recapitulated with 10 simulation replicates. If not stated otherwise, readouts were collected every 10 s, resulting in 1200 measurements from the 10 replicates.
Computational fluorophore distance measurements. We mapped experimental fluorophore arrays (25,29) onto the computational chromatin chain and measured Euclidean distances between selected fluorophore pairs.
Computational Hi-C and interaction frequency analysis. We generated Hi-C-like representations of chromosome conformations during our simulations. Instead of contact frequency, we display Euclidean distance between any two chromatin beads, averaged over 12 000 conformations collected at 1 s intervals from the 10 simulation replicates. To plot interaction frequency as a function of genomic distance, we considered two beads as interacting if their Euclidean distance was within 500 nm. The principle conclusions from this analysis were insensitive to the chosen cutoff. All bead pairs were binned according to their genomic separation to generate a frequency distribution. The frequencies are normalized to have a sum of one across all bins.
Computational volume measurements. To facilitate volume measurements, we divided our system into 3D voxels (cubes). Each voxel has a dimension of 100 nm. We define the volume of the chromatin chain as the total volume of occupied voxels.

Condensin clustering analysis.
Clustering is performed using a DBSCAN algorithm in the open-source python library sklearn.cluster. A cutoff distance of 100 nm between centers of the two feet of individual condensins is selected to reflect two condensin diameters in the model. A minimum number of condensins per cluster of 2 is chosen for the comparison between STORM data and the computational diffusion capture and loop extrusion models.
Simulation MSD measurements. In order to determine the MSD exponent of chromatin mobility in our simulations, we used the same approach as for the experimental data. Instead of the experimental fluorophore labelled chromatin locus, we tracked the position of the 158 condensin binding sites. We analyzed the MSD exponent for each 2 s window, collected every 30th second during the simulations. Since in the experiments we only observe a randomly oriented 2D projection of the full 3D fluorophore motion, we applied a 3D to 2D projection of particles in our simulations. We found previously that this projection does not alter the mean MSD exponent, but slightly broadens the distribution (24).
Simulation anisotropy measurements. Anisotropy of chromatin bead motion in our simulations was determined as described for the experimental data during the same time windows as the MSD exponents.

A biophysical model of diffusion capture and loop extrusion along the fission yeast chromosome I left arm
To study fission yeast chromosome condensation, we developed a biophysical model of a chromatin chain representing the length of the long left arm of fission yeast chromosome I. Our coarse-grained chromatin chain consists of 1,880 beads, each covering a ∼2 kb region corresponding to ∼10 nucleosomes, representing 3.76 Mb of genomic distance ( Figure 1A). A stochastic force (F sto ) is applied to every bead of the system, under the assumption that a chromatin bead follows Brownian motion in isolation. Any two consecutive beads interact via a spring-associated tension force (F ten ) following Hooke's law. This results in collective dynamic behavior of a joined chromatin chain. In addition, a repulsion term is employed when beads overlap (F rep ), taking into account the soft nature of the chromatin chain within each bead. We consider the behavior based on F sto , F ten and F rep to be that of a free chromatin chain (Figure 1A). To simulate diffusion capture and loop extrusion, we selected specific beads along this polymer chain as condensin binding sites, recapitulating the experimentally observed condensin distribution along fission yeast chromosome I ( Figure 1B)   other by stochastic movements they form a pairwise interaction with a defined probability via a condensin capture force (F condensin , Figure 1C). When multiple condensins spatially meet at a common place, they are able to form larger clusters, limited in size only by the geometric constraints of the system. In vivo, condensin shows dynamic turnover on chromosomes (33,34). While condensin in our diffusion capture simulations remains stably chromatin bound, its dynamic nature is regulated by the condensin-condensin association probability. This probability not only controls formation of new diffusion capture pairs, but also maintenance of these bonds at every simulation step.
