## Main

In mammalian cells, interactions between chromosomal sequences play important roles in fundamental processes such as DNA replication1, repair2 and transcriptional regulation by distal enhancers3. Chromosome conformation capture (3C) methods, which measure physical proximity between genomic sequences in fixed cells, revealed that chromosomal contacts are organized into submegabase domains of preferential interactions known as topologically associating domains (TADs)4,5 whose boundaries can functionally insulate regulatory sequences3. TADs mainly arise from nested interactions between convergently oriented binding sites of the DNA-binding protein CTCF, which are established as chromatin-bound CTCF arrests the loop extrusion activity of the cohesin complex6,7,8,9,10.

Determining the timing and duration of chromosomal interactions within TADs and their relationship with CTCF and cohesin is key to understanding how enhancers communicate with promoters11,12. Single-cell analyses of chromosome structure in fixed cells4,13,14,15, chromosome tracing experiments16,17,18,19, in vitro9,10,20 and live-cell21 measurements of CTCF and cohesin dynamics, and polymer simulations6,15,22, as well as live-cell imaging of chromosomal locations and nascent RNA23,24, all suggested that TADs and CTCF loops are dynamic structures whose temporal evolution might be governed by the kinetics of loop extrusion25. Recent live-cell measurements of a CTCF loop connecting two opposite TAD boundaries in mouse embryonic stem cells (mESCs) provided direct evidence that this is the case, and revealed that cohesin-mediated loops between CTCF sites located 500 kilobases (kb) away last 10–30 min (ref. 26). However, it is still unclear if contacts between sequences separated by genomic distances where enhancers and promoters interact within the same TAD occur on the timescale of seconds, minutes or hours. We also have little knowledge on whether and how rates and durations of such contacts are modulated by loop extrusion. We finally do not know if cohesin increases chromosome mobility and thus favors the encounters between genomic sequences by reeling them into loops, or if instead it provides constraints that decrease mobility and prolong the duration of such encounters. Both scenarios have been suggested to be possible theoretically27,28, but it is unclear which effect dominates in living cells.

Here we use live-cell fluorescence microscopy to measure chromosome dynamics and its dependence on cohesin and CTCF in mESCs. By combining two live-cell imaging strategies with polymer simulations, we reveal that loops extruded by cohesin constrain global chromosome motion, while also increasing the temporal frequencies and durations of physical encounters between sequences inside the same TAD. Convergent CTCF sites substantially stabilize contacts through cohesin-mediated CTCF-anchored loops that last around 5–15 min on average. Our results support the notion that chromosome structure within single TADs is highly dynamic during the span of a cell cycle and thus that long-range transcriptional regulation might rely on transient physical proximity between genomic sequences. They also reveal how contact dynamics and the temporal variability in chromosome folding are modulated by cohesin and CTCF in single living cells and provide a quantitative framework for understanding the role of folding dynamics in fundamental biological processes.

## Results

### Cohesin decreases chromosome mobility independently of CTCF

To study how cohesin and CTCF influence the global dynamics of the chromatin fiber independently of local chromatin state and structural differences, we examined the dynamic properties of large numbers of random genomic locations in living cells. We generated clonal mESC lines carrying multiple random integrations of an array of ~140 repeats of the bacterial Tet operator sequence (TetO) using piggyBac transposition29. These can be visualized upon binding of Tet repressor (TetR) fused to the red fluorescent protein tdTomato. To compare the motion of genomic locations that either block or allow the loop extrusion activity of cohesin, the TetO array was adjacent to three CTCF motifs (3 × CTCF) that could be removed by Cre-assisted recombination (Fig. 1a). Motifs were selected based on high CTCF enrichment in chromatin immunoprecipitation followed by sequencing (ChIP–seq) and each was confirmed to be bound by CTCF in nearly 100% of alleles at any time in mESCs using dual-enzyme single-molecule footprinting30 (R. Grand and D. Schübeler, personal communication), thus providing a close experimental representative of an ‘impermeable’ loop extrusion barrier.

3 × CTCF-TetO sequences were introduced in mESCs that stably expressed OsTir1 and where the endogenous Rad21, Wapl or Ctcf genes were targeted with an auxin-inducible degron (AID) peptide fused to eGFP31,32. This resulted in several mESC clones (three per degron condition) with different sets of genomic insertions of the 3 × CTCF-TetO cassette, where over 95% of any of the AID-tagged proteins could be rapidly depleted upon addition of auxin (Fig. 1b and Extended Data Fig. 1a). This allowed us to study chromosome dynamics following acute depletion of factors affecting cohesin-mediated chromosome structure (Extended Data Fig. 1b) at previously reported time points (90 min for RAD21 (ref. 33), 6 h for CTCF31 and 24 h for WAPL32) that minimize secondary effects such as defects in cell-cycle progression (Extended Data Fig. 1c).

Mapping TetO insertion sites revealed 10–20 insertions per cell line, with on average 1–2 heterozygous insertions per chromosome without any strong bias towards active or inactive chromatin (Extended Data Fig. 1d,e). Insertions were on average 10 kb away from the nearest endogenous CTCF binding sites (Extended Data Fig. 1f). 4C sequencing (4C-seq) confirmed that insertion of 3 × CTCF-TetO cassettes often led to the formation of ectopic interactions with endogenous CTCF sites, which were lost upon removal of 3 × CTCF sites or depletion of RAD21 (Extended Data Fig. 1g,h).

To measure the dynamics of 3 × CTCF-TetO insertions, we acquired three-dimensional (3D) movies (one z-stack of 10 µm every 10 s for 30 min) using highly inclined and laminated optical sheet microscopy34 (Fig. 1b and Supplementary Video 1). This resulted in ~270 cells per condition with over 8,000 trajectories from three clonal lines imaged with 3–4 biological replicates per condition. Detection and localization of TetO arrays as subdiffraction fluorescent signals35 enabled reconstruction of trajectories of individual genomic insertions (Fig. 1c and Methods). We then studied their mean squared displacement (MSD) as a function of time after correcting each trajectory for the confounding effect of cell movement, which we inferred from the collective displacement of all insertions in each nucleus (Fig. 1d, Extended Data Fig. 2a and Methods). Independently of the degron background, in untreated cells, genomic locations underwent on average a subdiffusive motion whose anomalous exponent (~0.6) and generalized diffusion coefficients (D) (~1.2 × 10−2 µm2 s−α) were in line with previous studies of specific genomic loci36,37 (Fig. 1d and Extended Data Fig. 2b). The MSD of radial distances (radial MSD) between insertions within the same nuclei showed the same scaling although statistics were less robust for long time intervals due to the shorter trajectories that could be built based on pairwise distances (Fig. 1d). Interestingly, removal of 3 × CTCF sites (Extended Data Fig. 1g) or degradation of CTCF (6-h auxin treatment) did not have a significant impact on MSD averaged over all genomic locations nor on its distribution across trajectories and cells (Fig. 1e,f, results for single clones in Extended Data Fig. 2c, P values in Extended Data Fig. 2e).

