Article

Structural Fluctuations of the Chromatin Fiber within Topologically Associating Domains Guido Tiana,1 Assaf Amitai,2 Tim Pollex,3 Tristan Piolot,3 David Holcman,4 Edith Heard,3,5 and Luca Giorgetti6,* 1

Department of Physics and Center for Complexity and Biosystems, University of Milano and Istituto Nazionale di Fisica Nucleare, Milano, Italy; Institute for Medical Engineering & Science, The Massachusetts Institute of Technology, Cambridge, Massachusetts; 3Institut Curie, CNRS UMR3215, INSERM U934, Paris, France; 4Institut de Biologie de l’Ecole Normale Superieure, Paris, France; 5Colle`ge de France, Paris, France; and 6Friedrich Miescher Institute for Biomedical Research, Basel, Switzerland 2

ABSTRACT Experiments based on chromosome conformation capture have shown that mammalian genomes are partitioned into topologically associating domains (TADs), within which the chromatin fiber preferentially interacts. TADs may provide threedimensional scaffolds allowing genes to contact their appropriate distal regulatory DNA sequences (e.g., enhancers) and thus to be properly regulated. Understanding the cell-to-cell and temporal variability of the chromatin fiber within TADs, and what determines them, is thus of great importance to better understand transcriptional regulation. We recently described an equilibrium polymer model that can accurately predict cell-to-cell variation of chromosome conformation within single TADs, from chromosome conformation capture-based data. Here we further analyze the conformational and energetic properties of our model. We show that the chromatin fiber within TADs can easily fluctuate between several conformational states, which are hierarchically organized and are not separated by important free energy barriers, and that this is facilitated by the fact that the chromatin fiber within TADs is close to the onset of the coil-globule transition. We further show that in this dynamic state the properties of the chromatin fiber, and its contact probabilities in particular, are determined in a nontrivial manner not only by site-specific interactions between strongly interacting loci along the fiber, but also by nonlocal correlations between pairs of contacts. Finally, we use live-cell experiments to measure the dynamics of the chromatin fiber in mouse embryonic stem cells, in combination with dynamical simulations, and predict that conformational changes within one TAD are likely to occur on timescales that are much shorter than the duration of one cell cycle. This suggests that genes and their regulatory elements may come together and disassociate several times during a cell cycle. These results have important implications for transcriptional regulation as they support the concept of highly dynamic interactions driven by a complex interplay between site-specific interactions and the intrinsic biophysical properties of the chromatin fiber.

INTRODUCTION In the nuclei of mammalian cells, each chromosome occupies a well-defined region of the nucleus corresponding to a chromosome territory (1) within which the chromatin fiber is organized in a nonrandom manner. Indeed, the three-dimensional (3D) structure of chromatin is thought to play an important role in fundamental biological processes such as DNA replication, maintenance of genome integrity, and gene transcription in particular. Precise spatial and temporal control of gene expression often relies on the presence of short distal sequences of DNA called ‘‘enhancers’’, which direct the activation of the gene when bound by specific combinations of

Submitted July 17, 2015, and accepted for publication February 1, 2016. *Correspondence: [email protected] Tim Pollex’s present address is European Molecular Biology Laboratory (EMBL), Heidelberg, Germany. Editor: Nathan Baker. http://dx.doi.org/10.1016/j.bpj.2016.02.003 Ó 2016 Biophysical Society

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transcription factors (2). Given that enhancers can be very distant from the genes they control along the genomic sequence, they are thought to exert their regulatory role by physically associating in the 3D space with the genes that they control (3). Understanding the conformational properties of chromatin, and in particular characterizing the cell-to-cell and temporal variability of chromosome conformation, will therefore be important to understand quantitatively how long-range transcriptional regulation works. This will ultimately allow fundamental questions to be addressed, such as: How do the dynamic properties of enhancer/promoter contacts relate to gene activation? What is the probability that a given gene is active at any given time point in any given cell in a population or tissue? Do the cell-to-cell and temporal variability in enhancer-promoter 3D distance correlate with transcriptional heterogeneity and dynamics? Chromosome conformation capture (3C) and related techniques such as 4C, 5C, and Hi-C (reviewed in de Wit

Structural Fluctuations within TADs

and de Laat (4)) have enabled the conformational properties of the chromatin fiber within chromosomes to be assessed at increasingly higher resolution. Chromosome conformation capture and its derivatives measure the contact probabilities between different parts of the chromatin fiber, using populations of cells where chromosomes have been cross linked by formaldehyde to promote covalent bonds between parts of the genome that are in molecular contact. In the case of 5C and Hi-C, contact probabilities can be visualized as a two-dimensional contact matrix M, where each entry Mij can be interpreted as the contact probability of two genomic loci i and j (i.e., the probability that these loci are sufficiently close in the 3D space to be cross linked by formaldehyde). 5C and Hi-C contact maps recently revealed that mammalian chromosomes are partitioned into topologically associating domains (TADs), which are contiguous submegabase genomic domains where the chromatin fiber preferentially interacts (5–8). Contact probabilities for sequences are considerably higher within a TAD than between adjacent TADs, which makes each TAD a relatively insulated genomic region. As the vast majority of currently known enhancer-promoter pairs take place within TADs (9), these domains appear to behave as structural scaffolds to instruct the long-range interactions between regulatory regions (10,11). TADs are not uniform domains of highly interacting genomic sequences, and several studies have shown that within single TADs a finer-scale network of specific interactions between specific genomic loci exists (12–14). Increasing evidence indeed suggest that intra-TAD-specific interactions might be at least in part orchestrated by DNA binding proteins and in particular CCCTC-binding factor (CTCF) and cohesin, which are thought to mediate intrachromosomal looping when bound to DNA (14–19), although the mechanisms that mediate these looping events are not understood and it is unclear whether CTCF could mediate intrachromosomal interactions alone (20) or needs to associate with the cohesin complex to do so (19). Moreover, even though many CTCF/cohesin binding sites within TADs colocalize with enhancers and promoters and are thought to mediate their interactions (13,16), a significant fraction of them do not in fact overlap with promoters and enhancers (17). These latter might rather act as structural elements by promoting secondary loops and indirectly finetune the interaction frequencies across an entire TAD, as we recently showed for a CTCF/cohesin-bound structural master locus within the TAD of Tsix in mouse embryonic stem cells (ESCs) (21). Understanding the statistical and dynamical properties of the chromatin fiber within single TADs, and determining which genomic elements/DNA binding factors ensure those properties, represents a fundamental goal for our understanding of variability and specificity in enhancer-promoter interactions. Despite their descriptive power, however, 5C