In the loop extrusion model, the condensin attachment force is repurposed as an extrusion force (F extrusion ). Condensin initially binds to a condensin binding site, from where its front and rear feet start translocating into op-posite directions. F extrusion sequentially targets chromatin beads next to the current bead of residence, resulting in symmetric loop extrusion ( Figure 1D). The two condensin feet remain connected to each other by a condensin structure force (F structure ). When two condensins encounter each other, movement of the colliding feet is stopped, while feet that are not in collision continue translocation, resulting in asymmetric loop extrusion until they also encounter another condensin (see our discussion for the case that condensins can pass each other (32)). Loop extruding condensins periodically have a chance to unload and load again at a free condensin binding site, thus ensuring dynamic loop formation and loop nesting. Parameters are chosen to match experimentally observed loop extrusion rates (4,32).
Fission yeast condensin accumulates in the nucleus in mitosis. During interphase, nucleo-cytoplasmic shuttling leads to condensin redistribution and equalization between the compartments (35,36). We have previously determined the nucleus-to-cytoplasm ratio in fission yeast to be 0.14 ± 0.05 (24). With therefore approximately 15% of nuclear condensin, we use 15% occupied condensin binding sites to represent in silico interphase, while we refer to 100% condensin binding site occupancy as in silico mitosis. Further details on the computational implementation of the diffusion capture and loop extrusion models can be found in the Materials and methods.

Axial chromosome compaction by diffusion capture and loop extrusion
Axial shortening is a hallmark of condensin-dependent mitotic chromosome formation in yeasts (25,29,(37)(38)(39). To inspect axial chromosome compaction, we monitored the distance of two fluorophore-marked loci at 1.8 Mb distance from each other in vivo and of similarly spaced in silicomarked loci in our model. The median in vivo interphase distance, projected onto a 2D plane, was 1.1 m in interphase, which shortened by ∼ 39% to 0.65 m in mitosis ( Figure 2A) (25). The distance of the same fluorophore pair was previously measured in 3D to around 1.8 m in interphase contracting to around 1.0 m (i.e. by 44%), in mitosis (29). Mitotic compaction in both studies depended on condensin.
We started our computational simulations of diffusion capture from a relaxed chromatin chain, with either 15% (interphase) or 100% (mitosis) of occupied condensin binding sites. Diffusion capture pairs start to form and the system approaches a steady state when the number of capture pairs and the in silico fluorophore distance fluctuates around a constant value (Supplementary Figure S1A). Figure 2B shows representative conformations of our computational chromosome in both conditions. The real time movements of the chromatin chain can be observed in Supplementary Movies S1 and S2, illustrating frequent exchange of diffusion capture pairs in the steady state. We recorded 1200 3D fluorophore distance measurements at regular time intervals from 10 independent simulation repeats. These measurements show a well-defined distribution with a median of 1.2 m in interphase and 0.89 m in mitosis, roughly compatible with experimentally observed 3D distances and corresponding to a 26% mitotic chromosome axis shortening due to diffusion capture.
We next turned to the loop extrusion model. Upon the initiation of loop extrusion using either 15% or 100% of condensin per loading site, loops rapidly form and an axial condensin accumulation becomes discernable over time (Supplementary Figure S1B and Supplementary Movies S3, S4). At the interphase condensin concentration, a relatively short axial structure forms with long chromatin loops (Figure 2C). The in silico fluorophore distance is influenced by where the fluorophores find themselves relative to the axis, with a median distance of 1.3 m in interphase. At the higher mitotic condensin concentration a greater number of loops, including a greater fraction of nested loops, are formed. This results in shorter loops and correspondingly a longer chromosome backbone. The fluorophore distance now depends on how the backbone arranges itself inside the chromosome, resulting in a simulated median Euclidean fluorophore distance of 1.2 m. This corresponds to an 8% chromosome arm shortening, less than what was achieved by diffusion capture.