By contrast, acute depletion of RAD21 (90-min auxin treatment) led to a significant increase in mobility both in the presence (Fig. 1g) and absence of 3 × CTCF sites (Extended Data Fig. 2d), with only a very minor impact on anomalous exponents (Extended Data Fig. 2b,e, P values in Extended Data Fig. 2e). In the presence of wild-type levels of RAD21, generalized diffusion coefficients were on average ~30% lower than in depleted cells, where RAD21 levels were low enough to prevent formation of cohesin-mediated structures (compare with Extended Data Fig. 1b). This outcome was consistent across three clonal cell lines with different TetO insertion sites and the small differences in the magnitude of the effect were likely due to location-dependent effects (Extended Data Fig. 2f). Importantly, the effect was specific for RAD21 degradation as we did not observe any changes in MSD behavior in control cell lines expressing OsTir1 but no AID-tag (Extended Data Fig. 2g). In addition, depletion of WAPL (24-h auxin treatment), which results in higher levels of DNA-bound cohesin32, caused a substantial decrease in chromosome mobility (Fig. 1h and Extended Data Fig. 2e). Together, these results indicate that increasing levels of DNA-bound cohesin decrease chromosome mobility, with only very minor effects (if any) mediated by the presence of even strong CTCF motifs (Fig. 1i).

### Loop extrusion can explain reduced chromosome dynamics

We next used polymer simulations to determine if loop extrusion alone could explain the observed global reduction in chromosome dynamics in the presence of cohesin and minimal effects from CTCF. We simulated the dynamics of a polymer with excluded volume, with or without loop extrusion and extrusion barriers whose linear arrangement and orientation were sampled from endogenous CTCF sites (Fig. 2a and Extended Data Fig. 3a). To mimic random insertion of 3 × CTCF sites, we also simulated the same polymers with additional loop extrusion barriers separated by 800 kb which were inserted at random positions in the polymer (magnified area in Fig. 2a). To emphasize their potential effects on chromosome dynamics, all barriers in the simulations were impermeable to loop extruders. Every monomer represented 8 kb of chromatin, corresponding to the genomic size of the TetO array. Simulation steps were approximated to real-time units by matching the time needed for a monomer to move by its own diameter with the time required by the TetO array to move by its estimated mean physical size (Methods). We sampled an extremely large range of extruder residence times and loading rates (4 orders of magnitude each) centered around a residence time of ~30 min and extruder densities of ~20 per Mb (in line with previous measurements38,39), and using two extrusion speeds corresponding to in vivo and in vitro estimates (~0.1 kb s−1 and ~1 kb s−1, respectively)20,38 (Fig. 2b and Extended Data Fig. 3b).

In the absence of loop extrusion, the polymer underwent subdiffusive behavior with anomalous exponent of ~0.6 (Fig. 2c), as expected from simple polymers with excluded volume40,41,42 (see Supplementary Information) and compatible with our experimental results on random TetO insertions (Fig. 1e). Strikingly, in line with experimentally measured effects of RAD21 (Fig. 1g), introduction of loop extrusion led to lower generalized diffusion coefficients and minor effects on anomalous exponents, independently of loading rate and residence time (Fig. 2c,d), extrusion speed (Extended Data Fig. 3c) or the presence of extrusion barriers (Fig. 2e,f). Interestingly, for extruder residence times of 5.5–11 min and unloading rates corresponding to extruder linear densities of ~20 per Mb, the predicted decrease in generalized diffusion coefficients was in quantitative agreement with the experimentally observed value of ~30% (Fig. 2d,f; extruder densities as in Fig. 2b; compare with Fig. 1g). Also, consistently with WAPL depletion experiments (Fig. 1h), increasing extruder residence times systematically resulted in larger reductions in generalized diffusion coefficients (Fig. 2d,f and Extended Data Fig. 3c).

Importantly, addition of barriers in the presence of loop extrusion led to substantially smaller changes in polymer dynamics compared with the effect of loop extrusion itself even when probed directly on the barriers (Fig. 2g,h and Extended Data Figs. 3c and 4a,b), in agreement with our experimental finding that CTCF degradation had no strong effect on MSDs of TetO insertions (Fig. 1e,f and Extended Data Fig. 2c,e). Similarly, insertion of additional barriers had little impact on MSD (Fig. 2i,j), thus recapitulating the negligible effect of removal of 3 × CTCF sites (Fig. 1e). Polymer simulations thus strongly support the notion that the observed decrease in chromosome mobility and lack of effects from CTCF is a macroscopic manifestation of the physical constraints imposed by cohesin in living cells.

### Cohesin and CTCF constrain the dynamics of sequences in cis

We next asked how cohesin and CTCF impact the reciprocal motion of two genomic sequences located on the same DNA molecule. To this aim, we simulated the dynamics of a polymer carrying two convergent impermeable extrusion barriers mimicking strong CTCF motifs separated by ~150 kb (Fig. 3a). This is comparable to median distances between convergent CTCF sites within TADs genome-wide in mESCs (141 kb, Methods) and also to the estimated average separation between enhancers and promoters in human cells (~160 kb)43. Simulations performed with extrusion parameters recapitulating the dynamic effects of RAD21 depletion (black square in Fig. 2f) predicted that radial MSDs should be lowest in the presence of loop extrusion and barriers (Fig. 3b) due to the formation of transient loops anchored by the barriers (Fig. 3c). Similar to MSDs (Fig. 2), radial MSDs should increase upon removal of extrusion barriers and become maximal when loop extrusion is also removed (Fig. 3b).