and Hi-C are population-averaged assays in which one measures contact probabilities over populations of millions of cells. These techniques do not provide immediate access to the cell-to-cell and temporal variability in chromatin conformation. To bridge this gap between 3C data and single-cell conformations, we recently developed a polymer model that can deconvolve single-cell information from population-averaged 5C or Hi-C maps (21), and used it to analyze the 5C data of Nora et al. (6). The model assumes that the chromatin fiber is at equilibrium (at least locally, on the genomic length scale of a TAD—i.e., a few hundred kilobase pairs), so that the probability of a chain configuration is proportional to its Boltzmann factor. In brief, we use iterative Monte Carlo simulations (Fig. 1 a) on a coarsegrained beads-on-a-string model of the chromatin fiber (Fig. 1 c), where beads can interact through hard sphere wells of range R and depth Bij (Fig. 1 b) that are used to approximate the potential energy between loci. The iterative Monte Carlo scheme optimizes the depth of interaction potentials Bij to maximize the similarity between the contact probabilities of the Boltzmann ensemble of the model polymer and the 5C contact map (Fig. 1, d–f). At the end of this procedure, one obtains the ensemble of conformations, which together constitute the 5C map and contain information on the original cell-to-cell variability in the chromosomal structures that gave rise to the original map. In Giorgetti et al. (21), we applied this method to the 5C dataset of Nora et al. (6) and in particular to the two TADs harboring the promoters of Tsix and Xist (Fig. 1 g), which are two of the key regulators of X-chromosome inactivation in female mouse ESCs (22). We showed that: 1) starting from 5C data, the model derives realistic structural ensembles of conformations that could be validated by measuring distances in single cells using DNA FISH; 2) the model correctly predicts the structural effect of experimental perturbations in the potentials within the Tsix TAD, which we generated by genetic manipulation of the corresponding genomic sequence; and 3) the model correctly recapitulates the extensive structural variability in chromosome structure within each TAD, which we verified experimentally in single-cell experiments. We additionally showed that cell-tocell variations in chromosome structure are correlated, at the single-cell level, with variations in transcription levels of Tsix and other transcripts within the same TAD (21). In this study we use our polymer models for the Tsix and Xist TADs to study further the statistical properties of the chromatin fiber within TADs, and to predict their dynamic behavior. We propose in particular that: 1) the chromatin fiber within TADs is able to easily fluctuate between different conformational states, which are hierarchically organized and which are not separated by major free energy barriers; 2) this can be explained by the fact that the chromatin fiber within TADs is at the onset of the coil-globule transition; and 3) contact probabilities at each locus within one TAD are determined in a complex manner not only by the balance

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transcriptional regulation, as they suggest that enhancer-promoter interactions may be highly dynamic events, driven by a complex interplay of specific and nonspecific long-range interactions within one TAD, and occurring as a by-product of the intrinsic behavior of the chromatin fiber as a polymer. Furthermore, this study, where the dynamics of a chromosomal locus within the Tsix TAD is investigated, provides the first direct measurement, to our knowledge, of the microscopic diffusion coefficient of chromatin in mouse ESCs at high temporal resolution, and provides insight into the dynamical properties of chromatin inside single TADs. This article focuses on the theoretical properties of our TAD models, but it should be noted that the latter were extensively experimentally validated in a previous study (21). There, we demonstrated not only that our structural models are realistic, but also that they correctly predict the effect of genetic perturbations in the Tsix TAD. We provide an overview of the experimental validation in the Results, and we invite the interested reader to kindly refer to Giorgetti et al. (21) for a full account of the experimental tests of the models. MATERIALS AND METHODS The chromatin fiber model

FIGURE 1 (a) A scheme of the computational algorithm that we employed to obtain the potentials that reproduce the experimental 5C contact map. (b) The interaction potential is the sum of two-body terms shaped as spherical wells of energy Bij. (c) The chromatin fiber is modeled as a chain of beads, each representing a segment of 3 kilobases along the chromatin fiber. (d) At each iteration of the algorithm, a Monte Carlo sampling records the conformations that are statistically relevant according to this set of energies Bij. (e) A contact map is backcalculated from the sampled conformations. (f) The experimental contact map returned by the 5C experiments, where the Tsix and Xist TADs can be easily identified, is compared with the backcalculated map through evaluation of the c2. The c2 is then minimized by updating the energies Bij and reweighting the conformations sampled in (d). After a given number of optimization steps, a new Monte Carlo sampling is carried out and the algorithm is iterated until convergence of the c2. (g) Simulated versus experimental contact maps for the Tsix and Xist TADs in mouse ESCs (adapted from Giorgetti et al. (21)). (Arrowheads indicate the frequent interactions between Linx, Chic1, and Xite described in Nora et al. (6)). To see this figure in color, go online.

