The theory of LEAR

There are now several technologies that break the Abbe resolution limit of microscopy [1]. All of these have different advantages and disadvantages in regard to the spatiotemporal-resolution available [27,61]. Naturally, these have led to additional post-processing methodologies to remove artifacts and to further improve resolution (a few examples: [15,28,39,45,51]). Here we describe a post-processing methodology to improve the resolution of chromatin tracing capitalizing on the fact that there are many distinguishable loci: Loci Enabled Advanced Resolution (LEAR).

For one of the imaging dimensions with L total loci, the observed locations of locus () for the N traces is . The observed locations result from the localization error, , distorting the true positions, , with  +  . Each element within follows a random variable whose specifics result from the particular experiment.

The overall goal is to obtain a better approximation of , given the observed localizations of all loci: . With the experimental data, we can empirically quantify the variance of the differences between a pair of loci ( and ); with the assumption that the localization errors are independent, the variance is the following:

(1)

We now have an equation that allows us to approach . Put another way, if the localization error of locus were minimized () the variance above would approach − ; we refer to this quantity as the ‘goal variance’ which we can calculate given an approximation of . We describe a few approaches one can use to estimate in specific cases in Sect 6 in S1 Text, but it is important to state that an estimation may need to be quantified experimentally.

The central idea is that we can minimize the localization error of locus by adjusting to produce a variance equal to the goal variance; again, where the localization error is minimized. More specifically, we can adjust the positions with the addition of generating the adjusted locations: , so that . The thought being that will better approximate as approaches (Fig 1A).

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Fig 1. Illustration of ideas behind LEAR:

A) There are two loci, each have true locations (illustrated as solid circles) and localization errors (illustrated with Gaussian like distributions and standard deviations). The true differences between the true locations is T¹–T². B) Calculating the variance between the localizations with the localization error (aka. the observed raw localizations) results in a variance made up of 3 terms owing to error propagation: one from the variance of the differences of the true locations and the other two equal to the actual variance of the localization errors (see equation 1). With an estimate of , we can determine the goal variance; the variance if . The main idea is that we can improve the resolution by adjusting the positions of loci¹ until the differences calculated with the adjusted positions (A¹) produce a variance equal to the goal variance. C) With a single pair of loci, there is only one goal variance, hence there are not many constraints. However, if there are many different loci, one can quantify a goal variance for each pair, drastically increasing the number of constraints to guide the method.

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To evaluate an ‘guess,’ we can calculate the probability of observing a specific goal variance; this is done with the central limit theorem, resulting in a normal distribution with a mean ( and standard deviation () for that pair of loci (Sect 2 in S1 Text).

(2)

That is, we seek a conformation of that maximizes the above probability density.

Problematically, with only one pair of loci, and hence one goal variance, many different guesses will give the same goal variance. Importantly, however, with an increasing number of imaged loci, the number of goal variances increases, providing more constraints on (Fig 1B). To incorporate these constraints, we can seek a conformation of that maximizes the following expression, corresponding to a likelihood maximization problem:

(3)

More clearly, we make the assumption that the likelihood of having a “good” configuration is proportional to the product over all available loci L. The fact that the above equation is specific to a single locus allows one to parallelize the approach. To maximize the above equation, we utilized a simple stochastic descent algorithm (Sect 1 in S1 Text).

Case 1: Demonstrating that LEAR improves localization error experimentally

We first wanted to test our methodology experimentally. To do this we utilized three previously published datasets where specific loci were repeatably imaged within the same cell, allowing us to quantify the localization error by quantifying the distances between the initially imaged and the re-imaged localizations (Sect 5 in S1 Text). Two of the datasets were from experiments within Drosophila from the work of Mateo et al. [52]; the first had a genomic resolution of 2kb and the second had a genomic resolution of 10kb. The third dataset was sequential chromatin tracing data within IMR90 human cells from the work of Su et al. [63] with a genomic resolution of 50kb. The number of imaged loci in each dataset varied as well as the number of loci that were repeatably imaged, with the 50kb resolution data having the most repeatably imaged loci . These loci were repeatedly imaged in each cell specifically to quantify the localization error. The chromatin organizations of these datasets greatly varied, with the 2kb resolution data showing the smallest distances overall (Fig 2Ai,Bi and Ci), allowing us to demonstrate our methodology with diverse situations.

