Main

The accuracy of genome inheritance depends on the faithful segregation of sister chromatids. Segregation without errors requires correct biorientation of chromosomes, achieved when microtubules from opposite spindle poles attach to kinetochores on sister chromatids until all of them come under tension1,2. Sister chromatids are physically linked by the chromosomal complex cohesin, which counteracts the pole-directed, spindle-generated forces required for biorientation and chromosome segregation3,4. Once biorientation is established, the cohesin complex is cleaved by separase and individual chromatids symmetrically separate towards opposite poles5.

The cohesin complex is composed of four core subunits, arranged to form a distinct ring-like architecture critical to its capacity to embrace DNA and establish sister chromatid cohesion6,7,8. The flexible Smc1Psm1 and Smc3Psm3 subunits (budding yeast nomenclature with fission yeast proteins used in this study in superscript) are connected at one end via the hinge domain, while at the other end lie the ATP-binding heads5 (Fig. 1a). The kleisin subunit Scc1Rad21 completes the cohesin ring by connecting the ATPase heads. Scc1Rad21 also mediates interactions with Scc3Psc3, key to recruiting and maintaining cohesin’s association with chromosomes, as well as with the cohesin loader Scc2Mis4–Scc4Ssl3 and regulatory proteins Pds5Pds5 and Wpl1Wapl (refs. 9,10,11).

Fig. 1: Topological loading of individual cohesins on DNA.
figure 1

a, Schematic illustration of the cohesin tetramer. b, Graphical representation of the assay for visualizing and applying force to the head domain of cohesin bound to λ-DNA. Loading of the cohesin tetramer onto DNA is performed in the presence of the cohesin loader Scc2Mis4–Scc4Ssl3 and ATP (not depicted). c, Example of a kymograph (left) and three examples of single-step photobleaching traces (right) of TMR-labeled single cohesin molecules on λ-DNA. Arrows point to bleaching events of three independent cohesins. d, Distribution showing the fraction of single salt-resistant monomeric cohesins on DNA (n = 112). e, Total number of cohesin molecules bound to all λ-DNAs in the presence or absence of the cohesin loader and ATP after washes at increasing NaCl concentrations. Absolute numbers shown correspond to all cohesins on all DNAs counted across three independent experiments (n = 3). f, Stills showing loading of the Smc3Psm3–kleisin cohesin without the crosslinker. Five DNA examples are shown at 50 mM and after 500 mM NaCl wash. DNA itself is not visualized. g, Quantification of Smc3Psm3–kleisin cohesin crosslinking. Without addition of the crosslinker, 52% of all salt-resistant cohesins were spontaneously released from the tethered DNA. Under the same conditions, following crosslinking, only 3% of Smc3Psm3–kleisin cohesin were released (n = 67, total molecules).

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The cohesin ring physically entraps DNA6,12. This activity is consistent with its function in mitosis, where cohesin must possess remarkable mechanical stability to hold sister chromatids against tensions of up to several hundreds of piconewtons13 to allow correct chromosome biorientation. However, whether a single cohesin complex can hold both sister DNAs is unknown, as is the force that one cohesin might be able to withstand. During interphase, cohesin is thought to translocate along DNA and extrude DNA loops14,15,16,17. This might entail interactions between cohesin and mechanical barriers such as RNA polymerases and the replication machinery, which can generate tens of piconewtons of force18,19. How cohesin behaves upon encountering force-generating molecular complexes is also unknown. In this article, we measured mechanical forces that disengage a single cohesin ring complex and showed that the disengagement leads to the dissolution of both cohesin-DNA and cohesin-mediated DNA–DNA interactions.

Results

Individual cohesins load topologically on DNA

To address how cohesin mechanically interacts with DNA, we devised an in vitro system in which we could monitor the response of individual DNA-bound cohesin complexes to external force, while visualizing the same cohesin molecules and DNA by total internal reflection fluorescence (TIRF) microscopy (Fig. 1b). We purified active fission yeast cohesin tetramers labeled with both a tetramethylrhodamine (TMR) fluorophore for visualization, and a biotin tag for binding to streptavidin-coated beads and force application, fused to the Smc1Psm1 and Smc3Psm3 head domains, respectively (Extended Data Fig. 1a–d)20.

In the presence of ATP and the cohesin loader, we loaded individual cohesin complexes onto the tethered λ-DNA such that ~80% of all loaded cohesin molecules were single cohesin complexes, as indicated from fluorescence intensity and single-step photobleaching (Fig. 1c,d and Extended Data Fig. 1e). To ensure that only those cohesins that topologically entrapped DNA remained bound, we performed washes with increasing NaCl concentrations. After washes containing 130 mM NaCl, ~30% of all initially loaded cohesin persisted on DNA. This fraction remained unchanged after further increasing the salt concentration up to 2 M (Fig. 1e). When, in a separate experiment, we added cohesin without the loader Scc2Mis4–Scc4Ssl3 and ATP, required for topological loading, almost no cohesin remained on DNA following the salt washes (Fig. 1e), supporting the idea that salt-resistant cohesin topologically interacts with DNA. To test the topological nature of cohesin loading further, we cleaved the λ-DNA at a single site using the restriction enzyme XhoI. Upon cleavage, all the examined cohesin molecules slid off and left DNA (n = 25), which confirmed the topological interaction between DNA and salt-resistant cohesin (Supplementary Video 1).

Finally, we asked whether, following loading, DNA was entrapped inside cohesin’s main ring6. To test this, we employed the SpyTag–SpyCatcher covalent crosslinking system21. We purified cohesin bearing two SpyTags, one attached to the Smc3Psm3 C-terminus and one to the kleisin N-terminus (‘Smc3Psm3–kleisin’ cohesin). This allowed for covalent closure of the Smc3Psm3–kleisin interface using a crosslinker consisting of two SpyCatcher modules, connected by a long and flexible unstructured polypeptide linker (Methods and Extended Data Fig. 1f). Before crosslinker addition, Smc3Psm3–kleisin cohesin loaded onto DNA in a salt-resistant manner, similarly to the wild-type complex (Fig. 1f). After loading onto DNA, addition of the crosslinker efficiently prevented spontaneous cohesin release from DNA in the presence of ATP, consistent with covalent closure of the Smc3Psm3–kleisin interface through which DNA is thought to unload22,23. Agreeing with the efficiency of spontaneous release determined earlier22, ~52% of non-crosslinked cohesin complexes were released from the DNA in the presence of ATP after 60 minutes of incubation. In comparison, only 3% of the Smc3Psm3–kleisin crosslinked cohesin were removed in similar conditions. This indicates crosslinking efficiency of over 90% (Fig. 1g) and shows that closure of the Smc3Psm3–kleisin interface of cohesin on DNA prevents it from unloading. Hence, we confirmed that DNA was topologically entrapped inside the cohesin ring after it was loaded onto DNA.