between specific interactions and the entropy of the chain associated with single contacts, but also by correlations with other loci along the fiber. Finally, using high-resolution live-cell measurements of the motion of a site within the Tsix TAD, combined with dynamic simulations, we predict that the conformational changes within a TAD are likely to occur on the timescale of tens of minutes. The results of the analyses that we present here should have important implications in the context of long-range

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The fiber is described as an inextensible chain of beads (108 for the Tsix TAD and 199 for the Xist TAD). The distance between beads a is set to be equivalent to 3 kilobases to match the experimental resolution of the 5C datasets. This was found to correspond to a ¼ 53 nm in Giorgetti et al. (21), when the 3D distances predicted by the model were compared with those measured by DNA FISH in real cells. Each pair of beads interacts with a spherical-well potential of hardcore radius rHC ¼ 0.6a and range R ¼ 1.5a. Exact values of R and rHC were determined as those that gave the best agreement between calculated and observed contact probabilities after optimizing the set of potentials Bij (see below). R and rHC were varied in the range 0.6a–2a, and 0.4a–0.9a, respectively (see Fig. S6A in Giorgetti et al. (21)). Note that the interaction radius R also coincides in our model with the distance that we use to define when two beads are in contact in the 5C experiment. This ensures the existence of a unique set of potentials Bij that maximizes the agreement between calculated and observed 5C contact probabilities (see Supporting Materials and Methods in the Supporting Material). The energy Bij of the well is obtained minimizing the c2 between the experimental contact probabilities given by the 5C data and those backcalculated from the model. The contact probability is defined in the model as the fraction of simulated conformations in which the two beads are closer than R. The conformational space is sampled with a Monte Carlo algorithm involving flip, pivot, and multiple pivot moves at the nominal temperature T ¼ 1 (in energy units, setting the Boltzmann constant to 1), regarded as room temperature. Each sampling lasts for 5  108 steps, recording 5000 conformations. After each sampling, the c2 is minimized through 1000 random changes of the matrix elements Bij, using the reweighting scheme of Norgaard et al. (23). This procedure is iterated until convergence of the c2, i.e., ~150 times. All the details of the algorithm and the code can be found in Tiana et al. (24). Replicate iterative optimizations result in independent and highly correlated sets of interaction potentials (Fig. S1), which represent alternative approximations to the set of potentials that solve the optimization problem (see Supporting Materials and Methods).

Structural Fluctuations within TADs

Alternative control models The results of the optimized model were compared with results obtained from a homopolymeric model, a random-energy model, and a mean-field model. The homopolymeric model is obtained setting all interaction energies Bij equal to B0. The value of B0 is obtained requiring that the average gyration radius of the homopolymer at T ¼ 1 is equal to that of the optimized model, giving B0 ¼ 0.08. The random-energy model is obtained reshuffling at random the energies of the optimized model, only requiring the symmetry of the interaction matrix. We call this system ‘‘random chain’’ or ‘‘random polymer’’ in the sections below. The mean-field model is obtained by associating to each pair of beads a mean distance d*ij displaying a logarithmic dependence on the 5C data (compare to Bau` et al. (25)). While remaining a heteropolymeric model, where each pair of beads interacts in a pair-specific manner, this is a mean-field model in the sense that fluctuations in the distance between each pair of beads are neglected, thus establishing a direct functional relation between the 5C counts and the mean intrabead distance. The parameters that define completely the logarithmic function are obtained fitting the average distances calculated from the optimized model and are d*ij ¼ 3.16  0.73 log (5C)ij for the Tsix and d*ij ¼ 3.32  1.26 log (5C)ij for the Xist TAD (expressed in units of distances between consecutive beads). For each pair of beads, a potential well with depth 1 is implemented with hard-core radius d*ij  1.3 and interaction range d*ij þ 1.3, and an annealing is carried out to optimize the potential.

Simulations of the dynamics of the chromatin fiber The kinetics of the chain was simulated within the Monte Carlo (MC) scheme described in Tiana et al. (26), using flips constrained to a width of 1 as elementary moves of the polymer chain. The relation between MC step and real time is obtained by equating the experimental diffusion coefficient to that obtained from the simulation, calculated from the mean-square displacement of the beads of the chain in the 3D space, fitting with respect to time the square root of the mean displacement of the beads in trajectories of 104 MC steps and resulting in 1 MC step ¼ 0.015 s. The parameter q(t) is defined for each initial conformation as the fraction of contacts of the initial conformation that is present in the conformation at time t, that is

qðtÞ ¼

1 nc

n X

  qðDri ðtÞ < RÞq Dri0 < R ;

i¼1

where Dri0 values are the distances between pairs of beads in the initial conformation; Dri ðtÞ are those in the conformation at time t; P nc ¼ i qðDri0 < RÞ is the number of contacts in the initial conformation; q(c) is a function that takes the value 1 if condition c is true and 0 if it is false; and R is a contact distance, which we set equal to the interaction range of the potential, that is, R ¼ 78 nm.

Live-cell imaging Live-cell imaging was carried out using a DeltaVision OMX Ver. 3 system (Applied Precision, Issaquah, WA) coupled to an EMMCD Evolve camera (Photometrics, Tucson, AZ) and a PlanAPO 100 oil-immersion objective (NA ¼ 1.4) for an effective pixel size of 79 nm (Olympus, Melville, NY). Speckles were removed by shaking of the multimode fiber at 4 kHz; the OMX system is equipped with an objective heater system and a heating stage (Applied Precision). Glass-bottom dishes with a 0.17-mm thickness (MatTek, Ashland, MA) were placed in a cell observation room and placed on a custom-made heating sample holder. Consecutive Z stacks consisting of 10 planes separated by 200 nm were acquired every 910 ms (30 ms exposure each plane) for a total of ~180 s. Images were shading-corrected and

normalized using ImageJ (National Institutes of Health, Bethesda, MD), then deconvolved using Huygens software (Scientific Volume Imaging, Hilversum, The Netherlands). Three-dimensional particle tracking was performed with the Mosaic ImageJ plugin (http://mosaic.mpi-cbg.de/ ParticleTracker/).