To further explore the relationship between genomic and Euclidean distances in the diffusion capture and loop extrusion models, we inspected chromatin beads at 0.1, 0.7, 1.2, 1.8 and 2.2 Mb distance, corresponding to previously experimentally observed fluorophore pairs (29). In silico interphase in either the diffusion capture or loop extrusion models recapitulated in vivo measured interphase distances reasonably well (Supplementary Figure S2A). Diffusion capture resulted in mitotic axial compaction in almost all observed cases, albeit not to the full extent that is observed in vivo (Supplementary Figure S2B). Thus diffusion capture makes a robust contribution to mitotic axial chromosome compaction. In contrast, loop extrusion often exhibited the opposite trend, generating increased mitotic Euclidean distances. In the loop extrusion model, additionally activated condensin in mitosis will shorten chromatin loops and add new loop anchors to the chromosomes backbone, thus resulting in the overall lengthening of the chromosome axis. Additional mechanisms might be required to achieve reproducible chromosome axis compaction in the loop extrusion model, for example additional condensin-condensin interactions (40).

In silico contact probability distributions due to diffusion capture or loop extrusion
Chromatin contact probability distributions, obtained from high throughput conformation capture (Hi-C) experiments, contain important information on chromosome architecture (41). Condensin is known to enhance longer-range chromatin contacts during mitotic chromosome condensation at the expense of local chromatin contacts (25,42,43). Figure 3A shows experimental Hi-C maps of the fission yeast chromosome I left arm in interphase and mitosis, as well as the Hi-C interaction frequencies plotted as a function of their genomic distance. This illustrates enhanced mitotic chromatin interactions in a distance range from ∼90 to 900 kb, which depend on condensin (25).
To generate Hi-C-like depictions of our computational chromosome conformations, we display Euclidean dis- tance maps, averaged over time and between simulation replicates ( Figure 3B and C). These maps reveal that, in both the diffusion capture and loop extrusion models, the increased mitotic condensin concentration results in increased longer-range proximities, as seen by an expanded mitotic diagonal. To analyze interaction frequencies as a function of genomic distance, we set an arbitrary Euclidean distance cutoff at 500 nm to convert proximity into 'in silico Hi-C interactions'. In the case of dif-fusion capture, the interaction frequency plot reveals increased mitotic interactions over a distance range of 120-1100 kb ( Figure 3B), in approximate agreement with the experimental observations. Loop extrusion also resulted in increased longer-range interactions, albeit at a somewhat shorter distance range of 60-600 kb ( Figure 3C). Thus, both in silico diffusion capture and loop extrusion recapitulate condensin-dependent mitotic chromatin contact changes.   To better understand the distance range of enhanced mitotic chromatin interactions, we titrated the condensin concentration in our simulations. In the case of diffusion capture, the interaction frequency plot of a free chromatin chain (0% condensin) showed only little difference from our interphase conditions (15% condensin). As soon as additional condensin binding sites were activated (25%), chromatin interactions in the 120-1100 kb distance range were augmented. Interactions increased further as more condensin was added, while the interaction distance distribution remained roughly constant ( Supplementary Figure S3A). Loop extrusion showed a different response pattern. Compared to the free chromatin chain, 15% of condensin resulted in an increase in chromatin interactions longer than 200 kb. This is likely explained by the formation of long chromatin loops in the presence of low condensin levels. As the condensin concentration increased, the distance range of chromatin interactions shortened, as expected from shorter chromatin loops. A close match to the experimental interaction frequency distribution was obtained at intermediate condensin levels (Supplementary Figure S3B). Thus, both the diffusion capture and loop extrusion models reproduce experimental interaction frequency distributions. The distance range of enhanced mitotic interactions is robust in the case of diffusion capture, but sensitive to the condensin concentration in the case of loop extrusion.