Cohesin disengages from DNA under force

Having identified conditions for the topological loading of individual cohesin complexes onto DNA, we investigated the mechanical stability of the cohesin ring entrapping the DNA. To this end, we attached streptavidin-coated beads to cohesins on λ-DNA and used optical tweezers to apply force by displacing the bead.

To ensure that force was applied to a single cohesin only, we optimized the number of cohesin molecules per DNA. The experimentally determined probability distribution of the number of cohesins per DNA and the distribution of the number of beads per DNA allowed us to calculate the distribution of cohesin complexes per individual bead on DNA (Methods and Extended Data Fig. 2a). In optimized conditions, this resulted in approximately 70% of all beads attached to DNA via a single cohesin molecule and the rest by two or more cohesin molecules.

After the bead attachment, we probed mechanical stability of cohesin by displacing the bead with respect to DNA. When force was applied along the DNA, cohesin slid on DNA almost freely, with little to no resistance (Fig. 2a and Supplementary Video 2). In contrast, when force was applied perpendicularly to DNA, after the initial DNA stretching, cohesin strongly resisted movement and detachment. Free movement along the DNA, but resistance to perpendicular removal from DNA, is consistent with the topological interaction between cohesin and DNA.

Fig. 2: Single cohesin on DNA resists ~20 pN forces.
figure 2

a, Forces exerted on cohesin as it moves along the tethered λ-DNA (x direction, left) or perpendicularly to it (y direction, right). Distance shows the relative position of the bead along the corresponding axis. Relates to Supplementary Video 2. b, Typical example of an FD curve (top) and the corresponding single-step photobleaching trace (bottom) showing the rupture of a single cohesin from the tethered λ-DNA at a force of 20 pN. Relates to Supplementary Video 3. c, Normalized distribution of the total rupture forces for the cohesin variant where force was applied via the head domain (n = 89). In c and e, solid lines show theoretical distributions of rupture forces with optimal parameters determined by fitting experimental data. Parameter values are shown in the main text. d, Cohesin’s fluorescent signal colocalizing with the bead before and after two pulling experiments, in which the rupture event was recorded (left), or when the applied force was insufficient (5 pN) for detachment. DNA is not visualized. Bottom graphs show corresponding FD curves for these experiments as a function of both distance and time. e, Normalized distribution of the total rupture forces for the cohesin variant where force was applied via the hinge domain (n = 21). The solid line is the theoretical distribution.

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We next collected force–distance (FD) curves by moving the bead at a constant velocity perpendicularly away from the DNA, which first stretched the DNA and then led to an abrupt detachment event (Fig. 2b and Supplementary Video 3). Imaging cohesin attached to the bead before force application revealed that cohesin bleached in a single-step manner, consistent with the single-step detachment signature observed in the FD curves and demonstrating that the measured detachment force was that of a single cohesin molecule bound to DNA (Fig. 2b and Extended Data Fig. 2b). Of the total detachment events observed, 70% (total n = 92) of the FD curves showed a single rupture event at an average force of ~20 pN (Fig. 2c), and the remaining FD curves showed multiple detachment peaks (Extended Data Fig. 2c). Thus, the observed ratio of the single-step detachments was in excellent agreement with the expected fraction of the beads bound to single cohesin molecules, which further supported our conclusion that the single-step detachment events corresponded to single cohesin ruptures.

To verify that the observed detachments were indeed due to the rupture of the cohesin ring and release of its DNA interaction, we visualized single cohesins on DNA before and after the force-induced rupture. The disappearance of the cohesin signal from DNA after the mechanical rupture event confirmed cohesin detachment (Fig. 2d). As a control, we applied 5 pN of force to cohesin for the same duration of the experiment, which was not sufficient for detachment. After releasing the bead without observing ruptures in the FD curves, cohesin remained visible on the DNA and attached to the bead in all cases (n = 12).

To further confirm that our assay reported on cohesin rupture, and not detachment of the biotin–avidin interaction or DNA unwinding, we replaced cohesin with a biotin that was directly covalently coupled to DNA and to which we attached an avidin-coated bead. We could not detach the bead from DNA, even with forces exceeding 80 pN, ultimately leading to the bead escaping from the trap but remaining bound to DNA (Extended Data Fig. 2d). We also recorded FD curves of the DNA by attaching its one end to the flow cell and the other end to the bead. This resulted in the typical DNA overstretching transitions at ~65 pN (Extended Data Fig. 2e), as would be expected for the double-stranded DNA (dsDNA)24, and very different from our observed discrete cohesin–DNA rupture events. These controls further bolstered our conclusion that the bead detachment from DNA in our experiments was indeed due to the rupture of the cohesin ring.

The cohesin ring ruptures at a specific interface

To determine whether the direction of force applied to cohesin influenced the rupture, we tested a recombinant cohesin complex bearing the biotin tag inserted at the Smc3Psm3 hinge domain, instead of the head (Extended Data Fig. 3). Combined data from single-step FD curves revealed a small but statistically significant decrease in the rupture force for complexes pulled via the hinge domain (median 18 pN) compared to those pulled via the head domain (median 24 pN, P = 0.0083—Kolmogorov–Smirnov test) (Fig. 2e).

To explore possible reasons for this difference, we took a modeling approach. Using Monte-Carlo simulations, we tested whether our experimental observations could be explained by a model in which the rupture occurs due to the disengagement of a cohesin ring entrapping DNA. We described the cohesin ring using two parameters: k0, the rate of spontaneous cohesin ring disengagement (interface opening) in the absence of force, and δ, a mechanical displacement parameter which determines how force affects the disengagement (Fig. 3a, Extended Data Fig. 4a–f and Methods). When we independently fitted the experimental distributions for cohesin variants pulled either via the head or the hinge domain to this model, we obtained the following values for k0 and δ:

$${k}_{0}^{\left({{\mathrm{head}}}\right)}={0.0027\;{\mathrm{s}}}^{-1},(0.0023,0.0036),{\delta }^{({{\mathrm{head}}})}=1.23{\;\rm{nm}},(1.17,1.29)$$
$${k}_{0}^{\left({{\mathrm{hinge}}}\right)}\!=\!{0.0025\;{\mathrm{s}}}^{-1},(0.0016,0.0034),{\delta }^{\left({{\mathrm{hinge}}}\right)}\!=\!1.61{\;\rm{nm}},(1.47,1.71)$$

where the values in brackets show 90% confidence intervals (Fig. 2c,e, solid lines).