Estimating the anomalous diffusion exponent a and the diffusion coefficient We computed the cross-correlation function using the formula (27)

CðtÞ ¼

Np t  1 X 2 Rc ðkDtÞ  Rc ððk þ tÞDtÞ Np  t k ¼ 1

for t ¼ 1.T  1, where Np is the number of points in the trajectory; and Rc ðtÞ is the location of the tagged locus at time t. For short times, CðtÞ increases as a power law

CðtÞ ¼ Ata ; where A > 0. To extract the coefficient a, we computed CðtÞ for each trajectory and fitted the first seven points of the curve to a power law. A chromatin or DNA locus has been previously characterized experimentally by a < 1 (28), while for normal diffusion a ¼ 1. To compute the diffusion coefficient of the tagged locus, we use the following empirical estimator (27):

D ¼

Np 1  1 X 2 Rc ðkDtÞ  Rc ððk þ 1ÞDtÞ : 4Dt k ¼ 1

For short time intervals, the locus motion can be approximated as Brownian and the diffusion coefficient is well approximated by this equation.

RESULTS Chromosome conformation within TADs shares features with random chains and structured polymers In Giorgetti et al. (21) we used the algorithm described in Fig. 1 a to model the chromatin fiber within the Tsix and Xist TADs, and generated conformational ensembles for each of the two TADs, which are a representation of the equilibrium properties of the fiber in the corresponding regions (see Materials and Methods). Importantly, in our previous work we have shown that these conformational ensembles are realistic, in the sense that 1) they correctly reproduce the 3D distances that we measured in single cells using high-resolution DNA FISH between seven loci within the Tsix TAD, and 2) they predict the structural effects of deleting genomic regions within the TAD (for example, a 4.5 kb deletion including a strongly interacting set of beads or ‘‘master locus’’ (21). This makes our model the only extensively validated and predictive model for chromatin conformation at sub-TAD resolution, to the best of our knowledge. Both the Tsix and the Xist TADs are particularly interesting from the structural point of view. Within the Tsix

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TAD, where several loci are transcribed in embryonic stem cells, the 5C map shows frequent interactions between three elements containing CTCF/cohesin sites located within the Xite, Chic1, and Linx transcripts (6) (Fig. 1 g, arrowheads). Two of these (Linx and Xite) overlap with enhancers, whereas the third (Chic1) is believed to act as a structural master locus that influences the interactions between the other two (21). On the other hand, the Xist TAD is devoid of any apparent strong interactions between distal sites and presents a rather uniform distribution of contact probabilities, which correlates with the general absence of transcription in this region in embryonic stem cells. We have previously shown that the conformational ensembles of both TADs contain highly variable structures (Fig. 1 d), the existence of which was confirmed by high- and super-resolution DNA FISH (21). Here we explore in more detail the biophysical properties of our TAD models and start by analyzing the distribution of the distance root-mean-square deviation (dRMSD) between pairs of conformations in the Tsix and Xist TAD models, defined as     dRMSD ria ; rib ¼

   2 X  1    ria  rja   rib  rjb  NðN  1Þ i < j

!1=2 ;

where fria g and frib g denote the coordinates of beads within conformations a and b. In Giorgetti et al. (21) we used the dRMSD to perform a hierarchical clustering of structurally similar conformations, and showed that the structural ensemble for the Tsix TAD is grossly subdivided into two large classes containing compact and elongated conformations. To investigate the stability and the statistical properties of these conformational classes in more detail, we extended our initial analysis to study separately the behavior of the full distribution of dRMSD across the Tsix and Xist structural ensembles. We found that for both TADs, the distributions of dRMSD display a unimodal peak at large values of dRMSD, which is typical of randomlike conformations of a polymer (Fig. S2, a and b). Indeed, the distributions are qualitatively similar to those of a homopolymer of the same length, and very similar to the distribution obtained in a bootstrap model where the interaction potentials that we optimized starting from 5C data were randomly reshuffled (Fig. S2, a and b). Moreover, the distributions of the fraction q of common contacts between pairs of conformations are remarkably similar to those of homopolymeric models and of reshuffled models (Fig. S2, c and d). In fact, even the average gyration radii of the Tsix and Xist models (Rg ¼ 219.9 5 68.3 nm and 313.7 5 96.4 nm, respectively) are similar to the gyration radii that ideal pffiffiffiffi chains of the same length would display (Rg ¼ 61=2 a N ,