Importantly, simulations also predicted that scaling exponents of radial MSD curves should be considerably smaller (~0.2) than those we previously observed for TetO arrays separated by several Mb or located on different chromosomes (~0.6, Fig. 1d). This is because correlations in the motion of two monomers are stronger when they are located closer along the polymer. Indeed, simulations predicted that scaling exponents fitted from radial MSD curves at short times should increase with increasing genomic distance and approach 0.6 for loci separated by several Mb (consistent with radial MSDs of randomly inserted TetO arrays) (Extended Data Fig. 5a) before saturating to stationary values at longer times. This holds true also without loop extrusion (theoretical analysis in Supplementary Information and simulations in Extended Data Fig. 5b).

To test these predictions, we turned to a live-cell imaging approach allowing us to measure the radial dynamics of two sequences located within the same TAD, in the presence and absence of cohesin and/or strong CTCF sites. We engineered mESCs carrying targeted integrations of two orthogonal operator arrays: ~140× TetO and 120× LacO separated by 150 kb (Fig. 4a), which could be visualized upon binding of TetR-tdTomato and a weak DNA-binding variant of LacI fused to eGFP (LacI**-eGFP)44. To minimize confounding effects from additional regulatory sequences such as active genes or enhancers, we targeted the arrays into a 560-kb ‘neutral’ TAD on chromosome 15 where we previously removed internal CTCF sites3 (Fig. 4a). The two operator arrays were directly adjacent to excisable 3 × CTCF site cassettes arranged in a convergent orientation (Fig. 4a). Cell lines were verified by Nanopore Cas9-targeted sequencing (nCATS)45 to contain a single copy of each targeting cassette (Extended Data Fig. 5c). We additionally targeted the endogenous Rad21 locus with a C-terminal HaloTag-FKBP fusion allowing the inducible degradation of RAD21 upon treatment with dTAG-13 (ref. 46) as confirmed by severely decreased protein levels (>95% after 2-h treatment, Extended Data Fig. 5d).

Capture-C with tiled oligonucleotides revealed that integration of operator arrays themselves did not lead to detectable changes in chromosome structure (Extended Data Fig. 5e). Convergent 3 × CTCF sites, however, led to the formation of a new CTCF-mediated interaction within the TAD (2.8× increase in contact probability after correcting the confounding contribution of the wild-type allele) (Fig. 4b), which was lost upon RAD21 depletion along with all other CTCF-mediated interactions across the locus (Extended Data Fig. 5f).

We imaged cells for 3 h every 30 s in three dimensions (Fig. 4c and Supplementary Video 2), either in the presence or absence of RAD21, and measured distances between the two arrays over time (Fig. 4d, n = 3–7 biological replicates for each condition, on average 220 cells per condition, Supplementary Table 1 and Methods). Doublet signals corresponding to replicated alleles occurred in a very minor fraction (3%, Methods) of trajectories, compatible with the late-replication profile of the ‘neutral’ TAD and the cell-cycle distribution (Extended Data Figs. 5g and 6a). In these cases, only trajectories that were initially closest across channels were considered. After correction of chromatic aberrations (Methods and Extended Data Fig. 6b), we estimated our experimental uncertainty on radial distances to be ~130 nm by measuring pairwise distances in control cells where multiple TetO insertions were simultaneously bound by both TetR-tdTomato and TetR-eGFP (Extended Data Fig. 6c–e).

In agreement with model predictions for locations separated by 150 kb, radial MSDs of the two arrays showed scaling exponents close to 0.2, much smaller than those observed with randomly inserted TetO arrays (Extended Data Fig. 7a,b). Also in line with model predictions (Extended Data Fig. 5a), presence of RAD21 and 3 × CTCF sites led to the most constrained radial mobility, whereas RAD21 degradation and deletion of CTCF sites resulted in the least constrained motion (Extended Data Fig. 7c). These measurements thus verified the model prediction that genomic sequences located at short distances (150 kb) experience stronger physical constraints than sequences located at larger genomic distances26 (Fig. 1 and Extended Data Fig. 7b), and that loop extrusion provides constraints that are further reinforced by convergent CTCF sites.

Consistently with their more constrained radial MSD behavior, we finally observed that distances between TetO and LacO signals were smallest in the presence of convergent CTCF sites and cohesin. In these conditions, distances between TetO and LacO arrays tended to remain close to the ~130-nm experimental uncertainty with only occasional fluctuations toward larger values in the course of the 3 h of imaging (Fig. 4d,e). Removal of 3 × CTCF sites led to increased radial distances and variability within single trajectories, which were further increased upon degradation of RAD21, irrespective of the presence or absence of CTCF sites (Fig. 4d,e and Supplementary Video 3). Thus, constraints imposed by extruding cohesin and convergent CTCF sites reduce not only average physical distances between sequences but also their variability in time (Fig. 4f), also supported by analysis of distance changes (jumps) as a function of time (Fig. 4g).

Finally, to test whether the effects of cohesin on chromosome motion would be different in the presence of active transcription at nearby locations, we measured looping dynamics this time in a parental mESC line before the removal of resistance cassettes. In this line, both the TetO and LacO arrays were immediately flanked by mouse Pgk1 promoters47 driving the transcription of resistance genes (Extended Data Fig. 6f,g). In line with previous studies48,49, we found that active transcription led to slightly decreased radial MSD. Cohesin depletion resulted in similar amounts of increased radial mobility irrespective of the presence or absence of active promoters (Extended Data Fig. 7c).

### Chromosomal contacts are transient

We next set off to quantify changes in distances over time and determine whether despite the experimental uncertainty on 3D distances (Extended Data Fig. 6e) we could observe transitions between two states: a ‘proximal’ state with small radial distances (presumably including cohesin-mediated loops between convergent CTCF sites), and a generic ‘distal’ state with larger spatial distances corresponding to other configurations of the chromatin fiber. This was motivated by the expectation that any polymer with site-specific attractive interactions, such as those mediated by cohesin at convergent CTCF sites, should in principle result in two-state thermodynamic behavior. We thus fitted a two-state hidden Markov model (HMM) on the ensemble of trajectories obtained in cells where both convergent 3 × CTCF sites and RAD21 were present (Fig. 5a,b). Interestingly, distances in the proximal state inferred by HMM largely overlapped with those detected on perfectly colocalizing signals in control experiments where TetR-eGFP and TetR-tdTomato were bound to the same set of randomly inserted TetO arrays (149 versus 130 nm on average, respectively) (Fig. 5b and Extended Data Fig. 6e). The proximal state thus corresponds to configurations of the chromatin fiber where the two arrays were in very close physical proximity, also including (but not restricted to) cohesin-mediated loops between CTCF sites. For simplicity, we refer to the proximal state interchangeably as ‘contact’, without implying a direct molecular interaction between the two DNA fibers. Radial distances in the distal state (288 nm on average) instead were similar to those measured in cells where both CTCF sites had been removed (291 nm) (Extended Data Fig. 8a,b). Thus, the distal state largely overlapped with chromosome conformations where specific cohesin-mediated CTCF loops were lost.

