Supporting InformationBarbieri et al. 10.1073/pnas.1204799109SI TextSI Materials and Methods. Model and its parameters. In the “stringsand binders switch” (SBS) model, a chromatin filament is repre-sented as a self-avoiding polymer chain (1). Here, the chain ismade of n ¼ 512 spherical sites, each s0 bases long (total lengthL ¼ n · s0). Because of their scaling properties, modeling ofshorter or longer polymers displays similar behaviors (1–3).The polymer has a fraction, f , of binding sites (here, we considermainly the case f ¼ 0.5; see ref. 1 for the general case) and inter-acts with a concentration, cm, of diffusing molecules (binders),which have an affinity EX for those polymer sites. Real transcrip-tion factor binding energies range from approximately 2 kBT fornonspecific binding sites to approximately 20 kBT for specificones (kB is the Boltzmann constant and T is the temperaturein kelvin). Here, for simplicity, we set EX ¼ 2 kBT for all sites(see ref. 1 for the general case). In real situations, the numberand location of binding sites depend on the specific locus consid-ered. For definiteness, we consider here the simplified case wherethey are evenly distributed along the polymer chain.

The system is investigated by Metropolis Monte Carlo (MC)simulations, and for computational purpose the polymer inhabitsa cubic lattice with lattice spacing, d0, equal to the linear length ofa polymer site. Diffusing molecules and polymer sites randomlymove from one to a nearest-neighbor vertex on the lattice, withsingle occupancy. Polymers obey a nonbreaking constraint: twoproximal sites can sit either on next or nearest-next neighboringlattice sites. Chemical interactions are only permitted betweennearest-neighbor particles. Our binding molecules have a bindingmultiplicity of six, after considering the characteristics of knownchromatin organizers, such as CCCTC-binding factor (CTCF),capable of multiple binding, or other organizers, such as tran-scription factories. MC averages are over up to 103 runs, each runbeing up to 1011-step long. The linear length of a polymer site, d0,can be roughly estimated as: d0 is approximately ðs0∕GÞ1∕3D0 ,where D0 is the nucleus diameter and G is the genome content.In mammals, we have approximately G ¼ 6 Gb and D0 of ap-proximately 5 μm. Thus, for a polymer of L ¼ 10 Mb, we haves0 ¼ L∕n of approximately 20 kb, and we derive d0 of approxi-mately 0.1 μm. The fraction, c, of molecules per lattice site isrelated to their molar concentration, cm∶c ∼ cm d0

3 NA, whereNA is the Avogadro number. We explored the system behavioras a function of cm (given EX ¼ 2 kBT) and analogous findingsare encountered by changing EX (given cm).

Monte Carlo simulations. In our MC simulations, we employ aMetropolis algorithm and periodic boundary conditions to reduceboundary effects (4). Each MC step moves every single moleculeor polymer bead in random order once, on average, with a transi-tion probability proportional to exp(−ΔH∕kBT) (4), where ΔH isthe energy barrier encountered in the move. The rate of each trialis given by the Arrhenius factor r0 expð−ΔH∕kBTÞ, where r0 isthe bare reaction rate. We run our algorithm long enough to ap-proach its stationary state, where the relaxation of the normalizedgyration radius as a function of MC time is reported (in the casewhere cm ¼ 25 nmol∕L).

The squared radius of gyration is defined as: Rg2 ¼

1∕½2nðn − 1Þ�∑i;j¼1;nðri − rjÞ2, where ri is the position of beadi ∈ f1;…; Ng. Here, we normalize Rg by the average squaredgyration radius of a randomly floating self-avoiding walk(SAW) chain of size n. Pictorially,Rg is the radius of an “average”sphere enclosing the polymer: It attains a maximum when thepolymer is open in a SAW conformation, and a minimum when

it becomes a compact globule. We also record the polymer “con-tact matrix,” a matrix whose entries are defined by the rationi;j∕n, which is the average relative number of times sites i andj along the polymer enter in contact (i.e., are within a distance of2.5 d0 from each other). Analogously, the average contact prob-ability, PcðsÞ, of two sites is defined as having a distance s alongthe polymer.