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where a is the bead distance and N the number of beads), i.e., 234 and 305 nm, respectively, using a ¼ 53 nm as in Giorgetti et al. (21). In contrast, the dRMSD distributions are markedly different from a mean-field model, where the 5C data are implemented as constraints to derive a consensus structure (similar to Bau` et al. (25); Fig. S2). Thus from the point of their global properties, the optimized models of the chromatin fiber inside the Tsix and Xist TADs are more similar to random coils or globules than to structured polymers with strong site-specific interactions, which would instead display multimodal distributions of dRMSd and q. Not surprisingly, the Xist TAD is the one that is most similar to a random polymer (Fig. S2 b) due to the absence of particularly frequent interactions between distant sequences, as opposed to the TAD of Tsix. On the other hand, the optimized models for both the Tsix and the Xist TADs also have properties that are clearly not typical of random polymers, notably the fact that the contact probabilities do not depend uniquely on the genomic distance as in a homopolymer. This is a prominent feature of the Tsix TAD in particular, which displays a nonrandom enrichment in contact probabilities at the Linx, Chic1, and Xite loci (Fig. 1 g, arrowhead). Indeed, the contact maps obtained with the optimized models for the Tsix TAD are in much better agreement with the experimental 5C data than with homopolymer models, or even with any randomized models that result from a bootstrap procedure on the interaction potentials (p < 2  103 against 500 bootstrapped models) (Fig. S3). In particular, the optimized models correctly account for the presence of specific (nonrandom) long-range contacts, such as the ones among the Xite, Chic1, and Linx loci in the Tsix TAD, which would be obviously absent in the case of a homopolymer. Moreover, in the case of the Tsix TAD, we could show that the optimized model predicts the cell-to-cell distribution of distances between pairs of loci better than any homopolymer or randomized model, proving that the experimentally observed conformational properties of the TAD can only be recapitulated with a model that includes site-specific interactions (21). In the case of the Xist TAD, the relevance of nonrandom long-range contacts is smaller but, as will become apparent later in the text, this TAD also possesses hybrid characteristics that place it in between a random and a structured polymer. In summary, the Tsix and (to a lesser extent) the Xist TADs, as described by the corresponding optimized models, display a defined pattern of site-specific interactions embedded in a highly fluctuating polymer. The presence of site-specific interactions between distant regions along the chain, such as in the Tsix TAD, are able to fine-tune the structural properties of the TAD to a certain extent, but this is not sufficient to stabilize the polymer into stable states, so that the extensive properties of the conformational ensembles remain similar to those of homo- and random polymers.

Structural Fluctuations within TADs

Contact probabilities within TAD arise from sitespecific attractive interactions and nonlocal correlations We next asked to what extent the properties of the TAD conformational ensembles are determined by the interaction potentials that were optimized to maximize the fit to the experimental contact data, and in particular to account for the long-range contacts among key structural elements such as Linx, Chic1, and Xite within the Tsix TAD. We first analyzed whether the interaction potentials are directly correlated with contact probabilities within the two simulated TADs. For each pair i and j of 3-kilobase beads in our TAD models, we plotted the Boltzmann weight associated only to their mutual interaction energy Bij (assuming here a constant binding entropy, as in random globules) as a function of their contact probability in the simulations (Fig. 2, a and b) and found no obvious relationship between the two quantities. However, the more strongly interacting pairs of beads seem to be located close to each other along the linear sequence both in the Tsix and the Xist domains (Fig. S4), suggesting a dependence of the binding entropy on the linear distance between the beads. To better account for the effect of linear distance between beads in determining interaction probabilities, we calculated the ideal contact probability that every pair of interacting beads would have if they were isolated from the rest of the chain, and only linked by an ideal chain. This is given by (29)

a

b

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FIGURE 2 The contact probabilities of each pair of beads are plotted against the associated interaction energies Bij in the case of the Tsix (a) and Xist (b) domains, and against the contact probability obtained from an ideal-chain model of the Tsix (c) and Xist (d) domain. To see this figure in color, go online.

 exp  Bij  1:5 logji  j j

 Pideal ði; jÞ ¼ 1 þ exp  Bij  1:5 logji  j j and is plotted in Fig. 2, c and d, for the Tsix and Xist domains, respectively, against the actual contact probability derived from the 5C data. The ideal probability is well correlated to the actual contact probability (r ¼ 0.73 and r ¼ 0.86 for Tsix and Xist, respectively). However, the absolute magnitude of actual contact probabilities is significantly higher than in the ideal scenario, which implies that actual probabilities in the TAD models are affected by nonlocal correlations between different parts of the chromatin fiber. Thus, the actual contact probability between any two beads within the Tsix and the Xist TADs is nontrivially determined by 1) their mutual interaction energy, 2) their distance along the chain, and 3) nonlocal correlations with distal parts of the chromatin fiber inside the TAD. In particular, the frequent long-range contacts among Xite, Chic1, and Linx loci (circled in blue in Fig. S4 a) are driven only in part by attractive interactions (possibly mediated by CTCF/cohesin protein complexes that bind DNA specifically in those locations), and the actual high contact probabilities of the three loci appear to be determined by the cooperativity associated with the mutual correlations among them and with other interacting parts of the fiber. The latter may arise due to nonspecific interactions between other DNA-bound proteins, or occur because of differential chromatin modifications along the genomic sequence within the TAD. Indeed, the main difference between the conformational ensembles generated by the optimized TAD models and their ideal counterparts is not so much in the properties of each single contact, but rather in the correlations between the appearances of multiple simultaneous contacts. To investigate this, we focused on the 15 most stable contacts between pairs of beads, as obtained from the simulation (separated from each other by at least 10 other beads) within both the Tsix and the Xist TADs, irrespective of whether they correspond to particularly frequent interactions (in the case of Tsix) or to relatively weak interactions (for Xist). For each pair of contacts, defined by four beads (one contact being defined by a pair of beads), we calculated the Pearson correlation coefficient r between the simultaneous formation of each pair across the structural ensemble (Fig. S5 a). In the homopolymer model as well as in energybootstrap models, we found that r is always 80% of pairwise interactions between CTCF sites take place between loci where the CTCF binding motifs occur in a head-to-head orientation (14,18). In the case of the Tsix and Xist TADs, however, we did not detect stronger interaction energies for pairs of sites in a head-to-head orientation (such as Linx-Xite and Linx-Chic1) with respect to sites in the opposite orientation (such as Chic1-Xite). This may be due to the fact that our model is unable to distinguish whether two of those three DNA loci colocalize because they attract each other, or because they are both attracted by a third locus (as both scenarios would result in a similar contact map and similar correlations among the three loci). Alternatively, these three CTCF-bound loci may belong to the small subset of interactions that do not take place between sites that occur in a head-to-head orientation. In summary, although genomic sites bound by CTCF and cohesin appear to mutually attract each other within the