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Fig 2. Validating the methodology experimentally:

A, B and C) Are the analyses with the listed chromatin tracing dataset. i) The mean distances between the loci for the listed dataset. ii) The quantified localization error using the repeatably imaged loci from the dataset; the localization error is shown for each dimension and for whether it was from the initially imaged or the repeatably imaged. Error bars are equal to the standard deviation. iii) The quantified relative localization errors after the application of the methodology for the individual loci and along each dimension. Note, this was only done for the initially imaged loci.

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Here we pause to state that we filtered out some localizations that were reasoned to be “overwhelmingly” off target (discussed extensively within the Sect 8 in S1 Text, Figs A–D in S1 Text). Previous work has also filtered out localizations depending upon how the observed localizations deviated from a specific model [38,42]. However, little evidence was given to justify the discarding of certain localizations beyond a better agreement with chromosome capture contact frequencies [42]. Importantly, we provide multiple lines of evidence justifying ‘filtering’ within Sect 8 in S1 Text. A more complicated filtering approach rooted in polymer theory could be further developed taking into consideration the nature of off target localizations quantified here [42]. We must also state that due to this problem, the error improvement from LEAR is likely underestimated (Sect 8.2 in S1 Text). Throughout we only tested the application of LEAR on the initially imaged loci, for the repeat imaged loci showed a higher proportion of off target localizations (Figs C and D in S1 Text). Nonetheless, overall, we saw a general improvement with the application of LEAR no matter the dataset (Fig 2Aiii, Biii, and Ciii) as shown below.

We first quantified the amount of localization error for each of the datasets along each dimension; again, defined as the standard deviation of the localizations about their true localizations (). Using a simple process, we were able to determine the mean localization error for the initially imaged loci and the repeat imaged loci for the raw data (Sect 5 in S1 Text). The motivation for this came from the work of Su et al., where they noted a difference in the brightness of the repeat imaged localizations [63]—and hence, re-imaging the same locus in the same cell could be fundamentally different in terms of localization error. Our analysis supports this reasoning and quantifies the difference: we found that the repeat imaged localizations (generally) had a worse localization error (Fig 2Aii, Bii, and Cii). Of note, there is a drastic difference in the localization error for the Z dimension within the dataset of Su et al. On its own, this analysis is important regarding the actual resolution of an experiment—the localization error is apparently not accurately quantified using only the distances between initially imaged and repeat imaged loci. Simply put, for these chromatin tracing protocols, repeatably imaging the same locus can result in a different localization error than when the locus was initially imaged.

We then quantified the relative error of the initially imaged localizations after the application of our methodology. To do this we used the same logic as above to quantify the localization error but with the adjusted localizations of LEAR, and then quantified the relative localization error; STD with LEAR divided by the STD with raw localizations. We show the relative localization error for each loci for each dimension individually in Fig 2Aiii, Biii, and Ciii. Generally, we found that the larger the error and the smaller the distances between the loci, the more effective the error improvement from LEAR (discussed later). For the 2kb genomic resolution data, the Z dimension showed the greatest improvement (), while the other dimensions showed an improvement of (Fig 2Aiii). The same trend was seen with the 10kb genomic resolution data, but with lower improvements as calculated with only two loci (Fig 2Biii). The dataset of Su et al. showed consistent small improvements throughout, again in line with the fact that a lower genomic resolution (larger distances between loci) decreases the effectiveness of the method. Altogether, these results show that LEAR works on experimental data improving the localization error by 10-20% for high genomic resolution chromatin tracing data.

Case 2: Applying LEAR on the Sox2 locus shows drastic improvements to contact-frequencies

Although we demonstrated LEAR using experimental data, the fact that the true underlying locations were unknown prevented us from determining how LEAR performed in relation to commonly used metrics like the contact-frequency. Additionally, it was likely that off-target localizations resulted in an underestimation of LEAR’s error improvement with the experimental data, as described above. Therefore, we tested the methodology through simulations using realistic multi-loci microscopy data, where the ground truth was known.