Fig. 3: Mechanical force disengages the cohesin ring at the hinge interface.
figure 3

a, Schematic representation of the cohesin ring disengagement under external force for cohesin variants with the biotin tag for force application attached to either the heads or the hinge domain. The formulas show how disengagement rates for different geometries depend on force, \({k}_{0}\) and δ. b, Normalized distribution of the total rupture forces for Smc3Psm3–kleisin crosslinked cohesin (n = 24). c, Typical example of the FD curve showing a single rupture event at a high force (~70 pN) for the hinge-crosslinked cohesin. d, Normalized distribution of the rupture forces for the hinge-crosslinked cohesin (n = 25). In b and d, solid lines show theoretical distributions of rupture forces with optimal parameters determined by fitting experimental data. Parameter values are shown in the main text.

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This shows that the datasets for cohesin variants where force was applied either via the head or the hinge domain were best described with the same k0, but different δ. Bootstrap analysis confirmed that the sample size was sufficient, and that a further increase in the number of measurements would not affect this result (Extended Data Fig. 4g–i). This finding suggests that, when an external force is applied at two different locations, the cohesin ring opens at the same interface (characterized by the same spontaneous opening rate k0), while the displacement parameter δ varies because the different force direction leads to the bond rupturing along a different trajectory25. Indeed, a ring would always be expected to break at the same, weakest interface, irrespective of where the force is applied.

The cohesin ring opens at the hinge interface

To investigate where the cohesin ring breaks under external force, we began by examining the Smc3Psm3–kleisin interface, through which DNA is thought to pass during cohesin’s enzymatic, ATP-dependent unloading from chromosomes. As described above, we used the SpyCatcher-based crosslinker to covalently close this interface with over 90% efficiency after loading of Smc3Psm3–kleisin cohesin onto λ-DNA in the flow cell (Fig. 1g). We then applied force from the head domain of the Smc3Psm3–kleisin-crosslinked complexes. The histogram of the rupture forces revealed an average value of rupture force of ~20 pN, with a distribution similar to that of the wild-type complex (Fig. 3b). Moreover, when we fitted model parameters to this distribution, we obtained values that were statistically indistinguishable from those describing wild-type cohesin: \({k}_{0}^{({{\mathrm{head}}}/X)}\) = 0.0022 s−1, (0.0011, 0.0035), \({\delta }^{({{\mathrm{head}}}/X)}\) = 1.22 nm, (1.1, 1.35) (Fig. 2c). Since covalent bonds can resist forces over 1,000 pN, the rupture events could not have occurred at the crosslinked Smc3Psm3–kleisin interface in this case. Therefore, cohesin ring disengagement at ~20 pN must have occurred at a different interface.

Next, we examined cohesin’s hinge domain. The hinge presents a relatively small contact area between the Smc1Psm1 and Smc3Psm3 subunits, and earlier experiments have observed that it may transiently open8,26,27. To test whether the hinge could represent the weakest interface in the cohesin ring, we again used the SpyTag–SpyCatcher system. This time, we directly covalently crosslinked cohesin’s hinge domain by inserting a SpyTag into Smc3Psm3 and SpyCatcher into Smc1Psm1 (Extended Data Fig. 5a). The resulting hinge crosslinking efficiency was ~70% (Extended Data Fig. 5b,c). We found that the hinge-crosslinked complex could topologically load onto DNA, albeit with slightly reduced efficiency compared to wild-type cohesin (Extended Data Fig. 5d), consistent with recent observations of the budding yeast cohesin complex28. Considering both crosslinking efficiency and the reduced ability of crosslinked molecules to load onto DNA, approximately half of the DNA-loaded cohesin molecules are expected to be successfully crosslinked under these conditions (Methods).

Force application to the DNA-loaded cohesin complexes resulted in a bimodal distribution of rupture forces. Approximately half of the cohesin molecules detached at ~20 pN and half at ~70 pN (Fig. 3c,d), revealing a marked impact of hinge crosslinking on cohesin’s force response. Fitting the observed rupture force distribution to our model showed that the peak at 20 pN could be explained by the disengagement of the same interface as for wild-type cohesin and described with the statistically identical set of parameters: \({k}_{0}^{(20)}\) = 0.0021 s−1, (0.001, 0.0034), \({\delta }^{(20)}\) = 1.23 nm, (1.1, 1.37). Thus, the peak at 20 pN probably corresponds to the population of cohesin molecules that failed to crosslink at the hinge and therefore behave as wild type. In turn, the peak at 70 pN points to the rupture of a different interface, characterized by a distinct set of parameters indicating much stronger link: \({k}_{0}^{(70)}=2.7\times {10}^{-5}\) \({\text{s}}^{-1}\), (\(1.6\times {10}^{-5}\), \(3.9\times {10}^{-5}\)), \({\delta }^{(70)}\) = 0.8 nm, (0.77, 0.83). Since this peak was absent from all previous distributions, it must have arisen from cohesin molecules with successfully crosslinked hinge domain. As covalent crosslinking requires forces above ~1,000 pN to break, the 70 pN peak could not be the result of a breakage of the crosslink, but rather represents the disengagement at another, the second weakest, cohesin ring interface.

These experiments showed that in all cohesin constructs in which the hinge can disengage, the ruptures occur at ~20 pN, and much higher forces are required to break the cohesin ring when the hinge is covalently closed. Thus, we established that the hinge is the weakest interface in cohesin ring and that it breaks at ~20 pN.

A single cohesin complex establishes DNA–DNA interactions

Next, we addressed whether a single cohesin complex could entrap two DNA molecules and, if so, what force such an interaction withstands. After adding fluorescently labeled circular DNA plasmids, we found that both single-stranded DNA (ssDNA) (41 events) and dsDNA plasmids (36 events) could be captured by single cohesins preloaded onto λ-DNA. While second dsDNA capture is inefficient in bulk assays compared to ssDNA29, the sensitivity of our single-molecule experiments allowed us to observe rarer dsDNA capture events. Capture was evident from colocalization of cohesin and the second DNA (Fig. 4a,b), which persisted even after 500 mM NaCl washes (Extended Data Fig. 6a). Confirming the topological nature of second DNA capture, we found that the plasmid was lost in 16 out of 21 cases after addition of the restriction enzyme PacI, which recognizes the plasmid but not the λ-DNA (Fig. 4c). The few plasmid molecules that persisted following PacI addition were probably the result of either incomplete PacI cleavage in our reaction buffer, or of inefficient diffusion of the cut plasmid out of the sterically restricting cohesin ring.