Chromatin volume compaction in mitotic chromosomes
A visually striking aspect of mitotic chromosome condensation is the volume reduction during the conversion of diffuse interphase chromatin into distinct chromosome bodies (44). In human cells this entails a ∼2-fold volume compaction (45). Indeed, chromosome compaction was one of the first described roles of the fission yeast condensin complex (37). To quantify fission yeast chromosome compaction, we measured the chromatin volume in interphase and mitosis by 3D reconstructing serial z-stacks of fluorescent microscopy images of DNA stained with 4 ,6-diamidino-2-phenylindole (DAPI). The median interphase chromosome volume was 2.06 m 3 which decreased in mitosis to 1.64 m 3 , a 20% volume reduction ( Figure 4A). Mitotic compaction depended on the condensin complex and was no longer observed following condensin depletion using a combined transcriptional shut-off and auxin-inducible degron strategy (46).
To measure chromatin volume in our simulations, we divided the nuclear volume into 100 nm-sized cubic voxels. We counted a voxel as occupied if it contained at least one chromatin bead. The chromosome I left arm accounts for ∼20% of the fission yeast genome. Its in silico volumes were somewhat larger than the corresponding fraction of the experimentally measured DNA volume. This is likely the case because chromosomes lie close together in the yeast nucleus, reducing their apparent occupied volume at our microscopic resolution. Despite the different numerical values, the diffusion capture model resulted in a 16% volume reduction when comparing interphase and mitosis ( Figure 4B). Volume reduction was condensin concentration-dependent (Supplementary Figure S4). This observation demonstrates that establishment of stochastic pairwise interactions be-tween condensin binding sites along a chromatin chain can contribute to chromatin volume compaction. In contrast, loop extrusion resulted in only marginal volume changes. Specifically, the median volume increased by 0.09% during simulated mitosis ( Figure 4B and Supplementary Figure 4). While interactions between distant parts of the genome are established by loop extrusion, the intervening chromatin is extruded, which limits the potential for volume compaction. Increased loop nesting, beyond that achieved based on simple probability, might be able to achieve increased levels of compaction in this model.

Diffusion capture reduces mitotic chromatin mobility
During mitotic chromosome formation, condensin imposes constraint on the free movement of the chromatin chain (25). To experimentally study chromatin movements, we track a chromatin locus in the middle of the chromosome I left arm, marked by tandem lac operators bound by a LacI-GFP fusion protein. We then plot its mean squared displacement (MSD) over time. During interphase, we find that the MSD exponent over short time intervals is 0.49 ± 0.02 (mean ± 95% confidence interval, n = 595), consistent with a polymer chain whose diffusive behavior is only slightly constrained by a small amount of condensin ( Figure 5A) (24). In mitosis, the exponent is markedly reduced to 0.28 ± 0.02 (n = 271). Looking more carefully at the distribution of MSD exponents from individual chromatin traces, the interphase distribution is well described by a single Gaussian fit. The mitotic sample, however, showed a bimodal distribution that likely arose from contamination with a small number of interphase cells. A pure mitotic MSD exponent might be lower than 0.28, possibly as low as 0.25 (Supplementary Figure S5A). Exemplar trajectories of the GFP-marked locus over time illustrate the reduced mitotic chromatin mobility ( Figure 5A). We next explored the consequences of condensindependent in silico diffusion capture or loop extrusion on chromatin mobility. Similar to experimental observations, we track chromatin beads in simulation replicates and plot their MSD over time. In the diffusion capture model, the interphase MSD exponent was 0.52 ± 0.03 (mean ± 95% confidence interval, n = 1320), close to the experimentally observed value. The exponent was reduced to 0.38 ± 0.02 during in silico mitosis. The mitotic mobility reduction is reminiscent of our in vivo observations, although the extent of the MSD exponent reduction did not fully reach the experimental observation. An example trajectory of a chromatin bead exemplifies the constrained mitotic mobility due to diffusion capture ( Figure 5B). In the loop extrusion model, the interphase MSD exponent was 0.54 ± 0.03. The MSD exponent remained almost unchanged under mitotic conditions when it persisted at 0.51 ± 0.03. A representative bead trajectory further illustrates the largely unchanged mobility ( Figure 5C). This suggests that the structural flexibility and dynamics of the chromatin chain is constrained by diffusion capture but remains largely unaltered during loop extrusion.