Approach to stationarity. MC Metropolis simulations at stationar-ity correctly describe the general equilibrium state of a system (4);here, we focus on such equilibrium states. In the current prevail-ing interpretation (in systems dominated by Brownian motion,ref. 4), a MC Metropolis dynamic is also accepted to describecorrectly its general long-term evolution. We therefore testedwhether the orders of magnitude of the time scales to approachequilibrium, predicted by our MC dynamics, fall within the rangeexpected biologically (5). TheMC time unit corresponds to a timet0 ¼ 1∕r0 (4), which is related to the polymer diffusion constantDand to the lattice spacing constant d0 : D ¼ S2∕ðd2

0∕4t0Þ, whereS2 is the mean-square displacement of the polymer center ofmass per unit of MC time. We measure S2 and impose that Dis of the order of magnitude of the measured diffusion constantof mammalian DNA loci (D ¼ 1 mm2∕h) (6) to derive t0. In thisway, we can map the MC time steps into real time. The foldingtime predicted by MC is of the order of magnitude of fraction ofhours, which, interestingly, is in the biological expected range.

Globule formation and topological domains.With the SBS model wealso explored the mechanisms of formation of distinctly foldedpolymer domains, as recently discovered in chromatin studies(7, 8). To this end, we considered a variant of our model (Fig. 3A)where there are two kinds of equally spaced binding sites(f ¼ 1∕6) along the polymer chain (n ¼ 152 beads long). Thefirst type of binding site, shown in red, is located in the first halfof the polymer chain, whereas the second type, shown in green,lies in the second half. Importantly, the two types of site interactwith a distinct molecular binder each (here, each present in a con-centration cm ¼ 25 nmol∕L, and with EX ¼ 4 kBT), “red” siteswith “red” molecules and “green” sites with “green” molecules(here, molecules do not interact with each other).

We explored such a system and now discuss the case whereboth the red and green binders are in a concentration highenough to drive red and green polymer sites in their compactfolded states (i.e., cm ¼ 25 nmol∕L, andEX ¼ 4 kBT). As statio-narity is approached, two globules spontaneously appear alongthe polymer, one composed of red and the other of green sites,as reported in the pictures of Fig. 3B. For such a system we alsocomputed the contact matrix (i.e., the probability that any twosites along the polymer are in contact). This is shown in Fig. 3C:Two distinct domains appear in the contact matrix, with strongintradomain and much weaker interdomain interactions, simplyexplained here by the action of the two distinct, specific bindingmolecules. Interestingly, the formation of such separated glo-bules closely resembles the “topological domains” observed inchromatin by recent genome-wide chromosome conformationcapture (Hi-C) and Carbon-Copy Chromosome ConformationCapture (5C) studies (7, 8). The presence of the two domains isalso manifested in the average contact probability, PcðsÞ, plottedin Fig. 3D, which shows a shoulder and a change of slope at geno-mic distances, s, corresponding to the crossover from one to theother polymer region (roughly around s ¼ n∕2). Analogously,after an early increase with s the mean-square distance, R2ðsÞ,

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develops a plateau, corresponding to the compact folded struc-ture of the polymer domains, as seen in Fig. S6.

In Fig. S7 A and B, we also show the equilibrium distancedistribution of four specific sites along the polymer chain: sitesA, B, C, and D. It appears that A and B have a short distance andnarrow distribution because they belong to the same domain(analogously, C and D), whereas the distance distribution, for in-stance, of A and C (or A and D) is broader and centered aroundmuch higher values because A belongs to a distinct domain withrelation to C (and D).

Associating domains. Associations between different domains canbe generated if they share binding sites of a given kind. In thesame way that associations of sites within distinct topological do-mains (red/green) are determined (e.g., by the presence of sharedbinding molecules), different domains can associate with eachother when some of their binding sites have common binding mo-lecules (e.g., “yellow”). Topological associated domains (TADs)can be produced in this way, as recently seen in Hi-C and 5C ex-periments (7, 8).

Looping out of specific sites from their domains. To investigate themechanisms of looping out of a given polymer site and exploredomain formation, we also took advantage of the two-binding sitepolymer system considered in Fig. 3. For this purpose, we let thetwo (red and green) domains form before changing the three redbinding sites centered on site A to be in an inert state (i.e., weimpose that they are no longer able to interact with diffusingbinding molecules; see Fig. S7 C–F, where the sites that changedstate are highlighted in blue).