Fig. 4: A single cohesin complex captures two DNAs.
figure 4

a, Example of kymographs showing LD655-labeled cohesin colocalizing with a MFP488-labeled single-stranded plasmid (n = 36 in at least five independent repeats). b, Example of kymographs illustrating LD655-labeled cohesin colocalizing with a MFP488-labeled double-stranded plasmid, also stained with Sytox Orange (Sytox) (n = 41 in at least five independent repeats). c, Fraction of second DNA molecules that remain attached to λ-DNA after addition of restriction enzyme PacI (n = 21) or unloading complex Pds5Pds5–Wpl1Wapl (n = 6). Connection with the second DNA is stable when no additional proteins are added (n = 24). d, Three examples of single-step photobleaching traces of single LD655-labeled cohesin molecules that performed second DNA capture. Arrows point to bleaching events of three independent cohesins. e, Example of kymographs of second dsDNA capture by cohesin before and after Pds5Pds5–Wpl1Wapl addition (n = 6 in three independent repeats).

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Single-step photobleaching traces (Fig. 4d and Extended Data Fig. 6b) confirmed that single cohesin complexes were responsible for mediating the majority of both dsDNA–ssDNA (12 out of 13 cases analyzed) and dsDNA–dsDNA interactions (7 out of 10 cases analyzed). We observed that ssDNA capture was labile (Extended Data Fig. 6c and Supplementary Video 4) and that we could convert the second ssDNA to dsDNA using T7 DNA polymerase (Extended Data Fig. 6d), in agreement with previous observations29.

Next, we interrogated the dependency of the reaction on ATP and the cohesin loader Scc2Mis4–Scc4Ssl3 (ref. 29). Virtually no second DNA capture was recorded in the absence of either ATP (1 event out of 151 cohesin-decorated λ-DNAs) or Scc2Mis4–Scc4Ssl3 (0 events out of 236 λ-DNAs), compared to 77 second DNA capture events on 203 λ-DNAs in the presence of both. Moreover, introduction of the cohesin unloading complex Pds5Pds5–Wpl1Wapl following second DNA capture (Extended Data Fig. 6e) led to second dsDNA dissociation (Fig. 4c,e), confirming the potency of the cohesin unloader in dissolving DNA–DNA interactions11. Thus, the requirements for second DNA capture, its stable topological nature and sensitivity to a biological unloader suggest that single cohesin molecules have reconstituted physiologically relevant DNA–DNA interactions in our experimental setup.

Cohesin ring disengagement dissolves DNA–DNA interaction

Having established that individual cohesins can link two DNA molecules, we measured the force that a single cohesin complex can resist when holding together two DNAs (Fig. 5a). To do so, we used covalently biotinylated second DNA substrates to which we attached streptavidin-coated beads and recorded FD curves until we detected DNA–DNA ruptures (Fig. 5b and Supplementary Videos 5 and 6). Most ruptures appeared as a single peak, both for dsDNA substrates (24 out of 28 measurements) and for ssDNAs (17 out of 20 measurements) (Fig. 5c and Extended Data Fig. 7a). For those single-peak rupture events, we confirmed that the two DNAs were tethered by a single cohesin ring, as indicated by single-step photobleaching of cohesin’s fluorescent label, either before (Fig. 4d) or after binding to the beads (Fig. 5d and Extended Data Fig. 7b).

Fig. 5: The DNA–DNA interaction ruptures due to mechanical cohesin ring disengagement.
figure 5

a, Graphical representation of the assay developed for visualizing and applying force to cohesin holding together two DNAs. b, Schematic illustration (top) and snapshots (bottom) showing detachment of the second dsDNA from the λ-DNA, which remains attached to the surface (n = 24 in at least three independent repeats). Relates to Supplementary Video 6. c, Characteristic FD curve showing detachment of the second dsDNA at ~20 pN. d, Three examples of single-step photobleaching traces of single bead-bound LD655–cohesins linking two DNAs. Arrows point to bleaching events for three independent molecules. e, Normalized distribution of rupture forces for cohesin between two DNAs (both ssDNA and dsDNA combined, n = 41). The solid line shows theoretical distribution of rupture forces. Parameters are the same as in Fig. 2c,e. For details, see text. f, Schematics illustrating different events during cell cycle when cohesin can be removed from DNA, either biochemically (bottom) or mechanically, via disengagement of the cohesin ring (top).

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Average rupture forces recorded with both second ssDNA and dsDNA were ~16 pN (Fig. 5e). There was no statistically significant difference between forces recorded with a second ssDNA or dsDNA (Extended Data Fig. 7c). However, the rupture forces recorded between two cohesin-linked DNAs were overall smaller than those measured when beads were attached directly to cohesin (P = 3.8 × 10−5—Kolmogorov–Smirnov test). To understand this difference, we investigated the consequence of including the second DNA in our computational ring disengagement model, assuming the parameters of the cohesin ring rupture were exactly the same as those previously determined (Fig. 2c,e) and that disengagement happened randomly in either the hinge-like or head-like orientation. These simulations indeed resulted in the reduced predicted rupture forces consistent with our measurements (Fig. 5e). This is because stretching more DNA, which now includes both the first and the second DNA molecules, takes up additional time and reduces the effective force ramping rate, which increases the likelihood that cohesin ruptures at a slightly lower force (Extended Data Fig. 7d–f). Thus, the measured rupture force distribution of DNA–DNA interactions is explained by the rupture of the cohesin ring at the same interface and therefore is consistent with being determined by the same mechanism as in experiments where the force was applied directly to cohesin. Taken together, our experiments demonstrate that single cohesin ring holds together two DNA molecules and withstands forces up to 20 pN. All our results can be accounted for by a model in which the cohesin–DNA interaction breaks under external force due to cohesin disengagement at its weakest interface, the hinge domain.

Discussion

In this work we have shown that the mechanical stability of the cohesin–DNA interaction for cohesin complexes topologically entrapping DNA inside its main ring compartment is determined by the physical stability of the cohesin hinge domain. Our simulations confirmed that the simplest model that can account for all our data consists in the cohesin ring characterized by two interfaces that can disengage under mechanical force, leading to the release of DNA. We showed that the weakest interface that disengages first is the hinge with a stability limited by ~20 pN and the other is presumably the Smc3Psm3–kleisin, which is substantially stronger, but also eventually disengages at forces of ~70 pN (Extended Data Fig. 8). We have also demonstrated that single cohesin ring can hold two DNA molecules and that the stability of the DNA–DNA interaction under external force maintained by the single cohesin complex is limited by the physical stability of the cohesin ring.

