Toy model, with TUs coloured randomly (the random string).

(A) Overview. (i) Yellow, red, and green TFs (25 of each colour) bind strongly (when in an on state) to 100 TUs beads of the same colour in a string of 3000 beads (representing 3 Mb), and weakly to blue beads. TU beads are positioned regularly and coloured randomly, as indicated in one region of the string. TFs switch between off and on states at rates and αon = αoff/4 (τB Brownian time, which one can map to 0.6 6 10−3 s, see SI). (ii) The sequence of bars reflects the random sequence of yellow, red, and green TUs (blue beads not shown). (B) Snapshot of a typical conformation obtained after a simulation (TFs not shown). Inset: enlargement of boxed area. TU beads of the same colour tend to cluster and organize blue beads into loops. (C) Bridging-induced phase separation drives clustering and looping. Local concentrations of red, yellow, and green TUs and TFs might appear early during the simulation (blue beads not shown). Red TF 1 – which is multivalent – has bound to two red TUs and so forms a molecular bridge that stabilizes a loop; when it dissociates it is likely to re-bind to one of the nearby red TUs. As red TU 2 diffuses through the local concentration, it is also likely to be caught. Consequently, positive feedback drives growth of the red cluster (until limited by molecular crowding). Similarly, the yellow and green clusters grow as yellow TF 3 and green TF 4 are captured. (D) Bar heights give transcriptional activities of each TU in the string (average of 100 runs each lasting 8 105τB). A TU bead is considered to be active whilst within 2.24σ ∼ 6.7 × 10−9m of a TF:pol complex of similar colour. Dashed boxes: regions giving the 3 clusters in the inset in (B). (E) Pearson correlation matrix for the activity of all TUs in the string. TU bead number (from low to high) is reported on axes, with pixel colour giving the Pearson value for each bead pair (bar on right). Bottom: reproduction of pattern shown in (A,ii).Boxes: regions giving the 3 clusters in the inset in (B).

Simulating effects of mutations.

Yellow TU beads 1920, 1950, 1980, 2010, 2040 and 2070 in the random string have the highest transcriptional activity. 1-4 of these beads are now mutated by recolouring them red. (A) The sequence of bars reflects the sequence of yellow, red, and green TUs in random strings with 1, 2 and 4 mutations (blue beads not shown). Black boxes highlight mutant locations. (B) Typical snapshots of conformations with (i) one, and (ii) 4 mutations. (C) Transcriptional-activity profiles of mutants (averages over 100 runs, each lasting 8 105τB). Bars are coloured according to TU colour. Black boxes: activities of mutated TUs. (D) Activities (+/-SDs) of wild-type (yellow) and different mutants. 3 mutations: TUs 1950, 1980 and 2010 mutated from yellow to red. (E) Typical kymographs for (i) wild-type and (ii) 4-mutant cases. Each row reports the transcriptional state of a TU during one simulation. Black pixels denote inactivity, and others activity; pixels colour reflects TU colour. Blue boxes: region containing mutations. (F) Pearson correlation matrices for wild-type and 4-mutant cases. Black boxes: regions containing mutations (mutations also change patterns far from boxes).

Reducing the concentration of yellow TFs reduces the transcriptional activity of most yellow TUs while enhancing the activities of some red TUs.

(A) Overview. Simulations are run using the random string with the concentration of yellow TFs reduced by 30%, and activities determined (means from 100 runs each lasting 8 105τB). (B) Activity profile. Dashed boxes: activities fall in the region containing the biggest cluster of yellow TUs seen with 100% TFs, as those of an adjacent red cluster increase. (C) Differences in activity induced by reducing the concentration of yellow TFs. This plot is obtained by subtracting the transcriptional activity of the wild-type, Figure 1D, from that of the current system in panel B. (D) Pearson correlation difference matrix. This plot is obtained by subtracting the Pearson correlation matrix of the wild-type, Figure 1E, from that of the current system. Boxes: regions giving the 3 clusters from Figure 1B, inset.

Clustering similar TUs in 1D genomic space increases transcriptional activity.