To study the effects of diffusion capture and loop extrusion on chromatin mobility further, we again turned to condensin titration in our simulations. The mean MSD expo- nent of the free chromatin chain was 0.53 ± 0.03 consistent with that of an unconstrained Rouse polymer chain with excluded volume (24). Condensin titration in the diffusion capture model sequentially led from an interphase MSD exponent to more and more constrained mobility at full condensin binding site occupancy (Supplementary Figure S5B). The effect of loop extrusion was also condensin concentration-dependent, however did not result in a mean MSD exponent reduction <0.50 at any of the investigated concentrations. These observations uncover diffusion capture as a powerful mechanism that confines chromatin movements and that could contribute to the striking mobility reduction observed during mitotic chromosome condensation in vivo.

Mitotic chromatin movements gain anisotropy
In addition to overall constrained mitotic chromatin mobility, expressed in a reduced MSD exponent, we investigated whether mitotic chromosome condensation impacts on the freedom of the directionality of movement, i.e. its anisotropy. We employed an anisotropy metric η( t) that evaluates whether diffusive movement is equal in x and y directions of the microscope plane, or is constrained in one of the directions more than the other ( Figure 6). In effect, η( t) corresponds to a difference between the diffusion constants in both directions. This metric is most meaningful over short times to probe local directionality constraints. Over longer times, the system locally tumbles resulting in apparent isotropic behavior. For this reason, we focus on the average anisotropyη over delays of up to 0.1 s. As a benchmark of our expectations for an isotropic polymer, we first analyzed the anisotropy of our simulated free chromatin chain. We expect η( t) → 0 as t → 0, though the finite time resolution of our experiment gives us a finite value forη. Under our sampling conditions, we findη = 0.019 ± 0.005 (Supplementary Figure S6), which means that by random chance we find diffusion in one direction being roughly 75% of that in the orthogonal direction. Applied to our experimental chromatin trajectories, this analysis revealed that chromatin movements in interphase showed greater anisotropy (η = 0.048 ± 0.005) compared to the isotropic simulated polymer (diffusion in one axis being 64% of that in the orthogonal direction). The anisotropy became more pronounced in mitosis (η = 0.08 ± 0.01), i.e. movement in one direction was now only 56% of that in the other. This increase in mitotic anisotropy depended on condensin ( Figure 6A). It should be noted that the actual experimental anisotropy could be greater, since our microscopy recordings project 3D movements to a 2D plane, effectively removing any possible difference in diffusivity along the z-axis. We interpret these observations to mean that condensin adds local directional constraint to the diffusive behavior of the chromatin chain in mitosis.
We next applied the anisotropy metric to our simulated chromatin movements. Compared to the free chromatin chain, interphase concentrations of condensin slightly increasedη in both the diffusion capture and loop extru- sion models (Figure 6B and C). Increasing condensin towards mitotic levels barely affected the anisotropy of movements in the loop extrusion model. In contrast, it resulted in a dose-dependentη increase in case of the diffusion capture model (Supplementary Figure S6). Qualitatively, therefore, diffusion capture led to increased anisotropy of chromatin movement. Quantitatively the resultant anisotropy remained below that experimentally observed. We imagine that condensin binding site clustering in the diffusion capture model results in a chromatin topology that constrains chromatin chain movement in certain directions, thus creating local anisotropy that we can experimentally and compu-tationally detect ( Figure 6D). The effect might be more pronounced in vivo where additional chromosome-bound proteins might augment any constraints. The overall more dynamic nature of the chromatin chain in the loop extrusion model did not create a similar phenomenon.