These results provide a framework for understanding the mechanical forces acting on sister chromatids during cell division. Mechanical tension is the main regulator of mitotic progression from metaphase to anaphase. Various estimates put the maximum forces applied to chromosomes from the spindle of up to ~700 pN (ref. 13) and the absolute number of cohesins at ~200 molecules per centromere30, with cohesins located close to the microtubule attachment sites probably bearing the majority of the physical load exerted by the spindle.

Cohesin depletion studies have reported that ~20% of chromosome-bound cohesin (that is, ~40 molecules) is sufficient to sustain sister chromatid cohesion31,32, and that the apparent excess of cohesin molecules at the centromere might serve to avoid cohesion fatigue32. If individual cohesins entrap both sister chromatids in vivo, and assuming that a maximum load of ~700 pN is exerted by the spindle on the chromosome, each cohesin would have to resist ~18 pN, a value close to the force measured in our experiments. These estimates suggest that the minimal necessary number of cohesin molecules at centromeres is probably limited by the mechanical stability of the cohesin ring.

Our findings can explain the mechanism behind the transient splitting of chromosomes in pre-anaphase known as ‘centromere breathing’33. The build-up of tension generated by spindle forces could force the cohesin molecules that experience the most strain to undergo mechanical disengagement at the hinge. However, the connection with the DNA could be re-established by either dynamic cohesin loading or simple hinge closure.

Since the hinge domain is connected to the head domains only via long flexible coiled coils, it is unlikely that the purely mechanical disengagement of cohesin at the hinge would be affected by the chemical state of its heads. One important modification of cohesin is the acetylation of the Smc3Psm3 head domain34 that stabilizes the cohesin–chromosome association in the absence of external force. While this modification reduces the enzymatic turnover of cohesin on DNA, given the large distance and flexibility between the hinge and head domains, we speculate that the physical stability of cohesin under external force determined by the hinge would presumably be unaffected by acetylation.

Mechanical opening of the cohesin ring may also have implications in other processes that involve interaction between chromosomes and cohesin, including transcription and DNA replication. RNA polymerases are known to act as barriers for cohesin movement17. Although the stall force for RNA Polymerase II is ~8 pN, associated proteins increase this by a factor of two18, which might enable the transcription machinery to physically open cohesin molecules and assist in polymerase translocation past immobile cohesin, for example, when bound to CCCTC-binding factor17. Replicative helicases in turn generate forces in excess of 20 pN (ref. 19). Their encounter with cohesin could result in transient hinge disengagement, allowing the replication machinery to mechanically open up cohesin rings during the establishment of sister chromatid cohesion. This may lead to cohesin removal altogether or may allow the replication or transcription machinery to pass beyond the disengaged cohesin ring, which would remain on DNA in its open form and possibly close back forming the ring again after allowing the bulky complexes to move past it (Fig. 5f).

In conclusion, we show that physical force is a physiologically possible mechanism for disengagement of the cohesin ring, in addition to cleavage by separase and the enzymatic removal action of Pds5Pds5–Wpl1Wapl1 (Fig. 5f). Mechanical disengagement of the cohesin ring is likely to play roles during both interphase and mitosis. In the future, it will be interesting to explore the effect of conditionally closing cohesin interfaces in vivo to probe the consequences of altering cohesin’s mechanical stability in the context of transcription, replication and chromosome biorientation.

Methods

Detailed methods describing generation of constructs, protein purification, labeling and western blotting can be found in Supplementary Information.

Topological cohesin loading assay

Topological loading of cohesin onto DNA was performed following a previously described protocol20 with some modifications: 25 nM cohesin, 50 nM Scc2Mis4–Scc4Ssl3 and 3.3 nM pBluescript II KS dsDNA were combined on ice in reaction buffer (35 mM Tris–HCl pH 7.5, 1 mM MgCl2, 1 mM tris(2-carboxyethyl)phosphine (TCEP), 15% (w/v) glycerol and 0.003% (w/v) Tween20). Addition of 0.5 mM ATP initiated the reaction, which was incubated at 30 °C for 120 min, shaking at 1,000 rpm. The reaction was terminated by adding 500 µl of ice-cold Wash Buffer A (35 mM Tris–HCl pH 7.5, 500 mM NaCl, 0.5 mM TCEP, 10 mM ethylenediaminetetraacetic acid (EDTA), 5% (w/v) glycerol and 0.35% (w/v) Triton X-100). Anti-Pk antibody was added to protein A-conjugated Dynabeads (ThermoFisher) and allowed to adsorb by rotating the beads at 4 °C for at least 1 h. The anti-Pk-coated magnetic beads were added to the terminated reactions and allowed to bind while rocking at 4 °C for 2 h. The beads were then washed once with Wash Buffer B (35 mM Tris–HCl pH 7.5, 750 mM NaCl, 0.5 mM TCEP, 10 mM EDTA, 5% (w/v) glycerol and 0.35% (w/v) Triton X-100) and three times with Wash Buffer A. The cohesin-bound DNA was eluted from the beads using 15 µl of elution buffer (35 mM Tris–HCl pH 7.5, 50 mM NaCl, 1 mM EDTA, 0.75% sodium dodecyl sulfate (SDS) and 1 mg ml−1 Protease K (TaKaRa)) and heating the sample at 50 °C for 20 min. The cohesin-bound DNA was separated from the magnetic beads and the DNA was analyzed using 0.8% agarose gel electrophoresis in Tris acetate EDTA (TAE) buffer, then stained with SYBR Gold (Thermo Fisher Scientific). Gel images were recorded using a Typhoon FLA 9500 biomolecular imager (GE Healthcare), and ImageJ was used to quantify the band intensities.

Microfluidics device preparation

To perform single-molecule experiments, a microfluidics system was adapted from previously described work16,17. Flow cells were assembled using parafilm to generate microfluidics channels between glass slides and a hydrophobic coverslip. Gentle heat after assembly ensured slight melting of the parafilm to keep the channels in place. Each glass slide (VWR, 26 × 76 × 1.0 mm) was drilled with holes, into which metal tubings (New England Small Tube Corp) were inserted and glued with epoxy (Devcon), with the metal tubings forming the inputs and the outputs for the channels. The drilled glass slides were reused for experiments after a cleaning procedure: sonication in 100% acetone for 15 min, sonication in 100% ethanol for 15 min, blow-drying using compressed nitrogen air and plasma cleaning for 5 min.

The coverslips (Marienfeld, high-precision, 24 × 60 mm) were first cleaned by sonication in 100% acetone for 15 min, and sonication in 100% ethanol for 15 min followed by ten rinses with purified water. The coverslips were then blow-dried using compressed air (80% nitrogen) and plasma cleaned for 5 min on each side. To make them hydrophobic, the coverslips were silanized using a solution of 5% dichloromethylsilane (Sigma-Aldrich) dissolved in heptane (Sigma-Aldrich) for an incubation time of 60 min at room temperature. The silanized coverslips were then washed using two rounds of 5-min sonication in chloroform (Sigma-Aldrich) followed by a 5-min sonication in water. After a final sonication in chloroform, the coverslips were blow-dried with compressed nitrogen air.