We next fitted the HMM to all experimental conditions while keeping the same proximal state as in cells with 3 × CTCF sites and RAD21 (Fig. 5a). This showed that in the presence of RAD21, the LacO and TetO arrays spent ~78% of the time in contact (that is, in the proximal state) when 3 × CTCF sites were present. This was 2.3× higher than the 33% of time they spent in contact when the 3 × CTCF sites were removed (Fig. 5c), in agreement with the corresponding 2.8-fold difference in contact probability inferred from Capture-C (Fig. 4b). The fraction of time spent in contact decreased markedly upon depletion of RAD21 to ~23% in the presence of 3 × CTCF sites and 11% in the absence (Fig. 5c). Both the average duration of contacts and their rate of formation were maximal in the presence of RAD21 and 3 × CTCF sites, where they lasted around 16 min and reformed every 5 min on average (Fig. 5d,e). Contacts became substantially shorter (6 min) and rarer (one every 10 min) when 3 × CTCF sites were removed, and even more so upon RAD21 depletion (lasting 2 min and occurring every 22 min on average). Interestingly, these results were not affected by the presence of actively transcribed promoters in the immediately flanking regions (Extended Data Fig. 8b–f), in line with the lack of changes in contact probability measured in Capture-C (Extended Data Fig. 6g). Thus, both cohesin and CTCF impact both the duration and the probability of formation of chromosomal contact events between loci separated by 150 kb within an ‘empty’ TAD.

To understand if these results could be rationalized in terms of loop extrusion, we compared them with polymer simulations with convergent impermeable loop extrusion barriers separated by ~150 kb. Simulations were performed using loop extrusion parameters spanning a finer-grained 25-fold range around experimentally realistic values that reproduced the dynamic effect of RAD21 degradation (compare with Fig. 2d, black square) and with both in vitro and in vivo estimates of extrusion speeds20,38. In a large region of the parameter space, distances between convergent barriers were bimodally distributed, supporting the expectation that the polymer can be approximated as a two-state system (Extended Data Fig. 9a). To allow direct comparison with experimental distance-based HMM states, we applied random errors matching experimental uncertainty levels to radial distances generated by the models (Extended Data Fig. 9b). We called proximal and distal states using the same HMM strategy as with experimental data. Importantly, for a large number of parameter combinations, distances in the proximal state largely overlapped with the corresponding distribution observed experimentally in the presence of convergent CTCF sites and cohesin (Extended Data Fig. 9c and Supplementary Fig. 1a).

We then compared the distance, duration and fraction of time spent in the proximal state with those experimentally observed in the presence of RAD21 with or without 3 × CTCF sites. We found that their similarity was maximal for extruder densities ranging from 8 to 32 per Mb (Supplementary Fig. 1e) and residence times of 2.8–11 min, with extrusion speeds of both 0.1 and 1 kb s−1, all of which were in the range of previous estimations of experimental values20,21,38,50 (Fig. 6a and Extended Data Fig. 9d). Considering the five best-matching scenarios (red- and yellow-marked values in Fig. 6a), the two locations spent 45–55% of the time in the proximal state with an average contact duration of around 10–17 min, which reduced to 18% and 8 min in the absence of extrusion barriers (Fig. 6b,c). Similar to the effects observed experimentally upon depletion of RAD21, decreasing extruder densities (for example, by decreasing loading rates) led to decreased fractions of time and shorter durations of the proximal state (Fig. 6d,e, shown for the best case, general trends in Supplementary Fig. 1d–d). Thus, the duration and the fraction of time spent in the proximal state, and most importantly how these quantities change upon removing cohesin and/or CTCF sites, can be understood in terms of a simple loop extrusion model.

The HMM-based proximal state likely provides an overestimation of the duration of underlying CTCF-CTCF loops mediated by stalled cohesins, since it also contains a fraction of CTCF-independent proximity events that cannot be distinguished from loops. To estimate the duration and times the two loci spent in a cohesin-mediated CTCF-CTCF looped conformation, we quantified occurrences in the simulated polymer where the two monomers formed the base of an extruded loop (Fig. 6f and Methods). As expected, these events were rarer and shorter than contacts detected by HMM on polymer simulations (Fig. 6b,c), with two monomers spending ~20–31% of time at a loop base for 5–15 min on average in the presence of extrusion barriers (Fig. 6g,h). Finally, transient cohesin-dependent loops that are not stabilized by CTCF sites should occur much more rarely (1–3% of the time) and lasted less than a minute on average (Fig. 6g,h). Comparison of polymer simulations with HMM states thus suggests that the dynamics of chromosome contacts detected at a range of 150 nm are generated by faster and rarer cohesin-mediated CTCF loops (Fig. 6i).

## Discussion

Our study provides quantitative measurements of chromosome folding dynamics in living cells and reveals how they are controlled by cohesin and CTCF. Two experimental strategies allow us to minimize biological variation from specific regulatory and structural genomic contexts and enable direct comparison with polymer models. By studying large numbers of random genomic locations, we average over local differences in chromosome mobility and reveal the global dynamic effects of cohesin. By visualizing and manipulating two locations within a ‘neutral’ genomic environment, we unravel how cohesin and CTCF impact chromosome looping within a single TAD. We show that although higher extrusion speeds could in principle result in increased chromosome motion (Extended Data Fig. 9e,f), physiological extrusion rates rather generate transient constraints that decrease chromosome dynamics, in line with previous measurements of histone mobility49. Similar to previous reports51, we observe that constraints introduced by cohesin reduce spatial distances between genomic sequences in cis and increase the chances that they interact. We now, however, reveal that this entails an increase in both the rate of formation and the duration of contacts. Convergently oriented high-affinity CTCF motifs lead to higher contact frequencies and substantially longer contact durations, somewhat similar to the effect of insulator elements in Drosophila23. Comparison with polymer simulations reveals that in mESCs this can be understood in terms of stalling of loop-extruding cohesins. This observation also suggests that asymmetries in contact patterns established by CTCF motifs genome-wide might also lead to temporal asymmetries in physical interactions, notably between regulatory sequences. We additionally observe that constraints introduced by cohesin and CTCF sites lead to reduced temporal variability in physical distances, arguing that loop extrusion increases the reproducibility of chromosome folding at selected genomic sites.