Condensin cluster formation within mitotic chromosomes
Given the above contrasting observations of how diffusion capture and loop extrusion impact on mitotic chromosome behavior, we wanted to directly visualize the emergent 3D organization of condensin inside mitotic chromosomes. To this end, we performed stochastic optical reconstruction microscopy (STORM) to visualize condensin within fission yeast mitotic chromosomes at high spatial resolution. We arrested fission yeast cells in mitosis by transcriptional repression of the Slp1 activator of the anaphase promoting complex (29). Condensin's Cut14 subunit was fused to a SNAP tag, which we labeled with an Alexa Fluor 647 dye following cell fixation and permeabilization. STORM imaging now allowed us to determine the location of con-densin molecules within the fission yeast nucleus. The particle count per nucleus was 1114 ± 110 (median ± S.E.M, n = 19), in line with the expected number of condensin molecules (47,48). Qualitatively, condensin molecules appear to cluster in small groups that are widely scattered throughout the nucleus ( Figure 7A; a partial volume corresponding to the chromosome I left arm is depicted in Figure 7B). To quantitatively describe condensin clustering, we performed DBSCAN cluster analysis of the condensin distribution (see Materials and methods). This revealed a predominance of small clusters with two to four condensin molecules while larger clusters with 10 or more condensins were also detected, but less frequently ( Figure 7C). We next performed a similar analysis of the condensin distribution in our simulated mitotic chromosomes formed by diffusion capture or loop extrusion. Qualitatively, diffusion capture led to the formation of condensin clusters of various sizes, spread throughout the chromosome volume. Loop extrusion, in contrast, resulted in the formation of an apparent condensin backbone, consisting of approximately evenly spaced condensin molecules ( Figure 7B). When we subject these condensin distributions to the same quantitative cluster analysis, we find that diffusion capture results in a broad distribution of cluster sizes, skewed towards large clusters. In contrast, rarely more than two condensin molecules were found to cluster during loop extrusion (Figure 7C). Both simulated distributions differ from the experimental observation. While the experiment and the diffusion capture model show a range of cluster sizes, the median cluster size in the diffusion capture simulations was 12, which is distinctly larger than the experimentally observed median cluster size of 4. We cannot exclude that STROM imaging counted a subset of condensin molecules more than once while they were blinking, potentially increasing our estimate of clusters with only a few condensins. Despite of this caveat, if condensin clusters form by diffusion capture in vivo, a mechanism likely exists that limits their size.
As another quantitative metric to compare the condensin distributions within native and simulated chromosomes, we plotted the distances of each condensin molecule from its nearest neighbor. The condensin clustering observed in the experimental data, as well as in the diffusion capture simulations, mean that the majority of condensins possess a close neighbor. The median nearest distance was 29 nm in our STORM data and 34 nm in the diffusion capture simulations ( Figure 7D). Condensins in the loop extrusion model were spread out along the chromosome backbone with a greater median distance from their nearest neighbors of 107 nm. This analysis confirms a clustering pattern that is generated by the diffusion capture mechanism that is lacking from the loop extrusion simulations.

DISCUSSION
We computationally examined the consequences on chromosome formation of two prevalent models of condensin function, loop extrusion and diffusion capture. At their essence, both models result in the establishment of loops between distant sites along a chromatin chain. Only the mechanisms by which these loops form differ. In the case of loop extrusion, active movement of the chromatin chain results in loop growth. Diffusion capture, in contrast, takes advantage of stochastic loop formation by Brownian motion. Condensin in the latter case merely acts to stabilize such loops for a period of time. These parallels and distinctions result in similarities between chromosomes that form by both mechanisms, but also in a number of differences. In our simulations, we assume that one condensin is active per every ∼20 kb of chromatin. This estimate stems from experimentally observed condensin ChIP distributions (25,30,39) as well as quantitative estimates of ∼1000 condensin complexes per fission yeast cell nucleus (47,48).