Single-molecule experiments

Flow cells were first extensively washed with a total of 600 µl of phosphate-buffered saline (PBS). Solutions were introduced at a rate of 100 µl min−1 unless otherwise stated. For visualization-only experiments, flow cells were incubated for at least 30 min with 150 µl of anti-digoxigenin antibody (Anti-Digoxigenin-AP Fab fragments, Roche, 150 U) diluted 50 times in PBS. For optical tweezers experiments, flow cells were incubated for at least 30 min with 150 µl of Avidin-DN (VectorLabs) diluted 50 times in PBS. Flow cells were again washed with 600 µl of PBS. To limit background fluorescence derived from nonspecific protein interactions with the surface, the flow cells were passivated using a 1% solution of Pluronic F-127 (Sigma-Aldrich) in PBS for 10 min, washed with PBS and further incubated with 10 mg ml−1 β-casein (Sigma-Aldrich) in PBS for at least 2 h.

Before binding of the 48.5 kb λ-DNA (New England Biolabs) to the surface, the nucleic acid was modified as follows. For visualization-only experiments, λ-DNA was labeled with digoxigenin at its termini in a 50-µl reaction mixture containing 0.25 mg ml−1 of λ-DNA, 1× Standard Taq buffer (New England Biolabs), 0.5 µl of Taq DNA polymerase (New England Biolabs), 1 mM dATP, dCTP, dGTP (Promega) and dUTP–digoxigenin (Jena Bioscience), then incubated at 72 °C for 30 min.

For optical tweezers experiments, λ-DNA was modified by annealing the following biotinylated oligonucleotides to the ends of the DNA:

$${\rm{aggtcgccgccc}}[{\rm{BiotinTg}}]$$
(MR1)
$${\rm{gggcggcgacct}}[{\rm{BiotinTg}}]$$
(MR2)

Modified λ-DNAs were purified to remove unincorporated nucleotides and primers using a spin column (Bio-Rad, Micro Bio-Spin P30). The concentration of the modified λ-DNA was determined by measuring absorbance at 260 nm using a Nanodrop ND1000 spectrophotometer.

After passivation of the surface, 150 µl of a 10 pM solution of either biotinylated (optical tweezers) or digoxigenin-labeled (visualization-only) λ-DNA diluted in PBS was introduced into the channel at a rate of 20 µl min−1 using a syringe apparatus (Harvard Apparatus, Pico Plus Elite 11). Excess λ-DNA was washed out at a rate of 50 µl min−1 with 150 µl of PBS and to block any free streptavidin molecules on the surface (optical tweezers), we incubated 5 mM biotin for 10 min. The flow cell was then equilibrated with a further 150 µl of R buffer (35 mM Tris–HCl pH 7.5, 50 mM NaCl, 1 mM MgCl2, 15% (w/v) glycerol, 1 mM TCEP, 1% (w/v) glucose, 0.2 mg ml−1 glucose oxidase and 35 µg ml−1 catalase).

For single cohesin loading of the cohesin variants where the force was applied directly to the complex, either via the heads or via the hinge domain, 0.5 nM TMR-labeled cohesin, 1 nM Scc2Mis4–Scc4Ssl3 and 0.5 mM ATP in R buffer supplemented with 0.1 mg ml−1 β-casein were applied. For Smc3Psm3–kleisin cohesin and for hinge-crosslinked cohesin, 0.5 nM or 1.5 nM cohesin, respectively, 2 nM Scc2Mis4–Scc4Ssl3, 0.5 mM ATP in R buffer supplemented with 0.1 mg ml−1 β-casein were used. The reactions were introduced into the flow cell at a rate of 15 µl min−1 and incubated for 15 min. The flow cell was then washed at a rate of 30 µl min−1 with 150 µl of R buffer, followed by 150 µl of S buffer (35 mM Tris–HCl pH 7.5, 500 mM NaCl, 1 mM MgCl2, 15% (w/v) glycerol, 1 mM TCEP, 1% (w/v) glucose, 0.2 mg ml−1 glucose oxidase and 35 µg ml−1 catalase) and 150 µl of T buffer (35 mM Tris–HCl pH 7.5, 130 mM NaCl, 1 mM MgCl2, 15% (w/v) glycerol, 1 mM TCEP, 1% (w/v) glucose, 0.2 mg ml−1 glucose oxidase and 35 µg ml−1 catalase). Only for experiments in which the Smc3Psm3–kleisin variant was crosslinked after cohesin loading onto DNA, 30 nM of the SpyCatcher–SpyCatcher crosslinker was introduced into the flow cell at a rate of 15 µl min−1 and incubated for 10 min. Imaging was performed in U buffer (35 mM Tris–HCl pH 7.5, 75 mM NaCl, 1 mM MgCl2, 15% (w/v) glycerol, 1 mM TCEP, 1% (w/v) glucose, 0.2 mg ml−1 glucose oxidase and 35 µg ml−1 catalase) unless otherwise stated. In a subset of experiments, the tethered λ-DNA was stained by adding 2 nM Sytox Orange (Invitrogen) to U buffer.

For second DNA capture experiments, cohesin loading was performed using 1.5 nM LD655-labeled cohesin, 2 nM Scc2Mis4–Scc4Ssl3, 0.5 mM ATP in R buffer supplemented with 0.1 mg ml−1 β-casein, introduced into the flow cell at a rate of 15 µl min−1. After a 15-min incubation, excess cohesin was washed at a rate of 30 µl min−1 with 150 µl of S buffer followed by 150 µl of T buffer. Then 0.5 nM of the second DNA substrate was mixed with 2 nM Scc2Mis4–Scc4Ssl3 and 0.5 mM ATP in U buffer and incubated at 30 °C for 5 min. The reaction mixture was introduced at a rate of 10 µl min−1 and incubated into the flow cell for 10 min. The flow cell was then washed at 10 µl min−1 with 150 µl of either S or T buffer and imaged in U buffer, unless otherwise stated.

The DNA substrates used for second DNA capture were 7.2 kb M13mp18 ssDNA (New England Biolabs) and M13mp18 I RF dsDNA (New England Biolabs), which were covalently modified with either MFP488 fluorophores (visualization-only) or biotin labels (optical tweezers), using Label IT nucleic acid labeling kits (Mirus). The MFP488 fluorophore (488 nm excitation) on second DNA substrates allowed for simultaneous visualization with LD655–cohesin (647 nm excitation) and, where necessary, with λ-DNA stained with Sytox Orange (532 nm excitation). The protocol supplied for labeling was modified by reducing the amount of Label IT reagent to 0.1 µl µg−1, which resulted in roughly 1 fluorophore/biotin label per 120 bp.