Our study also provides estimates of the frequency and duration of chromosomal contacts at genomic-length scales that represent enhancer–promoter communication genome-wide. In our study, contacts are defined by physical distances (~150 nm) that might be comparable to those where signals arise in 3C methods11. For sequences separated by 150 kb, such contacts assemble and disassemble over minutes. This provides many opportunities in a single cell-cycle for regulatory sequences in a TAD to contact each other, and suggests that long-range regulation by distal enhancers might rely on transient interactions. We note that despite accurate correction of chromatic aberrations, shorter-range and thus potentially faster proximity events remain inaccessible in our experimental set-up52. Estimates based on comparison with polymer simulations further suggest that cohesin-mediated interactions between convergent CTCF sites might last around 5–15 min on average and at least for sequences located 150 kb apart occur around 27% of the time. This is in good agreement with recent estimates of the duration of a 500-kb loop in mESCs (10–30 min on average)26, which, however, occurs more rarely (3.5–6% of the time). This is in line with the predictions from polymer simulations that increasing the genomic distance between convergent CTCF sites should substantially decrease the frequency of CTCF-mediated interactions, but not their duration (Extended Data Fig. 10). Taken together, our data establish firm quantitative bases for understanding the dynamics of chromosome folding within TADs and provide temporal constraints for mechanistic models of chromosome structure and its impact on fundamental biological processes such as long-range transcriptional regulation.

## Methods

### Culture of mESC lines

All cell lines are based on the E14Tg2a parental mESC line (karyotype 19, XY, 129/Ola isogenic background; E14 for brevity). E14 CTCF-AID-eGFP (clone EN52.9.1) was published by Nora et al.31. E14 WAPL-AID-eGFP and E14 RAD21-AID-eGFP were published by Liu et al.32. The latter were kindly provided by Elzo de Wit (Netherlands Cancer Institute). All cell lines for the dual-array imaging approach are based on the double-CTCF knockout cell line described by Zuin et al.3. Cells were cultured on gelatin-coated culture plates in Glasgow Minimum Essential Medium (Sigma-Aldrich, G5154) supplemented with 15% fetal calf serum (Eurobio Abcys), 1% l-Glutamine (Thermo Fisher Scientific, 25030024), 1% Sodium Pyruvate MEM (Thermo Fisher Scientific, 11360039), 1% MEM Non-Essential Amino Acids (Thermo Fisher Scientific, 11140035), 100 μM β-mercaptoethanol (Thermo Fisher Scientific, 31350010), 20 U ml−1 leukemia inhibitory factor (Miltenyi Biotec, premium grade) in 8% CO2 at 37 °C. Cells were tested for mycoplasma contamination regularly and no contamination was detected. For Hi-C, Capture-Hi-C, 4C-seq, western blot and imaging experiments, cells were cultured in standard E14 medium supplemented with 2i (1 μM MEK inhibitor PDO35901 (Axon, 1408) and 3 μM GSK3 inhibitor CHIR 99021 (Axon, 1386)). For live-cell imaging experiments, cells were cultured in Fluorobrite DMEM (Gibco, A1896701) supplemented with 15% fetal calf serum (Eurobio Abcys), 1% L-Glutamine (Thermo Fisher Scientific, 25030024), 1% Sodium Pyruvate MEM (Thermo Fisher Scientific, 11360039), 1% MEM Non-Essential Amino Acids (Thermo Fisher Scientific, 11140035), 100 μM β-mercaptoethanol (Thermo Fisher Scientific, 31350010), 20 U ml−1 leukemia inhibitory factor (Miltenyi Biotec, premium grade) and with 2i inhibitors (1 μM MEK inhibitor PDO35901 (Axon, 1408) and 3 μM GSK3 inhibitor CHIR 99021 (Axon, 1386)).

### Generation of mESC lines carrying random integrations of TetO arrays

To generate clonal cell lines carrying random integrations of the TetO array in the degron cell lines (E14 Rad-AID-eGFP, E14 CTCF-AID-eGFP and E14 WAPL-AID-EGFP), 0.5 × 106 cells were transfected with 2 μg of PB-3 × CTCF-TetO vector, 200 ng of PB-TetR-tdTomato and 200 ng of pBroad3_hyPBase_IRES_tagRFPt (ref. 53) with Lipofectamine3000 (Thermo Fisher Scientific, L3000008) according to the manufacturer’s recommendations. Cells were cultured in standard E14 medium for 5 d and subsequently sorted by FACS for fluorescence emission at 581 nm (tdTomato) on 96-well plates to isolate clonal lines. Sorted cells were kept for 2 d in standard E14 medium supplemented by 100 μg μl−1 primorcin (InvivoGen, ant-pm-1) and 10 μM ROCK inhibitor (STEMCELL Technologies, Y-27632). At 10 d after sorting, the plates were duplicated by detaching with accutase (Sigma-Aldrich, A6964) and re-seeding in full E14 culture medium. One-third of the cells were replated onto Corning High-Content Imaging Glass Bottom Microplates (96-well, Corning, 4580). At 2 d after re-seeding, clonal lines were screened by microscopy for >10 insertions of TetO per cell and a good signal-to-noise ratio (SNR). Selected clones were expanded and genotyped by PCR for the absence of random integration of the piggyBac itself. Primers used for genotyping are listed in Supplementary Table 3.