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TADs that we have simulated, the precise interaction frequencies within these TADs (and in particular at sites bound by CTCF/cohesin themselves) are not uniquely determined by these attractive interactions but also depend on nonlocal correlations. The chromatin fiber can easily move among different conformations and lies at the onset of the coil-globule transition Despite the high degree of conformational variability in the equilibrium ensembles for the Tsix and Xist TADs, we wished to investigate whether these ensembles contain some conformational states that are relatively more frequent than some others. This is an important question because however energetically unstable they may be, these differential conformational states may correspond to alternative biological states of the chromatin fiber (for example, they could promote alternative localizations of enhancer sequences within one TAD with respect to target promoters). To address this question, we performed a cluster analysis of the conformations in both the Tsix and Xist TAD ensembles, using the number nq of shared contacts as a similarity measure between conformations instead of the dRMSD as in Fig. S2. We reasoned that this measure would enable us to assess the presence of a shared subset of molecular contacts rather than evaluating overall shape similarity as in the case of the dRMSD, and would therefore represent a more powerful predictor of differential biological function mediated by long-range interactions between regulatory sequences. As shown in Fig. 3 a, nq clearly discriminates the Tsix and Xist models from both the homopolymeric and randomly reshuffled models, thus confirming that the Xist TAD is also markedly different than a purely random polymer. We performed hierarchical clustering of the conformations of both TADs based on nq using the complete linkage algorithm (Fig. 3, c and f, respectively), and compared the results with those obtained using the corresponding homopolymeric models (Fig. 3, d and g) and models arising from a random reshuffling of the interaction potentials (Fig. 3, e and h). Both conformational ensembles display a large number of unclustered conformations (orphans) as well as many conformations that cluster together because they share few contacts. However, unlike the control models, the optimized models of the Xist TAD, and (to an even greater extent) the Tsix TAD, display clusters of conformations with >60 common contacts (nq > 60). Each of these clusters is not sharply defined, but is instead made of largely variable conformations. The fraction of non-orphan clusters as a function of the threshold on nq is displayed in Fig. 3 b. At nq ¼ 63, where all conformations are orphans in the homopolymer and randomized models, 10% of the conformations in the Tsix TAD still populate 72 nontrivial clusters, while in the Xist TAD, 6% of the conformations populate 97 nontrivial clusters. Each of these

Structural Fluctuations within TADs

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h

FIGURE 3 Results obtained by applying the complete-linkage clustering algorithm to the equilibrium conformations of the Tsix and Xist TADs (red and black graphs, respectively). The number nq of common contacts between pairs of conformations was used as a similarity measure. (a) Distribution of mutual similarity nq in the case of the optimized models (solid curves), homopolymeric models (dashed curves), and the model with reshuffled interaction energies (dotted curve). (b) The fraction of nonorphan clusters is plotted as a function of the threshold on nq. At nq ¼ 63, all conformations are orphans in the homopolymer and randomized models. Clusters of conformations in the Tsix and Xist TADs (c and f) are represented as a dedrogram and compared with the associated negative models (d–h). To see this figure in color, go online.

clusters is characterized by the formation of a specific subset of the limited number of most intense long-range contacts observed in the experimental 5C map of Fig. 1 g. Taken together, our results support the existence of a conformational space formed by a large number of states that are not separated by large free-energy barriers. This is in line with our observation that other extensive properties of the models (such as the distributions of dRMSD and of common contacts q, as well as the gyration radii, see above) are very similar to those of an ideal coil or globule. This can be explained by the fact that the interaction potentials Bij that were obtained through the optimization procedure set the coil-globule transition of the chromatin polymer (i.e., its q-point) very close to room temperature (Fig. 4) for both TADs. This suggests that conformations belonging to both phases (elongated and compacted in Giorgetti et al. (21)) can be explored by the fiber, which can easily move among structurally diverse conformations. Dynamics of the chromatin fiber in the Tsix TAD To study the eventual presence of free-energy barriers in the conformational space of the chromatin fiber, we set out to simulate the dynamics of the fiber within the Tsix TAD with an appropriate Monte Carlo scheme (26). This scheme

FIGURE 4 The specific heat and the gyration radius of the models of the Tsix and Xist TADs (red and black graphs) are plotted as a function of the simulation temperature. Room temperature, corresponding to where the experimental 5C data were simulated, is conventionally defined as T ¼ 1. To see this figure in color, go online.

enables the investigation of the dynamics of a polymer, provided that the elementary timescale of the diffusion of the system is known, which is possible if the microscopic diffusion constant D of the chromatin fiber is determined. To determine D at the genomic region that we simulated, and in the same cell type, we used female mouse ESCs carrying a homozygous insertion of a Tet operator (TetO) array (consisting of 224 singe TetO) within the Tsix TAD, which can be visualized upon binding of Tet repressor-EGFP (Fig. 5 a) (33,34). We performed time-lapse 3D imaging of the TetO locus in undifferentiated embryonic stem cells (ESCs) at 900 ms resolution, followed by particle tracking to determine the evolution of the position of the locus, over a period of ~180 s (Movie S1). To rule out confounding effects arising from directional cell and/or nuclear movement during the duration of a movie, we performed principal component analysis on each trajectory to identify and analyze separately the transverse and longitudinal components of the motion, which likely correspond to the internal motion at the locus and the directional movement of the entire cell nucleus, respectively (Fig. 5 b). Analysis of cross-correlation functions calculated over the trajectories (see Materials and Methods) revealed that the TetO locus undergoes subdiffusive motion in the transversal (intrinsic) direction with an anomalous coefficient of at ¼ 0.56 5 0.16 (n ¼ 107) (Fig. 5 c). This indicates the presence of strong correlations across the chromatin fiber in this region (35) and is in the same range as recent measurements performed in pro-B cells (36). We measured a slightly larger anomalous exponent (al ¼ 0.66 5 0.19) in the longitudinal direction, as expected if cell nuclei undergo directional movement (which is indeed detected by visual inspection of movies; see Movie S1). Although the motion of the TetO locus appears to be subdiffusive, on short timescales it can be approximated to a Brownian behavior (see Materials and Methods), which allows the extraction of an