For ssDNA-to-dsDNA conversion experiments, the ssDNA used was a 3 kb phage-derived pBluescript II KS (+) plasmid29, modified by annealing the following biotin-containing oligonucleotides for conjugation of three fluorescent labels:

$${\rm{caaccaagtca}}[{\rm{BiotindT}}]{\rm{tctgagaatagtgtatgc}}$$
(MR3)
$${\rm{tcagctcca}}[{\rm{BiotindT}}]{\rm{ggtcc}}$$
(MR4)
$${\rm{cttgaag}}[{\rm{BiotindT}}]{\rm{ggtggcctaactacgg}}$$
(MR5)

In these experiments, oligonucleotides were annealed to the DNA as opposed to chemically attaching several fluorescent labels to minimize steric hindrance for DNA polymerase.

The plasmid was then incubated with Qdot 525 Streptavidin Conjugate (Invitrogen) overnight at 25 °C. Following, 200 µM biotin was incubated for 60 min at 25 °C with the plasmid and excess Qdot-streptavidin/biotin eliminated using an Illustra S-400 HR MicroSpin column (GE Healthcare). Second DNA capture was performed as described above. After removing excess ssDNA, 120 µl of a mixture containing 0.1 unit µl−1 of T7 DNA polymerase (New England BioLabs) and 1 mM deoxynucleotide triphosphates (Sigma-Aldrich) in U buffer was added to the flow cell at a rate of 15 µl min−1. The mixture was incubated for 15 min, and the flow cell washed with 150 µl of S or T buffer supplemented with 2 nM of Sytox Orange.

Optical tweezers

Optical tweezer experiments were carried out on a JPK NanoTracker 2 system integrated with a custom-made TIRF microscope. For binding of cohesin or second DNA substrates to beads, biotin–streptavidin interactions were selected as they form the strongest known noncovalent interactions and can resist very high forces (160–400 pN) (refs. 35,36,37). The beads used for the experiments were either 0.5 µM streptavidin-coated beads (Bangs Lab) (1% solids w/v) or 1 µm polystyrene beads coated with streptavidin (Bangs lab) (1% solids w/v), which attached to the covalently conjugated biotin label on either cohesin or on the second DNA substrate, respectively. Before introduction into the flow cell, the beads were diluted 20 times into T buffer supplemented with 0.1 mg ml−1 of β-casein, washed three times (spun at 10,000g, at 4 °C for 15 min) and resuspended into the same buffer.

Beads were introduced into the flow cells at a rate of 5 µl/min and incubated for at least 15 min before being washed out with T buffer at the same rate. All optical trapping measurements were performed and recorded in T buffer. Force ramp measurements were all conducted at a constant speed of 0.16 µm s−1 in the y direction. Most experiments were performed in the absence of any DNA dye, but in experiments were the DNA and beads were visualized (such as force application on the second DNA), 2 nM Sytox Orange was added to the T buffer used to wash out excess beads. Addition of low concentrations of Sytox Orange were also used to confirm the integrity of the λ-DNA, without unduly altering its physical properties. In experiments performed in the presence of ATP, 1 mM of the nucleotide was added to T buffer as excess beads were washed out of the flow cell.

Data collection

FD data were processed using JPK Processing software (JPK NanoTracker version 6.1). For imaging, fluorescence emission was collected by an Andon iXon Life 888 EMCCD camera running at maximum electron multiplying gain. Typical exposure time was 50 ms. The Andor Solis software (version 4.31) was used to acquire and record microscopy data.

Experiments not requiring optical trapping were performed on a Nikon Eclipse Ti2 commercial TIRF microscope with a sCMOS camera (Photometrics, Prime 95B). The Nikon NIS-Elements software (version 5.41) was used to acquire microscopy data. The frame rate used varied depending on the experiment, ranging from 0.5 frames s−1 to 1 frame every 2 s. Images were saved as TIFF files without compression and further analyzed using ImageJ.

Data processing

Images and videos were analyzed using FIJI ImageJ software. The length of the DNA molecules was manually measured converting distance in pixels into kb, and kymographs were generated to observe colocalization of differentially labeled fluorescent molecules. Photobleaching data were first visually analyzed using ImageJ and later quantified using a custom-made MATLAB script generated to extract EMCCD counts following subtraction of background fluorescence. Analysis of the FD curves was performed using the JPK Processing Software and illustrated using MATLAB. Forces of rupture of each experiment were manually tracked and plotted using MATLAB illustrating contributions for both x and y directions. For statistical comparison between different groups of force rupture distributions, MATLAB was used to run a Welch t-test (unequal variance t-test) as the populations followed a normal distribution but did not have the same variance. Images were processed using Adobe Illustrator CC.

Monte-Carlo simulations

To simulate the distribution of rupture forces, we considered the λ-DNA attachment geometry as occurring in our single-molecule assay (Extended Data Fig. 4a). A DNA molecule with a persistence length of 50 nm and a contour length of 16.32 µm was assumed to be tethered to the surface by its two ends. The distance between the ends (D) was assumed to be randomly distributed with mean value 8.8 µm and standard deviation 0.5 µm, both of which were inferred from experimental data. Cohesin was assumed to be positioned at the center of the DNA during force-application experiments. The time step in simulations was varied between 0.1 and 0.01 s. At each time step, DNA was moved with respect to cohesin in the direction perpendicular to the DNA attachment points, with the experimental force loading rate of 0.16 µm s−1 (Extended Data Fig. 4a–c).

The resulting DNA length at the time point was calculated from the updated position of cohesin:

$${L}_{{{\mathrm{DNA}}}}=2\sqrt{{\left(\frac{D}{2}\right)}^{2}+{x}^{2}},$$
(1)

where x is the distance between the cohesin anchor point and the segment connecting the DNA tethering points (Extended Data Fig. 4a).

Next, we calculated DNA tension and the force acting on cohesin at each time step:

$${F}_{{{\mathrm{DNA}}}}=\frac{{k}_{\mathrm{B}}T}{P}\left(\frac{1}{4{\left(1-\frac{{L}_{{{\mathrm{DNA}}}}}{{L}_{0}}\right)}^{2}}-\frac{1}{4}+\frac{{L}_{{{\mathrm{DNA}}}}}{{L}_{0}}\right),$$
(2)
$${F}_{{{\mathrm{cohesin}}}}=2\times {F}_{{{\mathrm{DNA}}}}\sin \alpha,$$
(3)

where P is DNA persistence length, LDNA is current DNA length, L0 is DNA contour length, and \(\tan \alpha =2\frac{x}{D}\).