### Generation of dual-array (TetO-LacO) mESC line

Integration of the TetO array into the genomic locus on chr15:11,647,372: the vector containing the guide RNA (gRNA) sequence was available from a previous study (PX459-chr15_gRNA/Cas93). The gRNA sequence can be found in Supplementary Table 3. E14 mESCs already containing a double-knockout for CTCF sites (clone D6 in ref. 3) were transfected with the targeting vector pMK-3 × CTCF-TetO-Rox-PuroR-Rox and the gRNA vector PX459-chr15_gRNA/Cas9 using nucleofection with the Amaxa 4D-Nucleofector X-Unit and the P3 Primary Cell 4D-Nucleofector X Kit (Lonza, V4XP-3024 KT). Then, 2 × 106 cells were nucleofected with 1 μg of TetO targeting vector and 1 μg of PX459-ch15_gRNA/Cas9) as described above and treated with 1 μg ml−1 puromycin (InvivoGen, ant-pr-1) 48 h after transfection for 3 d to select cells for insertion of the TetO cassette. Cells were then cultured in standard E14 medium for an additional 7 d and subsequently sorted by FACS on 96-well plates as described above to isolate clonal lines. At 10 d after sorting, the plates were duplicated by detaching with accutase (Sigma-Aldrich, A6964) and re-seeding in full E14 culture medium. Genomic DNA was extracted on-plate by lysing cells with lysis buffer (100 mM Tris-HCl pH 8.0, 5 mM EDTA, 0.2% SDS, 50 mM NaCl and 1 mg ml−1 proteinase K (Macherey-Nagel, 740506)) and 0.05 mg ml−1 RNase A (Thermo Fisher Scientific, EN0531) and subsequent isopropanol precipitation. Individual cell lines were analyzed by genotyping PCR to determine heterozygous insertion of the TetO cassette. Cell lines showing the corrected genotype were selected and expanded. Primers used for genotyping are listed in Supplementary Table 3. Targeted nanopore sequencing with Cas9-guided adapter ligation45 was performed on expanded clones to confirm single-copy insertion of the TetO cassette. Clone 2G5 was used for further engineering. Integration of the LacO array into the genomic locus on chr15:11,496,908: the gRNA sequence for the CRISPR–Cas9 knock-in of the LacO cassette was designed using the online tool https://eu.idtdna.com/site/order/designtool/index/CRISPR_SEQUENCE and purchased from Microsynth AG. The gRNA sequence can be found in Supplementary Table 3. The gRNA sequence was cloned into the PX330 plasmid (Addgene, no. 58778) using the BsaI restriction site. The clonal line carrying the TetO cassette (clone 2G5) was transfected with the targeting vector pUC19-ITR-NeoR-ITR-3 × CTCF-LacO and the gRNA vector pX330-chr15_LacO_gRNA/Cas9 using nucleofection with the Amaxa 4D-Nucleofector X-Unit and the P3 Primary Cell 4D-Nucleofector X Kit (Lonza, V4XP-3024 KT). A total of 2 × 106 cells were harvested using accutase (Sigma Aldrich, A6964) and resuspended in 100 μl transfection solution (82 μl primary solution, 18 μl supplement, 15 μg targeting vector and 5 μg of gRNA vector) and transferred to a single Nucleocuvette (Lonza). Nucleofection was performed using the protocol CG110. Transfected cells were directly seeded in pre-warmed E14 standard medium. At 48 h after transfection, 250 μg ml−1 G418 (InvivoGen, ant-gn-1) was added to the medium for 3 d to select cells for insertion of the LacO cassette. Cells were sorted and genotyped as described for the TetO integration. Primers used for genotyping are listed in Supplementary Table 3. Cell lines showing the corrected genotype were selected and expanded. Expanded clones were transiently transfected with 200 ng of PB-TetR-tdTomato and 200 ng of PB-LacI-eGFP using Lipofectamine3000 according to the manufacturer’s instructions (Thermo Fisher Scientific, L3000008) and 2 d after transfection validated for heterozygous insertion of the LacO cassette on the same allele as the TetO by microscopy. Targeted nanopore sequencing with Cas9-guided adapter ligation45 was performed on correct clones to confirm single-copy insertion of the LacO cassette. Clone 1F11 was used for further engineering. To visualize the operator arrays in live-cell imaging and remove the puromycin resistance gene used for selection during integration, 0.5 × 106 E14 TetO-LacO cells (clone 1F11) were transfected with 200 ng of PB-TetR-tdTomato, 200 ng of PB-LacI-eGFP and 200 ng of pBroad3_hyPBase_IRES_tagRFPt (ref. 54) with Lipofectamine3000 (Thermo Fisher Scientific, L3000008) according to the manufacturer’s instructions. At 7 d after transfection the cells were sorted (as described previously) for fluorescence emission at 507 nm (eGFP) and 581 nm (tdTomato). Sorted cells were cultured and genotyped as described for the random TetO integration. Primers used for genotyping are listed in Supplementary Table 3. Cell lines showing the corrected genotyping pattern were selected and expanded and a good and comparable SNR was selected for by microscopy. Clones 1B4 (+PuroR) and 2C10 (−PuroR) were used for further engineering.