To apply LEAR on realistic simulation data we utilized the empirical chromatin tracing data of Huang et al. [38], where they quantified the 3d positions of loci distributed around the Sox2 gene at a genomic resolution of 5kb within mouse embryonic stem cells (mESC). The contact frequency quantified from traces with a distance threshold of 150 nm is shown in Fig 3A; loci lacking a sufficient number of datapoints were excluded. It is important to state that for chromatin tracing contact frequency refers to the proportion of traces where two or more genomic loci are spatially within a defined distance threshold (e.g., 150 nm or 200 nm), which may be different from the contact frequencies determined with other experimental methodologies. The idea was that the empirical data would be treated as the ground truth and different amounts of localization error would be computationally introduced, the performance of LEAR could then be evaluated with the knowledge of the true underlying locations with no off target effects. Because localization errors can lead to artificial increases in mean distances [23], we first applied LEAR utilizing estimates of the localization error within the Huang et al. dataset (Fig F in S1 Text) to generate a more “realistic” dataset, especially in terms of ensemble measurements (Fig 3B, Sect 7 in S1 Text, validated throughout). This dataset was used as the ground truth for the following simulation experiments within the main text (Fig 3). We also performed the same analysis using the raw localizations as the ground truth (Fig 3A) and observed similar results but with slightly less error improvement due to the larger distances between the loci (Fig G in S1 Text, discussed later).

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Fig 3. Testing the Method with Simulation:

A) The contact frequencies using a distance threshold of 150 nm from the chromatin tracing data of Huang et al. [38]. B) The contact frequencies after the application of LEAR. This data was used as the ground truth. C) The contact frequencies after incorporating normally distributed localization error with standard deviations equal to 50nm or 100nm. The arrows direct one to the contact frequencies that result from LEAR. D) The observed contact frequency vs. the true contact frequencies for the results shown in C—the blue line is for reference. Note, the colors for the conditions throughout are shown at the bottom of the figure. E) The average relative localization error after applying the methodology. F) A linear interpolation of the data in B, approximating a dataset with a genomic resolution of 1kb. G) The relative error for the interpolated dataset with the two localization errors. H) Similar to D but for the interpolated 1kb genomic resolution dataset. I) The observed 3-way contact frequency vs the true 3-way contact frequency for the interpolated dataset (how often three different loci are within 150 nm of each other).

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We quantified the efficacy of LEAR by introducing normally distributed localization error and then testing how well we recovered the ‘ground truth’ post LEAR processing. We added the localization error using normally distributed random numbers with either or (Fig 3C, STD is the standard deviation). As expected, the contact frequencies were greatly perturbed with the 100 nm error showing the largest reductions. The application of the methodology resulted in contact frequency maps that nicely approximated the ground truth (Fig 3C, below arrows)—illustrating a clear benefit of the LEAR methodology. To show these results more clearly, we plotted the observed contact frequency vs. the true contact frequency for the different conditions (Fig 3D), where there was a clear improvement for both the 50 nm and 100 nm error, especially for the larger contact frequencies. To measure the improvement at the single molecule level we quantified the average error using the distances between each localization and their true localizations, and then calculated the relative error of LEAR; similar to before, the relative error is the average error with LEAR divided by the average error before the application of the method. This analysis showed that the 100 nm localization error had about 25% less error while the 50 nm error showed a lesser improvement of 12% (Fig 3E). We also show a plot of C vs. to further illustrate that LEAR works at the single molecule level (Fig H in S1 Text). The analysis suggests that the degree of improvement is sensitive to certain aspects of the data. Nevertheless, qualitatively one sees the off diagonal peak in Sox2 contact with the super enhancer with greater contrast after LEAR processing.