Implications for chromosome dimensions and chromatin density
Nucleic Acids Research, 2021, Vol. 49, No. 3 1309 During our mitotic loop extrusion simulations, all these complexes are equally active in extruding loops and in initiating nested loops, based on simple probability. This results in a chromosome backbone that is longer than experimentally observed and, notably, is longer in mitosis than in interphase. We note that a previous loop extrusion simulation study reported axial compaction, based on measuring the sum of pairwise distances between consecutive condensin molecules (11). Here, we measure Euclidean distances between in silico fluorophores, which we believe allows a more meaningful comparison to experimental values. To achieve native-like chromosome axis shortening by loop extrusion, it is possible that additional levels of condensin regulation tune loop intervals and loop nesting. While such mechanisms remain to be explored in organisms that rely on a single source of condensin, like fission yeast, the existence of two distinct condensins in other organisms could facilitate such regulation.
The condensin density on human chromosomes is similar to that in fission yeast (∼1 condensin per 20 kb (49)). If human condensin shapes chromosomes by loop extrusion, we should expect loop sizes and chromosome dimensions to be sensitive to changes in condensin concentration (11). Against this expectation, chromosome volume and shape are remarkably insensitive to substantial reductions in condensin levels (50)(51)(52). It will be interesting to further study the consequences of altered condensin concentrations on chromosome condensation in a defined experimental system (53).

Implications for chromatin mobility
Mitotic chromosomes are not a static end-product of chromosome condensation, they are dynamic entities whose integrity is maintained through continued condensin ATP hydrolysis cycles (54). In the loop extrusion and diffusion capture models, continued ATP hydrolysis maintains chromosome architecture in different ways that make distinct testable predictions about chromosome properties. In the diffusion capture model, condensin dissociation and reassociation gives condensin clusters the plasticity to evolve by merging or splitting. The net consequence of condensin clustering, however, is to limit chromatin movements and to impose anisotropy. In contrast, the loop extrusion model envisions that dissociating condensins initiate new loops that grow again by directional enlargement. In our simulations, loop extruding condensins turn over on average every 2 min. While condensin turnover on mitotic fission yeast chromosomes remains to be measured, 2 min corresponds to relatively stable association, when compared to budding yeast condensin or even human condensin II (33,34,55). Despite the therefore relatively slow turnover of condensin in our loop extrusion simulations, chromatin remains mobile and unconstrained in the directionality of movements, contradicting our experimental observations. While chromatin movements in interphase are close to what is expected from an unconstrained Rouse polymer with excluded volume (24), our experiments point to a markedly smaller MSD exponent in mitosis, potentially as small as 0.25. This reduction is partly reproduced by the diffusion capture model. An exponent of 1 / 4 has been described to arise from the behavior of long ring polymers in a melt or from ring polymers in a set of fixed obstacles (56,57), suggestive of a potential role of chromatin loops in the sub-diffusive behavior observed in mitosis. Understanding the quantitative nature of the observed diffusive behavior, in light of biophysical models of chromosome formation, remains an open challenge.

Condensin cluster formation within mitotic chromosomes
A predicted feature from the diffusion capture model is the formation of condensin clusters of variable sizes, spread throughout chromosomes. The loop extrusion model, in contrast, predicts that condensins are spaced out along a chromosome backbone. We could not discern such a chromosome backbone in our STORM images of mitotic fission yeast cells. Rather, condensin was found in dispersed small foci. While these foci are reminiscent of those predicted by diffusion capture, their median cluster size was smaller than observed in our simulations. We note that cluster size in our simulations is principally restricted by steric constraints created by the chromatin chain. These steric constraints can be expected to be greater in vivo, where numerous proteins in addition to histones decorate the chromatin chain. Such additional constraints offer one possible explanation for why cluster sizes might be smaller in vivo. Alternatively, other properties of condensin or of its chromosomal binding sites might limit cluster sizes. When we computationally limit cluster size to six condensin molecules, we obtain a good agreement with the experimentally observed cluster size range, while mitotic chromosome dimensions and dynamics remain comparable to those that result from unconstrained clusters (data not shown).