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a

b

c

d

a

b FIGURE 5 (a) Schematic view of the genomic locus including Tsix and Xist, showing the positions of the corresponding TADs as well as the position of the TetO array that has been targeted into the mouse ESC line from Matsui et al. (33) and Pollex et al. (34). The TetO array is inserted in the 30 untranslated region of the Chic1 gene. (b) A representative mouse ESC expressing Tet repressor-EGFP, visualized in live-cell fluorescence microscopy (maximum intensity projection over 10 Z planes). The two dots represent the position of the TetO array within each of the two X chromosomes. The 3D position of each chromosomal locus is determined over a period of ~180 s at 900 ms intervals, and principal component analysis is used to extract the longitudinal (l) and transversal (t) component of the trajectory (identified as the directions of maximum (longitudinal) and minimum (transversal) variance across the trajectory). (c) Transversal crosscorrelation function calculated over a sample trajectory showing subdiffusive behavior. (d) Histogram of microscopic diffusion constants of the TetO array locus, measured in n ¼ 107 individual cells (average value is D ¼ 0.0040 5 0.0019 mm2/s). To see this figure in color, go online.

instantaneous diffusion coefficient D ¼ 0.0040 5 0.0019 mm2/s. When possible, we tracked and analyzed separately both sister chromatids in S phase cells (that were identified thanks to the presence of fluorescent nuclear signal in cells that coexpressed PCNA-mCherry, which is known to be associated with DNA replication foci), although we found no significant differences in mobility when compared to G1 cells (data not shown). Based on the average diffusion coefficient that we determined experimentally, we next ran dynamic simulations starting from 10 different conformations. We repeated 100 independent runs and calculated the time dependence of the average similarity parameter qðtÞ, which measures the fraction of contacts present in the initial conformation that are preserved at time t. More details about the calculations and their interpretation are given in the Supporting Material. The relaxation of qðtÞ is shown in Fig. 6 a for four different initial conditions. At experimental temperature (black curves), the relaxation is subdiffusive, decaying as ta=2 with a < 1 (compare to right inset in Fig. 6 a), consistent with our experimental observations. This behavior suggests that the motion of the beads, although constrained by longrange correlations, overall does not have to cross large (i.e., RkT) free-energy barriers (see Fig. S7). The typical

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FIGURE 6 (a) Four representative plots of the relaxation of the mean fraction of contacts that is present in the initial conformation of the Tsix TAD, calculated for four different initial conformations, and averaged over 100 dynamic simulations for each initial conformation. (Red dashed curves) Power-law fits of the relaxation curves. (Inset) Distribution of the fitted power-law exponents for a total of 10 curves obtained from different initial conformations. (b) The relaxation of the fraction of contacts present in the initial conformation and restricted to the mostly interacting loci within the Tsix TAD at temperature T ¼ 1 (black curves) and T ¼ 0.8 (blue curves). (Dashed curves) Power-law fit to the simulated relaxation. To see this figure in color, go online.

time needed to lose most of the contacts present in the initial conformation is of the order of hundreds of minutes, although the precise duration of such structural transitions may depend on the specific genomic context, as suggested by the disparate values of the microscopic diffusion constant that were measured in the literature. However, this suggests that the contacts between enhancers and their target genes within the same TAD could be lost and reformed multiple times within a single cell cycle (which lasts ~16 h in mouse ESCs). To study specifically how fast contacts between the most frequently interacting Xite, Linx, and Chic1 loci are assembled and disassembled over time, we next considered only the 15 most frequently interacting beads within the TAD (as in Fig. S5) by defining a similarity parameter qm ðtÞ, which selectively measures how many of those 15 contacts that may be present in the initial conformation are preserved at time t. The value qm ðtÞ is also subdiffusive and decays on

Structural Fluctuations within TADs

the same timescale as qðtÞ (Fig. 6 b), suggesting that all contacts in the chain (even the strongest) are disrupted in hours in a barrier-free manner. At temperatures well above the coil-globule transition, the relaxation of both qðtÞ and qm ðtÞ is diffusive and very quickly reaches (in a few hours) a state with essentially no contacts at all (data not shown). At temperatures below the coil-globule transition, qðtÞ displays an overall subdiffusive relaxation only slightly slower and with exponents only slightly smaller than those observed at the experimental temperature (data not shown). However, the dynamics of qm ðtÞ is extremely slow (blue curves in Fig. 6 b), cannot be fitted satisfactorily by a power law, and does not change appreciably from its initial value in several days. Taken together, these results suggest that the timescale over which the chromatin fiber inside the Tsix TAD transitions from a compact to an elongated conformation (and vice versa) may be substantially shorter than the duration of a single cell cycle (>15 h in the ESCs used here). This implies that all contacts, including those among frequently interacting elements such as Xite, Linx, and Chic1, might appear and disappear dynamically, thus giving rise to the observed structural variability within a cell population. DISCUSSION In this study we used a physical model of the chromatin fiber within two TADs (Tsix and Xist in mouse ESCs) to understand the statistical and dynamic properties of the 3D interactions between genes and potential structural and regulatory elements (enhancers) within TADs. Our model adopts a thermodynamic interpretation of chromosome conformation capture data (37) and its main assumption is that the chromatin fiber can be described as a polymer at equilibrium, where contact probabilities between loci are governed as given by the Boltzmann distribution of polymer states. While for a whole chromosome structure, the time associated with conformational rearrangements may be too large to reach equilibrium during one cell cycle (38), it is not unreasonable to assume that at the level of single TADs the chain might move fast enough to be described at equilibrium. This is suggested by the available measurements obtained in living cells that place the microscopic diffusion constant of chromatin in the 0.0001–0.06 mm2/s range (39–41), and by our new measurements of the diffusion constant of a locus in the Tsix TAD in mouse ESCs (D ¼ 0.0040 5 0.0019 mm2/s; Fig. 5). In fact, this assumption is also supported by the fact that the predictions of our equilibrium model could be validated experimentally at the single-cell level in fixed cells (21). In an equilibrium polymer model, interaction probabilities are determined on the one hand by the balance between interaction energies and the entropic cost associated with the formation of contacts, and on the other hand by correlations in the simultaneous formation of different contacts. The