We then sought to probe what aspects of the chromatin tracing data influences performance. Most importantly, we hypothesized that the effectiveness of the methodology is proportional to the size of the error ()) and inversely proportional to the variance of the differences ( − ) (see Sect 3 in S1 Text); generally, the larger the distances between neighboring loci the worse the performance. To test this hypothesis, we compared the above results to computational experiments that utilized the actual raw localizations as the ground truth instead (Fig G in S1 Text)—the raw localizations have larger mean distances, and naturally, larger variances from the localization error. The improvement in the quantification of contact-frequency remained noticeable showing significant improvements (Fig Gd in S1 Text). However, and as expected, we observed less relative error improvement with for the 100 nm localization error and for the 50 nm (Fig Ge in S1 Text). For completeness, we then applied our methodology on chromatin tracing data collected at a genomic resolution of 30kb [10] which showed consistent results with this line of reasoning (Fig I in S1 Text). Note, we also investigated aspects of the methodology and chromatin tracing data that could affect the performance of the methodology Fig Ib–e in S1 Text, as discussed extensively discussed within the Sect 4 in S1 Text. Together, these results clearly explain the differences in performance for the diverse experimental datasets (Fig 2Aiii, Biii, and Ciii), and for datasets similar to that of the Sox2 data here, improvements in both contact-frequency calculations and resolution can be substantial.

We next sought to understand how the methodology performs in certain limiting conditions expected to deliver large improvements in resolution post-LEAR. Such conditions would be chromosomes where the number of imaged loci is large and the distances between neighboring loci is small. To computationally generate such a dataset, we performed a linear interpolation of the ground truth data (Fig 3B), generating interpolated chromatin tracing data at a genomic resolution of 1 kb (Fig 3F). As before, we introduced the two localization errors, applied the methodology, and found large improvements in the relative error for both the 50 nm and 100 nm (Fig 3G), with 50% improvement for both cases. And, in terms of 2-way contact-frequency, we found that the methodology was able to obtain the correct result no matter the true underlying contact frequency (Fig 3H). Lastly, to further highlight the effectiveness of LEAR, we decided to investigate 3-way contact frequencies (Fig 3I); that is, how often three different loci are within a certain distance of each other. The detrimental effect of the 100 nm localization error is shown in green in Fig 3I. Nicely, the methodology allowed us to obtain dramatically improved 3-way contact frequencies—clearly showing the benefit of LEAR.

In summary, this analysis demonstrates that LEAR greatly reduces the error in quantifying contact-frequencies and as the genomic resolution of the chromatin tracing data increases so does the benefit of applying LEAR. The improvement in quantifying contact-frequencies was especially apparent for multi-way contacts. Our investigations with the Sox2 data suggest that applying LEAR to similar datasets improves localization error by approximately 10-20%. And lastly, with future technological advances in chromatin tracing, where more loci are analyzed at even finer resolutions, our idealized system experiments indicate that LEAR could potentially reduce localization errors by up to 50%.

Case 3: Quantifying the impact of LEAR with polymer simulations

Next, we sought to apply and test LEAR utilizing polymer simulation data as the ground truth. The application of LEAR in Case 1 and Case2 was motivated from experimental data [38,52,63], indicating that LEAR will prove beneficial in a real life scenario. However, because of the potential experimental errors the use of these datasets as a ground truth for the chromatin structure may not represent the true underlying structure.

We decided to test LEAR utilizing the recent polymer simulation data of Jusuf et al. [43]. Though the 1 kb genomic resolution polymer simulation was originally performed at a chromosomal scale with several CTCF sites and loop extrusion, we decided to focus on the first 100 loci of the dataset with 500 individual traces within the main text (Fig 4A). We also analyzed a region with CTCF sites with similar results in S1 Text (Fig J in S1 Text). Because of the equal large improvements with both the 50 nm and 100 nm localization errors obtained with the interpolated 1 kb dataset (Fig 3G), we decided to further test the limits of LEAR by introducing normally distributed localization error with either or . We hypothesized that the improvements seen with LEAR would be less for the smaller error, as discussed above.