The rate of cohesin disengagement at the current tension was determined as

$${k}_{{{\mathrm{diss}}}}\left(F\,\right)={k}_{0}\exp \left(\delta \times {F}_{{{\mathrm{cohesin}}}}/{k}_{\mathrm{B}}T\right).$$
(4)

Finally, the detachment time was calculated as

$${t}_{{{\mathrm{detach}}}}=-\frac{1}{{k}_{{{\mathrm{diss}}}}(F\,)}\mathrm{ln}\left(1-r\right),$$
(5)

where r is a random number evenly distributed in the interval [0;1]. Cohesin was considered disengaged at the current timestep if \({t}_{{{\mathrm{detach}}}}\le {{\mathrm{d}}t}\).

Once the rupture occurred, the current value of the rupture force was recorded, and the simulation repeated over ~104 times to collect distribution of rupture forces. Average rupture forces (µ) and standard deviations (σ) corresponding to different sets of parameters were calculated for the force distributions for different k0 and δ values (Extended Data Fig. 4d–f).

To simulate rupture events when the force was applied to the second DNA we used a similar approach, but with some modifications. In this case, the relationship between DNA lengths and forces is given by the following system of equations:

$$\left\{\begin{array}{c}\begin{array}{c}{F}_{1}=\frac{{k}_{\mathrm{B}}T}{P}\left(\frac{1}{4{\left(1-\frac{{L}_{1}}{{L}_{0}}\right)}^{2}}-\frac{1}{4}+\frac{{L}_{1}}{{L}_{0}}\right)\\ {F}_{2}=\frac{{k}_{\mathrm{B}}T}{P}\left(\frac{1}{4{\left(1-\frac{{L}_{2}}{{L}_{0}}\right)}^{2}}-\frac{1}{4}+\frac{{L}_{2}}{{L}_{0}}\right)\end{array}\\ \begin{array}{c}{F}_{1}=2{F}_{2}\sin \alpha \\ \begin{array}{c}\frac{{L}_{2}}{2}\sin \alpha +{L}_{1}=x\\ {tg}\alpha =2\frac{x-{L}_{1}}{D}\end{array}\end{array}\end{array},\right.$$
(6)

where F1, L1, F2 and L2 are the forces and lengths of the first and the second DNA, respectively; \(\frac{\pi }{2}-\alpha\) is the angle between the two DNAs at the point of contact; and x is the distance between the cohesin point and the segment between the two tethering points of the first DNA (Extended Data Fig. 7d).

This nonlinear system of equations was solved numerically at each time step to calculate the tension applied to DNA, the DNA length for both molecules as well as the force applied to cohesin. Next, the disengagement rate and time of rupture were calculated as in equations (4) and (5).

For generating the parameters in Fig. 4e, k0 was 2 × 10−3 as in Fig. 2a. To simulate possible different orientations of cohesin, δ was chosen randomly to be either 1.23 nm or 1.61 nm with equal probability for each simulated FD curve.

Model fitting and parameter determination

Since our model is nondifferentiable, we used two-dimensional golden search algorithm to find sets of k0 and δ parameters that minimized the least square distance between the means and variances of the experimental and simulated distributions of rupture forces. This algorithm converges to a single optimal solution because the dependence of both mean and variance of the rupture forces on k0 and δ is monotonic (Extended Data Fig. 4e).

To determine the 90% confidence interval for the parameters, we calculated the maximum likelihood distribution for k0 and δ directly by sampling it using a Monte-Carlo approach. The individual probabilities were defined as

$${P}_{i}=\exp \left\{-\frac{{\left(\,{\mu }_{i}-\text{mean}\left(F\,\right)\right)}^{2}}{{\mathrm{var}}_{{{\mathrm{mean}}}}}-\frac{{\left({\sigma }_{i}-\text{std}(F\,)\right)}^{2}}{{\mathrm{var}}_{{{\mathrm{std}}}}}\right\}.$$
(7)

Here μ and σ are the mean and standard deviations of the simulated distributions, F is the set of experimentally measured forces, and \({\mathrm{var}}_{{{\mathrm{mean}}}}\) and \({\mathrm{var}}_{{{\mathrm{std}}}}\) are the experimentally determined variances in the mean and standard deviations, respectively.

To determine the confidence interval for the parameter k0, we sampled the partial distribution by calculating the probabilities directly using equation (7) for randomly samples values of k0 while keeping the value δ fixed at its optimum determined by the golden-search algorithm. 90% confidence interval was then determined directly from the distribution as a region under the curve covering 90% of the total area. For determining the confidence interval for δ we used the same algorithm except δ was now sampled and k0 was kept constant at its optimal value. Unless stated otherwise, values in the main text in brackets following the value of a parameter indicate 90% confidence interval determined by this method.

For Extended Data Fig. 4h the standard deviation in determining k0 and δ was calculated directly as the standard deviation between parameters determined for the different subsamples.

Confidence intervals determined by the Monte-Carlo sampling were consistent with the standard deviations determined for the subsampled data on Extended Data Fig. 4h;

P values in Extended Data Fig. 4i were calculated using the Kolmogorov–Smirnov test.

Determination of the average number of cohesin molecules per bead

To infer the distribution of the number of cohesins per bead from the experimentally obtained distributions for the number of cohesins and beads per DNA, we used the following algorithm. First, we introduced parameter λ—efficiency of the bead–cohesin interaction. Our sampling procedure typically consisted of 10,000 steps; and at each step, we picked a DNA with cohesin distributed according to a sample randomly chosen from the experimentally measured distribution (Extended Data Fig. 2a) and assumed that each cohesin can bind a bead with probability λ. In case of successful binding, the bead had one cohesin attached to it. If there were other cohesins on this DNA, we assumed that the same bead binds them with 100% efficiency if they were spatially closer than 1 µm to the original cohesin, which resulted in the increased the number of cohesins attached to this bead. If other cohesins on DNA were further away than 1 µm from the initial cohesin, we assumed that they could also bind to the same bead, but with probability λ, which again would increase the number of cohesins bound to the bead.

This sampling yielded distributions of the number of cohesins per bead and the number of beads per DNA as a function of λ. We then varied λ to match the number of beads per DNA to the experimentally observed value (Extended Data Fig. 2a), which yielded the value of λ corresponding to our experimental conditions. The number of cohesins per bead for this λ gave us the corresponding distribution of the number of cohesins per bead in our experiments (Extended Data Fig. 2, right).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.