The system starts from a conformation with the two compactdomains (red/green) discussed above and is allowed to approachthe new equilibrium state induced by the presence of the new“blue” sites, including site A, which are inert. A conformationof the system at stationarity and the new contact matrix are shownin Fig. S7E; the signature of the red and green domains is stillclear, but now the region around site A has lost contact with itsformer domain (compare to the contact matrix in Fig. 3C of thetwo-domain state previously discussed). Consistently, the dis-tance distribution of A from B opens up to become similar to thedistribution of distances from C or D (compare Fig. S7 F and B).

This example illustrates how looping out of a site from its domaincan easily be modeled within the SBS model.

Hi-C and TCC data analyses. Hi-C data (9) from GM06990 cellsmapped to hg18 were downloaded from the Gene ExpressionOmnibus (www.ncbi.nlm.nih.gov/geo/, dataset GSE18199,file GSE18199_RAW.tar). The data include three GM06990replicates (HindIII, HindIII Biological Repeat, NcoI). The threeGM06990 replicates were concatenated to calculate the contactprobabilities genome-wide and for individual chromosomesusing a similar method to the one described by Lieberman-Aidenet al. (9).

Hi-C and TCC paired-end reads from GM12878 cells (10)were downloaded from the National Center for BiotechnologyInformation Sequence Read Archive (NCBI SRA, www.ncbi.nlm.nih.gov/sra/, entry SRA025848). The data include twosequencing runs each of HindIII Hi-C and TCC replicates.Hi-C paired-end reads from IMR90 and H1–hESC cells (7) weredownloaded from SRA (entry SRP010370). The IMR90 data re-present eight sequence runs from two HindIII replicates, andthe H1–hESC data are from seven sequencing runs of two repli-cates. The sequence data for a given cell line were first combined,and the raw paired-end reads were mapped separately to hg18with Bowtie (11). Mapped reads were then joined and processedto remove all but one instance of duplicate read pairs.

To calculate contact probabilities Pc, intrachromosomal con-tacts were first extracted and sorted based on chromosome num-ber. Pairwise contacts were then sorted by genomic distance intoincremental bins of 100 kb, and the number of Hi-C sequencingreads in each bin was calculated. The contact probability Pc at agiven chromosomal distance (s) was determined by dividingthe Hi-C read count at that distance (Xd) by its correspondingnormalization constant (Nc) as follows: PcðsÞ ¼ Xd∕NcðsÞ.

Nc corresponds to the total number of possible site pairsseparated by a given s along a chromosome as follows: NcðsÞ ¼Lc − s.Nc values were determined for each chromosome by sub-tracting Ld from the total chromosome length (Lc) and calculat-ing the sum of all possible pairs separated by that chromosomaldistance. Genome-wide PcðsÞ values were calculated by dividingthe sum of read counts by the sum of expected probabilities fromall chromosomes at a given s.

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Fig. S1. Chromatin folding exhibits a variety of folding behaviors that are locus and cell-type specific. (A) FISH and Hi-C data. The table summarizes publisheddata on the behavior of the mean-square distance, R2ðsÞ, and contact probability, PcðsÞ, as a function of genomic distance, s, found in FISH and Hi-C experi-ments. The reference numbers given in this table refer to the list of the main text. (B) Summary of parameters commonly used to quantify chromatin foldingbehaviors. The figure illustrates the definition of linear genomic distance (s), spatial distance (R), contact probability of two loci [PcðsÞ], and radius of gyration ofthe polymer (Rg) representing the radius of the average sphere enclosing the polymer.

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Fig. S2. The polymer conformation displays switch-like responses to changes in binder concentration. The polymer equilibrium-squared gyration radius, Rg2,

is plotted as a function of binder concentration, cm (for EX ¼ 2 kBT ). Rg is normalized by the gyration radius of a random SAW chain of equal contour lengthN. Rg

2 has a step-like behavior. Below a transition value of Ctr (Ctr is approximately 10 nmol∕L for EX ¼ 2 kBT ), Rg2 is indistinguishable from the value found for

the random SAW polymer. Above Ctr , Rg2 takes a value corresponding to a chain folded in a compact spherical globule. (Insets) Schematic drawings of the

polymer conformation in the different states.