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Fig 4. Validating the methodology with polymer simulations:

A) The ground truth contact frequency map of the first 100 loci of the 1 kb polymer simulation data [43]. B) The observed 2-way contact frequencies vs. the true underlying contact frequencies for 25 nm or 50 nm localization error both with and without LEAR—again blue line is for reference. C) The same as in B but for all possible 3-way contact frequencies. D) The relative localization error for each individual locus after the application of LEAR for the 25 nm and 50 nm errors. E) The relative localization error for each individual trace of the polymer simulation data. F) Example of individual trace with underlying ground truth in purple, LEAR in black and with localization error in green.

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We found that LEAR performed similarly when compared to the interpolated 1kb dataset (Fig 3G). As before, LEAR was able to correctly determine 2-way and 3-way contact frequencies with both types of error (Fig 4B and 4C). Importantly, we observed that the relative localization error per locus still showed significant improvements for the 25 nm localization error, with about a 35% improvement (Fig 4D). The relative improvement for the 50 nm localization error was around 50%, as in the interpolated 1 kb data in Case 2. To make it more clear that LEAR is able to improve the localization error at the individual trace level, we quantified the relative localization error per trace (, for each trace i). As expected we saw similar improvements for the two localization errors.

Overall, Case 3 demonstrates that LEAR performs similarly on polymer simulation datasets. Of note, we also found that LEAR can lead to error improvements 35% for even small localization errors of , given the fine genomic resolution of the simulation. Together these further support the potential use of LEAR in diverse experimental situations.

Transcriptional bursting of Sox2 is related to E-P proximity in individual cells

The Sox2 gene and its super-enhancer (SE) represents one of the most investigated enhancer-promoter (E-P) pairs. However, the role of 3D chromatin organization in regulating Sox2 transcription remains unclear [3,16,19,24,38,57,65]. Although ensemble data shows a correlation between the contact frequency of the super-enhancer and Sox2 transcription [19,24,38], no clear relationship has been established between individual transcriptional bursts and the distance between the super-enhancer and promoter in single cells [3,38,57], raising questions about the relevance of E-P spatial proximity and transcription regulation. Specifically, live-cell microscopy has shown no relation between E-P distance and nascent transcription [3,57]. However, chromatin tracing has resulted in conflicting results [38]; a weak but significant change in the median E-P distance with nascent transcription was observed (a change of 18 nm), but there was no clear difference in the proportion of alleles showing active transcription as a function of E-P distance threshold. We therefore sought to use the improved localization error of LEAR to investigate the relation between Sox2 transcription and local chromatin organization with the chromatin tracing data of Huang et al. [38]—the dataset from Fig 3.

We first wanted to investigate the contact frequencies of the Sox2 locus with LEAR with a focus on the effect of a strong boundary element between the SE and the promoter. Previously, Huang et al. reported that insertion of multiple CTCF sites led to a reduction in the SE-P contact frequency and a decrease in the fraction of transcriptionally active Sox2 alleles [38]. Using the raw chromatin tracing data we show the contact frequencies (thresh = 200 nm) for the no boundary Sox2 allele, with the boundary, and then the difference between the two conditions to emphasize the change (Fig 5B, red indicates a loss of contact frequency, top row). As in Huang et al., we observed a loss in contact frequency between the promoter and the SE region (Fig 5B, top right), with an absolute change in contact frequency of .07. After applying LEAR, the contact frequencies dramatically increased for both conditions, and the absolute difference in chromatin structure was much greater (Fig 5B, bottom row). This indicates the effect of the boundary insertion on chromatin organization was more substantial than previously measured, suggesting that the observed transcriptional changes might be linked to broader conformational shifts that were previously masked by localization errors.

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Fig 5. Link between Sox2 transcription and local chromatin organization:

A) An illustration showing important factors around the Sox2 gene, also shows the indices used for the 5kb chromatin tracing experiments [38]. B) The contact frequencies for with and without the boundary inserted, and also with and without the methodology being applied (see labels). The right column shows the difference in contact frequency for the specific row. C) Contact frequency maps for the no boundary system with traces where an initial transcriptional burst was seen (left column) and from traces where no transcription was observed (center). The difference of the contact frequency maps with and without initial transcription is shown on the far right. The contact frequency maps using the raw data are shown on the top row (and labelled), the maps generated utilizing the method are on the bottom row. D) The same as C but for the boundary experimental dataset.