(A) Simulations involve toy strings with patterns (dashed boxes) repeated 1 or 6 times. Activity profiles plus Pearson correlation matrices are determined (100 runs, each lasting 8 105τB). (B) The 6-pattern yields a higher mean transcriptional activity (arrow highlights difference between the two means). (C) The 6-pattern yields higher positive correlations between TUs within each pattern, and higher negative correlations between each repeat.

TU transcriptional networks and demixing.

Simulations are run using the toy models indicated, and complete correlation networks (qualitatively reminiscent of gene regulatory networks) constructed from Pearson correlation matrices. respectively (above a threshold of 0.2, co(rres)ponding to a p-value ∼ 5 10−2). The complete network consists of n = 100 (A) Simplified network given by the random string. TUs from first (bead 30) to last (bead 3000) are shown as peripheral nodes (coloured according to TU); black and dashed grey edges denote statistically-significant positive and negative correlations, individual TUs, so that there are pairs of TUs couples; we find 990 black and 742 gray edges. Since p-value nc = 223, most interactions (edges) are statistically significant. Networks shown here only correlations (i) between red TUs, and (ii) between red and green TUs. (ii) (B) Average correlation (shading shows +/-SD, and is usually less than line/spot thickness) as a function of genomic separation for the (i) random, (ii) 6-, and (iii) 1-pattern cases. Correlation values at fixed genomic distance are taken from super-/sub-diagonals of Pearson matrices. Red dots give mean correlation between TUs of the same color (3 possible combinations), and blue dots those between TUs of different colors (4 possible combinations). Cartoons depict contents of typical clusters to give a pictorial representation of mixing degree (as this determines correlation patterns); see SI for exact values of θdem.

Comparison of transcriptional activities of TUs on different human chromosomes determined from simulations and GRO-seq.

(A) Overview of panels (A-C). The 35784 beads on a string representing HSA14 in HUVECs are of 4 types: TUs active only in HUVECs (red), “house-keeping” TUs – ones active in both HUVECs and ESCs (green), “euchromatic” ones (blue), and “heterochromatic” ones (grey). Red and green TFs bind strongly to TUs of the same colour, and weakly to euchromatin; neither binds to heterochromatin. (B) Snapshot of a typical conformation, showing both specialized and mixed clusters. (C) TU activities seen in simulations and GRO-seq are ranked from high to low, binned into quintiles, and activities compared. (D) Spearman’s rank correlation coefficients for the comparison between activity data obtained from analogous simulations and GRO-seq for the chromosomes and cell types indicated.

Small clusters tend to be unmixed, large ones mixed.

After running one simulation for HSA 14 in HUVECs, clusters are identified. (A) Snapshot of a typical final conformation (TUs, non-binding beads, and TFs in off state not shown). Insets: a large mixed cluster and a small demixed one. (B) Example clusters with different numbers of TFs/cluster (2, 10, 20, 30, 40) chosen to represent the range seen from all-red to all-green (with 3 intervening bins). Black numbers: observed number of clusters of that type seen in the simulation. (C) Average of the demixing coefficient θdem (error bars: SD), showing a crossover between demixed and mixed clusters with increasing cluster size. Values of 1 and 0 are completely demixed and completely mixed respectively. Grey area: demixed regime where θdem is > 0.5.

HiP-HoP model simulations: small clusters tend to be unmixed, large ones mixed.

(A) Snapshot of a configuration adopted by HSA14 in HUVECs, within the HiP-HoP model. Grey regions represent less accessible chromatin regions poor in H3K27ac, while cyan regions represent those enriched in H3K27ac. In addition, H3K27me3 and H3K9me3 peaks determine the chromatin binding sites for polycomb-like and heterochromatin proteins, and are represented in yellow and blue respectively. As in the previous DHS multicolour model, TUs only present in HUVEC are represented in red, while the house-keeping ones in green. (B-C) Example of clusters of proteins: large mixed cluster (B) and a small demixed one (C). (D)Average of the demixing coefficient θdem (error bars: SD). Values of 1 and 0 correspond to completely demixed and completely mixed clusters respectively. Grey area: demixed regime where θdem is > 0.5.