Fig. S3. The mean-square distance of subchromosomal regions from FISH data. FISH data on the mean-square distance, R2ðsÞ, in primary fibroblast chromo-some 11, spanning 80 Mb (8). The superimposed dashed line is the behavior predicted by the FG model, and the continuous line the behavior predicted by theSBS model in the compact state.

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Fig. S4. Contact probability across Hi-C and TCC experiments. Hi-C and TCC contact probabilities for different chromosomes in different cell types. (A) PcðsÞ isshown for different chromosomes separately from the published Hi-C dataset for GM06990 cell line (9). Chromosomes 11 and 12 follow the average behaviorenvisaged by Lieberman-Aiden et al. (9) in the 0.5–7 Mb region (shaded in grey), having an exponent α of approximately 1.08. Chromosomes 19 and X deviatefrom the average, with α exponents ranging from approximately 0.93 to approximately 1.30. (B–D) PcðsÞ analyses from Hi-C and TCC datasets from GM12878cell line (10) or IMR90 (7) show a behavior similar to the GM06990 cell line. (E) PcðsÞ analyses from Hi-C for H1–hESC (7): A distinct behavior is seen for H1–hESC,where all chromosomes appear to have an exponent close to 1.6, which corresponds to a more open conformation in the SBS model (see main text). The sameresults are seen in H1–hESC whether or not the sex chromosomes are included in the analyses. (F) Comparison between the genome average PcðsÞ observed inthe different cell lines considered, showing different exponents and behaviors.

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Fig. S5. Comparison of the contact probability of chromosome 18 and 19 within individual experiments of Fig. S4.

Fig. S6. Mean-square distance, R2, between two polymer sites having a genomic distance, s, in the two-domain state of the model pictured in Fig. 3A.

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Fig. S7. Modeling chromatin looping out of domains. (A) Scheme of the polymer of Fig. 3 with the position of sites A, B, C, and D illustrated. (B) Distancedistribution of the four specific sites A to D at equilibrium. Note that A and B, and C and D belong to red and green domains, respectively. (C) Schematicrepresentation of the polymer considered to study looping of site A from the red domain. Once the red/green polymer represented in A reaches equilibrium,sites represented in blue are modified to loose interaction with redmolecules—i.e., they change from red (with affinity to red binders) to blue (without affinityto red or green particles). (D) Picture of a looped-out configuration of site A and neighbor sites fromMC simulations. (E) The steady-state contact matrix of thepolymer system showing that site A is no longer inside the red domain. Color scale from blue to red indicates contact frequencies. (F) Distance distribution ofsites A–D, when site A becomes inert (blue). Site A loops out of its domain: In comparison to distances represented in B, distances between sites A and B are nowlonger, whereas C and D (for comparison) keep their relative positions into their green domains.

Table S1. Summary of exponents α for different cell lines and experimental approaches

GM06990 HiC* GM12878 HiC† GM12878 TCC† IMR90 HiC‡ H1–hESC HiC‡

Mean 1.1 1.2 1.3 1.1 1.6Chr 11 1.1 1.2 1.3 1.1 1.6Chr 12 1.1 1.2 1.3 1.1 1.6Chr 18 1.1 1.1 1.2 1.1 1.6Chr 19 1.3 1.2 1.3 1.3 1.6Chr X 0.9 1.0 1.0 0.9 1.6

Approximate values of the exponents α of the contact probability across different cell lines (GM06990,GM12878, IMR90, and H1–hESC) and different experiments (Hi-C and TCC) for the whole genome (mean) orspecific chromosomes (11, 12, 18, 19, and X). The precise value of α depends on the specific genomic regionconsidered. The exponents reported in this table correspond approximately to the 0.5–7.0 Mb region, asoriginally discussed in ref. 9 and illustrated in Figs. S4 and S5.*Lieberman-Aiden et al. (9).†Kalhor et al. (10).‡Dixon et al. (7).

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