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We then aimed to assess how chromatin structure changes with nascent Sox2 transcription at the single-cell level. To do this, we compared contact frequency maps when nascent RNA was detected versus when it was not. In Huang et al.’s study, a fluorescent protein (FP) was inserted directly after Sox2, allowing the RNA-FISH to differentiate between nascent RNA containing both FP and Sox2 signals (‘later burst detection’) and only Sox2 signal (‘initial burst detection’); the lack of FP signal indicates that RNA polymerase did not have enough elongation time to reach the FP sequence (illustrated in Fig Ka in S1 Text). For our analysis, we decided to only use traces with initial burst detection for the traces with active transcription, chromatin diffusion during transcription elongation distorts the E-P distances associated with transcription initiation [14]. Note, we also performed the same analysis without excluding traces where we observed a similar but lesser correlation between SE-P proximity and nascent transcription (Fig K in S1 Text). For the no boundary condition contact frequency maps and the difference between them, we observed some changes focused mostly around the promoter region, with one of the more notable changes being located at one of the recently characterized enhancer like element [16] (Fig 5C, top row). After applying LEAR, the contact frequencies increased, and the changes were slightly more noticeable (Fig 5C, bottom row). Though this result may suggest some correlation of chromatin structure and transcription regulation, the contact frequencies of the P-SE showed no real variation with nascent transcription. For the boundary condition (Fig 5D, bottom row), we observed a change centered around the inserted boundary element with transcription (Fig 5D, top row), which is an interesting result when considering the recent models of CTCF hubs enabling E-P regulation within different chromatin domains [35,41]. Notably, with LEAR, we found a noticeable change centered around the SE-P contact frequencies, indicating that with the insertion of the boundary element, the spatial proximity of the SE may be important for the initiation of Sox2 transcription. Importantly, without LEAR, this relation would have likely been missed.

Overall, we have demonstrated the usefulness of the methodology investigating the link between chromatin organization and Sox2 transcription regulation. Not only did we show that changes in chromosome contacts were greater than previously quantified, we also found a link between chromatin organization and individual Sox2 transcriptional bursts previously obscured by error.

Cooperativity in multi-way contacts is not explained solely by loop extrusion

Emerging evidence suggests that multiple enhancers work together as a system, necessitating accurate and sensitive methods for detecting multiway contacts for understanding enhancer biology [12,66,68]. Considering that multi-way contacts are difficult to detect by chromosome conformation capture and were especially sensitive to localization error (Fig 3I), we sought to utilize the LEAR methodology to probe higher order contacts vis-a-vis loop extrusion. Previously, the work of Bintu et al., obtained rich chromatin tracing datasets in HCT116 cells with and without loop extrusion [10], which was eliminated with the degradation of RAD21 using the auxin degron system [59]. For the Bintu et al. dataset, we found a sufficient number of traces with a detection efficiency above 95% (7591 traces without auxin and 2726 traces with auxin), allowing for a deep interrogation of higher order contacts with the application of our methodology.

We conducted an in-depth analysis of loop extrusion on two- and three-way contacts. To start, we quantified the effect of LEAR on two way contacts with and without loop extrusion (Sect 9 in S1 Text). The 2-way contacts with a distance threshold of 200 nm are shown in Fig 6A, without (top row) and with the application of LEAR. To more clearly show the effect of loop extrusion on 2-way contact frequencies, we plotted the frequency without auxin vs. with auxin (Fig 6B). Though LEAR generally showed greater differences in 2-way contact frequencies due to loop extrusion, the results were mainly the same. We found that larger fold changes (not due to error) were around a contact frequency for loci pairs in the no loop extrusion condition, in some cases approaching a ten-fold increase with loop extrusion (See zoom in for Fig 6B).