High resolution imaging of condensin in human chromosomes, using stimulated emission depletion (STED) microscopy, has also revealed condensin clusters instead of a continuous condensin backbone (49). While appearing overall scattered, these clusters were enriched towards axial positions inside human chromosomes. Loop extrusion is a powerful mechanism to explain axial enrichment. Expanding loops move outwards while pushing loop anchors towards the center. Could it therefore be that condensin shapes human chromosomes by a combination of diffusion capture and loop extrusion? To achieve loop extrusion, condensin has been proposed to employ an intrinsic motor, as observed in vitro (4,7). However, it remains uncertain whether condensin can extrude densely packed chromatin loops in vivo. We have therefore suggested in the past that loops that are established by diffusion capture could expand by means of an extrinsic motor, e.g. RNA polymerases that are known to reposition condensin along transcription units (39,58,59). Such an extrinsic 'loop expansion' mechanism (2) could similarly result in axial condensin cluster accumulation. If transcription indeed acts as an extrinsic motor that drives loop expansion, such a mechanism would be absent from chromosomes that form in the absence of transcription, e.g. in Xenopus oocyte extracts. It will be interesting to analyze the condensin distribution within such chromosomes (60).

Outlook
A perceived benefit of loop extrusion is that it provides a fool-proof mechanism to ensure that condensin-dependent chromatin interactions happen within the same chromatin chain, rather than between neighboring chromosomes. However, experimental observations suggest that condensin promotes interactions both within as well as between chromosomes (19)(20)(21). If diffusion capture is blind as to whether interactions are established within or between chromosomes, how can we explain condensin's ability to individualize chromosomes? There are a number of natural mechanism that promote individual polymer chains to separate, including excluded volume interactions (61) and entropydriven forces under confinement (62). These mechanisms will result in preferential diffusion capture interactions within each chromatin chain. Indeed, our previous simulations of diffusion capture along two small neighboring budding yeast chromosomes revealed how, despite repeated interactions between chromosomes, chromatin chains individualize over time (14). Diffusion capture thus provides an inherent mechanism that sufficiently explains chain compaction and individualization, at least of small chromosomes. In the case of larger chromosomes, loop expansion following diffusion capture could aid chromosome individualization as outwards moving loops repel each other (40,63).
Recently it was observed that, in vitro, loop extruding condensins can pass each other to form what was coined z-loops (32). We investigated the consequences of allowing condensins to pass each other to form such z-loops, while maintaining all other simulation parameters unchanged. The outcome was the emergence of very compact, highly entangled structures (data not shown). While z-loops therefore might overcome some pitfalls of the loop extrusion model, e.g. with regards to chromosome volume reduction, their formation would have to be carefully regulated. An excess of chromatin entanglement that the mechanism readily produces is unlikely to be physiological.
Lastly, our diffusion capture model is a specific case of a string-and-binder polymer model (64,65). A feature of such models is that they can lead to a collapse of the polymer chain into a dense ball. We found this to be the case only when condensin binding sites were much more closely spaced than experimentally observed. Using actual condensin spacing, local clusters form that are isolated from neighboring hubs by steric constraints. While clusters evolve over time by dynamic exchange of condensin binding sites, the overall chromosome remains in a stable steady state. In summary, we have provided arguments to suggest that diffusion capture can make an important contribution to mitotic chromosome formation. In how far this mechanism cooperates with intrinsic loop extrusion, or with extrinsic loop expansion, to shape chromosomes remains a fascinating question to address by further integrative computational and experimental studies.

DATA AVAILABILITY
The code for the biophysical simulation of chromatin chain behavior in the diffusion capture and loop extrusion mod-els is available in the GitHub repository (https://github.com/ FrancisCrickInstitute/Chromosome-Condensation).