### Live-cell imaging

First, 35-mm glass-bottom dishes (Mattek, P35G-1.5-14-C) were coated with 1–2 μg ml−1 Laminin (Sigma-Aldrich, L2020) in PBS at 37 °C overnight. Cells (1 × 106) were seeded in Fluorobrite medium (as described above) 24 h before imaging. For targeted degradation of RAD21, WAPL or CTCF in the degron cell lines, the medium was exchanged to medium containing 500 μM auxin (Sigma-Aldrich, I5148-2G) at the respective time required for complete degradation of the protein target before imaging (RAD21: 90 min, WAPL: 24 h, CTCF: 6 h). For targeted depletion of RAD21 using the FKBP degron system (dual-array cell lines), cells were cultured in Fluorobrite medium containing 500 nM dTAG-13 (Sigma-Aldrich, SML2601-1MG) 2 h before imaging. For fixed cell measurements to estimate the localization error, 1 × 106 cells were seeded onto Mattek dishes and incubated for 24 h at 37 °C, 8% CO2. The medium was removed and the cells were fixed in 4% paraformaldehyde (Electron Microscopy Sciences, 15710) in PBS for 10 min at room temperature. The cells were washed three times in PBS and Fluorobrite medium was added to the Mattek dish to achieve comparable background fluorescence levels. Cells were imaged with a Nikon Eclipse Ti-E inverted widefield microscope equipped with a Total Internal Reflection Microscopy iLAS2 module (Roper Scientific), a Perfect Focus System (Nikon) and motorized Z-Piezo stage (ASI) using a CFI APO TIRF 100 ×1.49 NA oil immersion objective (Nikon). The microscope was operating in highly inclined and laminated optical sheet mode34. Excitation sources were a 48-nm, 200-mW Toptica iBEAM SMART laser and a 561-nm, 200-mW Coherent Sapphire laser. Images were collected on two precisely aligned back-illuminated Evolve 512 Delta EMCCD cameras with a pixel size of 16 × 16 μm2 (Photometrics). Cells were maintained at 37 °C and 8% CO2 using an enclosed microscope environmental control set-up (The BOX and The CUBE, Life Science Instruments). Before the acquisition of movies for the dual-array set-up, TetraSpeck Microspheres, 0.1-μm beads (Thermo Fisher Scientific, T7279), were imaged to allow for correction of chromatic aberrations during image processing and analysis. Movies for measurement of random TetO integrations in degron cell lines were acquired every 10 s (exposure time: 50 ms) in 34 z-planes (10-μm stack, distance between consecutive z planes = 300 nm) with the Visiview software (Visiview 4.4.0.12, Visitron). Images for measurement of cell lines with the dual-array set-up were acquired every 30 s, with an exposure time of 50 ms, respectively, each in a sequential mode with 21 z-planes (6-μm stack, dz = 300 nm). For the measurement of the time it takes the operator arrays to displace by their own size, images were acquired continuously on a single focal plane over 10 s every 0.1 s with exposure times of 50 ms.

### Image processing

Raw images were deconvolved using the Huygens Remote Manager and a classical maximum likelihood estimation algorithm with a theoretical point-spread function. The initial SNRs were estimated from the images and images were deconvolved until one of the following stopping criteria was reached: the maximum number of iterations was performed (for random integrations: 20 cycles, for tdTomato and eGFP; in dual-color set-up: 15 cycles for tdTomato signal, 5 cycles for GFP signal) or a quality change criterion below 0.001 was returned. Representative image series shown in the main figures were deconvolved as described above, adjusted to display the same brightness and contrast, and interpolated using a bicubic interpolation. Movies were corrected for bleaching over time using an exponential fit. The two-dimensional (2D) projection of intensity changes over time was created using the Temporal Color Code in Fiji v.2.0. (https://github.com/fiji/fiji/blob/master/plugins/Scripts/Image/Hyperstacks/Temporal-Color_Code.ijm).

### Spot detection and localization of multi operator data

Our field of view typically contains approximately 25 mESC nuclei. Despite the fact that our mESC lines are clonal, background nuclear fluorescence intensities in each cell can vary substantially. This poses challenges to conventional threshold-dependent algorithms for spot detection and localization which perform unevenly across cells with different background intensities. To overcome these limitations, we implemented a two-step procedure for 3D spot detection and localization. To detect spots, we used deepBlink v.0.1.1 (ref. 35), a convolutional neural network-based spot detection and localization algorithm in two dimensions, which has been shown to be able to deal with different background intensities and to detect spots in a threshold-independent manner. To enhance our detection efficiency, we employed custom models trained on a combination of the following datasets: smFISH and SunTag datasets provided by deepBlink and in-house manually curated live-cell imaging images. To detect 3D spots, we applied deepBlink to all z-stacks separately followed by linkage of the spots across z-stacks using Trackpy53. The precise 3D coordinates of the spots were then determined using 3D Gaussian fitting using a voxel of size 6 × 6 × 4 pixels centered at the spot in the brightest z-stack. deepBlink models can be found at https://github.com/zhanyinx/SPT_analysis/tree/main/models. The parameters and models used for each cell line can be found in Supplementary Table 4. All scripts used for the analysis can be found at https://github.com/zhanyinx/SPT_analysis/.

### Tracking and cell motion correction of multi operator data

3D spots coordinates are fed into TrackMate for tracking using linear assignment problem (LAP) tracker. Each track is assigned to manually annotated cell masks (from max z-projection of frame 93) using a custom script (https://github.com/zhanyinx/SPT_analysis/blob/main/source/spot_detection_tracking/assign_cellids.py), which uses the majority rule. Motion correction is then performed using a roto-translation model. Specifically, for each pair of consecutive time frames, a set of matching spots in every cell is determined by solving the LAP using the Euclidean distance between spots as a measure of distance. Only spots that match across two consecutive frames are then used to estimate the roto-translation model which is then applied to correct for nuclear motion (six matching spots on average across all time frames, trajectories and movies, with a minimum of four spots per pair of time frames). All scripts used for the analysis can be found at https://github.com/zhanyinx/SPT_analysis/.

### MSD analysis of multi operator data

Tracks with fewer than ten spots are filtered out for follow-up analysis. To calculate the MSD, we first calculate the time-averaged MSD for each trajectory. We then calculate the ensemble average (across trajectories) MSD by pooling all replicates. The ensemble average is done in log space. We corrected the localization error effect on the MSD curve by estimating the standard deviation of the error distribution using fixed images as described by Kepten et al.55. To calculate the scaling (α) and the generalized diffusion coefficient (D) of each MSD curve, we fitted the ensemble average of the log-time average MSD between 10 and 100 s. To test the significance of differences between conditions, we fitted α and diffusion coefficient for each cell. The P value is calculated using Student’s t-test (two-sided). Since we are always comparing two conditions whose cell-cycle profiles are similar, we ignore the effect of sister chromatids. All scripts used for the analysis can be found at https://github.com/zhanyinx/SPT_analysis/. The specific Fiji and relative plug-ins can be found at https://github.com/giorgettilab/Mach_et_al_chromosome_dynamics/tree/master/Fiji.