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Fig 6. Higher order contacts with and without loop extrusion:

A) The 2-way contact-frequencies for the chromatin tracing data from Bintu et al. The addition of auxin leads to the removal of active cohesion eliminating loop extrusion. The same chromosomal region is shown with and without the methodology, and with and without the addition of auxin. B) The 2-way contact-frequency without auxin vs. that with auxin, with and without the application of the methodology, and below is a zoom in of the top plot. C) The 3-way contacts between triplets compared to the expected 3-way contact frequency if the 2-way contact frequencies contributed independently, with and without auxin. D) The same as C but with the methodology. E) Same as B but with 3-way contact frequencies. F) The difference between the method and the no method 3-way contacts for the no auxin condition vs the 3-way contact with auxin. G) The same as E but with 4-way contact frequencies. H) Same as F but with 4-way contact frequencies. I) An illustration pointing out that for certain 3-way contacts the loci are never in contact without loop extrusion, but with loop extrusion a notable number of 3-way contacts are seen.

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We next sought to investigate higher order multi-way contacts with and without loop extrusion. To do this, we first quantified the proportion of traces where all three loci were within 200 nm and compared the expected 3-way contact frequency; calculated assuming the 2-way contact frequencies contributed independently (Sect 10 in S1 Text). We found a strong enrichment above the expected independent 3-way contact frequency indicative of cooperativity (Fig 6C and 6D). While cooperativity in multi-way contacts has been seen in cells lacking loop extrusion, a careful quantification of the relation between loop extrusion and cooperativity was lacking [10]. Astonishingly, we found that the auxin and no auxin condition followed the same curve both without (Fig 6C) and with LEAR (Fig 6D). Probing further, we performed the same analysis but for 4-way contact frequencies and obtained a similar result (Fig L in S1 Text). These outcomes are significant because it indicates that the cooperative behavior primarily hinges on the frequencies of two-way contact frequencies. Put another way, the underlying mechanisms influencing individual two-way interactions do not alter the degree of cooperativity, suggesting loop extrusion is not the mechanism behind the cooperative behavior and that the true underlying mechanism is mainly dependent upon spatial proximity.

We next attempted to more carefully quantify how the higher order multi-way contacts vary with loop extrusion but from a different perspective. To do this, we plotted the multi-way contact frequency with loop extrusion vs. without loop extrusion, for both 3-way (Fig 6E) and 4-way (Fig 6G) contact frequencies. We observed a large range of values with LEAR, approaching a max of .5 and .3 for the 3-way and 4-way contacts respectively, suggesting that even though the distances between pairs of loci show a high degree of variability [29], a large proportion of traces exhibit higher order multi-way contacts. To highlight the difference in multi-way contact frequency with LEAR, we quantified the difference in multi-way contact frequency due to LEAR and plotted it vs the multi-way contact frequencies with auxin (Fig 6F and 6H). We found that the change due to LEAR was drastic, again showing just how sensitive these multi-way contacts are to localization error and demonstrating the need of the method.

Lastly, we focused our attention on frequencies that showed the most dramatic changes with loop extrusion (Fig 6E and 6G, inset). Interestingly, we observed that initially zero multi-way contact frequencies without loop extrusion could increase to .04 with loop extrusion (Fig 6E and 6G, inset). This result implies > 40-fold changes, considering we only quantified multi-way contact frequencies if there were at least 1000 distances between all pairs of loci (Fig 6E and 6G, zoom in). In other words, the analysis suggests that without loop extrusion certain multi-way contacts ‘never’ happen, but with loop-extrusion the frequency can dramatically increase (Fig 6I).

In summary, we have performed an in-depth analysis of cooperative multi-way contact frequencies with LEAR. We find that multi-way contacts were greatly perturbed by localization error and become more prevalent with the use of LEAR. We also found that loop-extrusion, while clearly altering 2 way contact frequencies, was not the mechanism leading to high-order cooperativity. Rather, we found that the changes in 2-way contact frequencies brought about by loop-extrusion could lead to drastic many-fold changes in multi-way contact frequencies. We propose this result for multi-way contact frequencies arises from the cooperativity mechanism being primarily a function of spatial proximity.