# Polymer Models of Chromatin Imaging Data in Single Cells
## 3 Results
### The Models Recapitulate Ensemble-Averaged Microscopy Data  
In a first validation of the models, we computed their median distance matrix that we compared against the corresponding map from multiplexed FISH data [15] (Figure 1). The median distance map is the ensemble median of the single-molecule distance maps, which, by definition, are symmetric square matrices reporting the Euclidean distances between all pairs of polymer sites. Those matrices are visually represented as a 2D heatmap with a color bar scheme to highlight contacting regions that are closer in 3D space (e.g., TADs or loops, colored in red in Figure 1) or those that have no significant interactions (blue regions in Figure 1). To efficiently compute the distance matrices, we used built-in functions within the Python SciPy package [28]. To reduce the noise within imaging data, we applied a Gaussian filter on single-cell distance maps (standard deviation of the Gaussian kernel = 1) and excluded single-cell conformations whose 3D coordinates have >80% missing values. To quantitatively estimate the similarity between microscopy and model distance matrices, we used the genomic distance-corrected Pearson correlation coefficient, r\({}^{\prime}\), which corrects the usual Pearson coefficient for genomic distance effects [12]. In brief, r\({}^{\prime}\) is the Pearson coefficient computed on distance matrices where each entry is subtracted by the mean value of its diagonal. We found that the different models recapitulate the complex TAD patterns observed in microscopy data, as well as specific pointwise interactions, as highlighted by the high r\({}^{\prime}\) correlation values: r\({}^{\prime}\) = 0.49, r\({}^{\prime}\) = 0.77 and r\({}^{\prime}\) = 0.70, respectively, for LE, SBS and LE+SBS [28]. To substantiate the statistical significance of the r\({}^{\prime}\) correlations of the models, we considered as null control model a self-avoiding chain with the same number of beads as the imaged conformations and found that it returns a significantly lower correlation value (r\({}^{\prime}\) = 0.11, which is, respectively, four and seven times lower than the values found for LE and SBS/LE+SBS). That indicates that both active processes, like loop-extrusion,  
Figure 1: The median spatial distance matrix of the IMR90 locus (Chr21: 28-30 Mb) from multiplexed FISH microscopy [15] is compared against the corresponding in silico matrices of the considered polymer models [28]. Both loop-extrusion (LE) and phase-separation based models (SBS and LE + SBS) well recapitulate the complex experimental pattern of contacts. Adapted from [28]. and passive mechanisms, e.g., thermodynamic polymer phase-separation, are consistent with the average structure of the considered IMR90 genomic region measured by bulk imaging data. Next, we checked whether our models also explain local properties of chromatin structure. To this aim, we computed the boundary probability genomic function, i.e., the probability for each genomic position across the studied IMR90 locus to appear as boundary of a single-cell TAD domain [15] (Figure 2). The boundary function has local peaks corresponding to the main TAD boundaries visible in the median distance map of the IMR90 locus and, interestingly, this function is non-zero across the entire imaged region, indicating a substantial cell-to-cell variability in the genomic position of TAD boundaries. We found that the different models return boundary probabilities with profiles consistent with imaging data, as quantified by the high Pearson correlations between models and experiment: r = 0.83, r = 0.63 and r = 0.65, respectively, for LE, SBS and LE + SBS [28]. Taken together, these analyses show that both loop-extrusion and phase-separation processes can quantitatively explain chromatin structure at the cell population-averaged level. In particular, loop-extrusion (LE) is the best to capture the boundary probability function, while phase-separation based models (SBS and LE + SBS) return overall higher correlation values with ensemble-averaged distance data.