### Chromatic aberration correction of dual-color data

To correct for chromatic aberration we took 3D image stacks of TetraSpeck Microspheres, 0.1-μm beads (Thermo Fisher Scientific, T7279), adsorbed on MatTek dishes in 1 × PBS at the beginning of every imaging session and used them to correct the corresponding set of movies. After detecting signals from single beads in each channel using deepBlink and determining their 3D location by Gaussian fitting, we first identified spots that are shared across channels by solving the LAP using the Euclidean distance between spots. We then used the common set of bead signals to compute a 3D roto-translation that we finally applied to xyz positions. This procedure corrects for x, y and z aberrations simultaneously. The same transformations accurately corrected chromatic aberrations in actual experiments in double-labeled mESCs (‘Control TetO’ in Extended Data Fig. 6g), with the exception of a small residual systematic shift (approximately 40 nm) along the z axis (‘TetO-LacO case’ in Extended Data Fig. 6g), which is likely due to 3D image anisotropies that cannot be measured using ‘2D’ bead images.

### Tracking and MSD analysis of dual-color data

To increase the ability to detect longer tracks, we used an in-house script to stitch multiple tracks belonging to the same cell (https://github.com/zhanyinx/SPT_analysis/blob/main/source/dual_channel_analysis/utils.py, stitch function). In short, if two tracks from the same cell overlap more than 50% in time, the shortest one is filtered out. We called cell masks using CellPose56 on the max z-projection of the middle frame of the movie using the GFP channel and used these masks to define cell identity. For tracks with overlaps lower than 50%, the overlapping part of the tracks are randomly removed from one of the two tracks. The resulting tracks are stitched if the distance across the time gap is smaller than 1.6 μm. To match tracks across channels, we used the following measure to calculate the distance between tracks across channels:

$$\frac{{\mathop {\sum}\nolimits_{i = 1}^3 { < \left( {x_{1i}\left( t \right) - x_{2i}\left( t \right)} \right)^2 > _{t \in T_1 \cap T_2}} }}{{\sqrt {\mathrm{len}\left( {t \in T_1 \cap T} \right)_2} }}$$

Where x1 are the coordinates from channel 1 and x2 are the coordinates from channel 2, T1 contains all the time frames from channel 1 and T2 contains all the time frames from channel 2, and len is a function that returns the length of an array. We solved the LAP using the distance measure above to match tracks across channels. Tracks with average distances across channels higher than 1 μm are filtered out. Matched tracks with lower than 25 time points are filtered out. For each matched pair of tracks, we calculate the pairwise distance using the Euclidean distance in three dimensions. We define noisy pairwise distance using the ratio of the pairwise distance in three dimensions and two dimensions. In particular, we defined as noisy the top 5% of this ratio and filtered them out. To calculate the radial MSD, we first calculate the time-averaged radial MSD for each pairwise distance ‘trajectory’. We then calculate the ensemble average (across trajectories) of the log of time-averaged radial MSD. We corrected for the radial localization uncertainty by estimating the standard deviation of the error distribution using fixed images as described by Kepten et al.55. To calculate the scaling (α) and the generalized diffusion coefficient (D) of each MSD curve, we fitted the ensemble average time average MSD between 30 and 300 s. Since we are always comparing two conditions whose cell-cycle profiles are similar, we ignore the effect of sister chromatids. All scripts used for the analysis can be found at https://github.com/zhanyinx/SPT_analysis/.

### Estimation of experimental uncertainty on radial distance

To estimate our uncertainty in detecting distances across channels, we used a cell line with multiple integration of TetO arrays that can be tagged with TetR-eGFP and TetR-tdTomato. Spot detection is done as for our dual-color lines. We corrected for chromatic aberration using TetraSpeck Microspheres, 0.1-μm beads (Thermo Fisher Scientific, T7279), and then matched spots across channels by solving the LAP using scipy.optimize.linear_sum_assignment function with the Euclidean distance between spots as a measure of distance. Spots across channels with distances higher than a threshold are filtered out to avoid mismatches. We used a threshold of 300 nm for matching the spots registration. We applied a second round of chromatic aberration correction using the set of registered points themselves. The resolution limit (uncertainty) is then estimated as the average distance between registered spots which corresponds to 130 ± 70 nm.

### HMM for detection of the proximal state

To detect the proximal state in a threshold-independent manner, we used an HMM with two hidden states (‘proximal’ and ‘distal’). We used a Gaussian model for the emission probabilities. Only distance trajectories with less than 20% missing values at any time point are kept. Missing values are filled with the first preceding time point with distance value. To more reliably detect the proximal state, we used all the trajectories from the experimental condition with both cohesin and CTCF sites to train an HMM. We then re-trained an HMM model for each experimental condition by using the proximal state (Gaussian mean and standard deviation) from the experimental condition with both cohesin and CTCF sites. Finally, we applied the experimental condition-specific HMM to every trajectory to estimate the contact duration and rate of contact formation for all the experimental conditions. The HMM model training can be found as a jupyter notebook (https://github.com/zhanyinx/SPT_analysis/blob/main/notebooks/HMM_experimental_data.ipynb). We modified the hmmlearn library to allow fixing proximal state during HMM training. The modified hmmlearn library can be found at https://github.com/zhanyinx/hmmlearn.

### Simulations

Polymer simulations were performed using LAMMPS57. We chose Langevin dynamics with the NVT thermostat. Arbitrary units were set such that thermal energy kBT = 1, where kB is the Boltzmann constant and T is room temperature, corresponding to 300 K. For every set of parameters, we performed ten independent runs. A run consists of an equilibration part of 107 simulation steps and a production part of 108 simulation steps. For subsequent analysis and calculation of contact maps, we recorded the data every 104 simulation steps. In simulations for Fig. 2, the chain length was 1,125 beads. In simulations for Figs. 5 and 6, the chain length was 1,000 beads. We used PyMOL software (v.2.3.3) to represent snapshots of polymer chain in Fig. 2a. Examples of initial conformations and simulation parameters can be find at https://github.com/giorgettilab/Mach_et_al_chromosome_dynamics, in the polymer simulations section.

To simulate the loop extrusion process, we developed and embedded in LAMMPS a package called ‘USER-LE’. The loop extrusion model contains extruders and barriers on the polymer. An extruder is represented as an additional sliding bond, which extrudes the loop in a two-sided manner. It can be loaded to the polymer between (i) and (i + 2) beads with a certain probability only when the bead (i + 1) is unoccupied by another extruder and is not a barrier. Each extruder can be unloaded from polymer with a certain probability. Every bead can be occupied by only one extruder. Extruders cannot pass through each other. When extruders meet each other on the polymer, they stall until one of them is released. Every extruder attempts to make an extruding step every N simulation steps.