interplay among energy, entropy, and correlations is complex, and as a consequence it is virtually impossible to predict a priori which combinations of those ingredients are required to reproduce a given contact map. Our iterative Monte Carlo scheme proved efficient in finding the set of interaction energies that, when combined with entropy and correlations, produces the experimental contact map measured by 5C in the region of Tsix and Xist. The finding that the chromatin fiber within the Tsix and Xist TADs can easily fluctuate between different conformational states in the absence of energy barriers (Figs. 4 and 5) implies that in single cells, conformations where promoters and long-range regulators are in contact may be relatively unstable states that can be easily created and destroyed. This is in contrast with the canonical notion of stable, energetically favored loops connecting two distal DNA sites (42) and supports a view of regulatory interactions as probabilistic events that take place in a subset of cells at any given time point, at least at the locus we have investigated (21). This also seems to be in line with the fact that the chromatin fiber within our models of the Tsix and Xist TADs lies at the onset of the coil-globule transition (Fig. 4), which implies the absence of very stable structures (Fig. 3) and the possibility of easily folding an unstructured configuration into a highly compact, globular structure. Interestingly, a previous study based on a more simplified model of the chromatin fiber (43) also suggested that a polymer state close to the coil-globule transition could explain the scaling behavior of contact probabilities as a function of genomic distance observed in Hi-C data. Even without making any a priori assumptions on binding energies, but rather by deriving them from the experimental data, we also detect the onset of this phase transition in our chromatin models (which, trivially, reproduce the correct scaling behavior measured in 5C as they reproduce contact probabilities by definition). Although our physical models are extracted from static, population-averaged 5C data, it is possible to use the interaction energies to run dynamic simulations and predict the temporal dynamics of the chromatin fiber within the Tsix and Xist TADs. Based on our experimental determination of the microscopic diffusion constant of a locus within the Tsix TAD (Fig. 5), we were able to predict that the folding (or unfolding) of any particular TAD configuration occurs on the timescale of tens of minutes (Fig. 6). This finding has important implication for transcriptional regulation as it suggests that enhancer-promoter interactions occur in the context of a highly dynamic chromatin fiber, which fluctuates between very different conformations over times that are much shorter than the duration of a cell cycle (~16 h for mouse ESCs). This supports the concept that multiple rounds of contact and disengagement occur between regulatory sequences. In the case of Tsix, dynamic interactions between its potential long-range regulators, located in the same TAD as its promoter (6,21), could occur in a single cell cycle. If proven, this type of dynamic interaction

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might impact on transcriptional variability (for example, by dynamically modulating the probability that a promoter is active and thus contributing to intrinsic transcriptional noise (44,45)), and play a role in stochastic decision-making during cell differentiation (46). In this perspective, it would be interesting to develop models that integrate dynamical structural fluctuations with transcriptional dynamics (47). In conclusion, our results suggest that the regulatory sequences within the TAD of Tsix in mouse ESCs may engage physical contacts in the context of a highly dynamic environment, and support the notion that in general, enhancer/promoter communication may be the result of dynamic, unstable fluctuations of the chromatin fiber within TADs. Our live-cell measurements of the dynamics of a chromosomal locus within the Tsix TAD represent, to our knowledge, the first characterization of the dynamical behavior of chromatin at high temporal resolution (~900 ms) in mouse ESCs, and provide an experimental basis to support our predictions of the timescale over which conformational changes take place within TADs. A direct test of these predictions would require mouse ESCs where multiple loci within the same TAD can be visualized simultaneously in live cells. Although technically challenging, this could be soon made possible thanks to the ever-increasing ability to engineer the mouse genome (48) and the advent of new live-cell imaging technologies (39). According to our chromatin models, the determinants of the equilibrium properties of the chromatin fiber, as well as of their dynamics, are on the one hand the interaction energies between distinct loci along the polymer chain (some of which could be mediated by protein complexes in cell nuclei, such as CTCF/cohesin), and on the other hand by the entropic cost of contact formation, which depends on the genomic distance between the loci forming the contact. However, as often happens for physical systems with heterogeneous interactions, correlations between the formations of different contacts play an important role in determining the properties of the system. As a consequence, one cannot predict the formation of a contact simply from the interaction energy and the associated genomic distance, as shown in Fig. 2. In other words, all sequences within the TADs we analyzed (including potential regulatory regions and strongly interacting structural loci such as the Xite, Linx, and Chic1 triad in the Tsix TAD) interact in a nontrivial manner and form contacts, the statistical and dynamical properties of which involve a great number of loci within their TADs, including those that do not form specific or particularly energetic interactions. Our analysis strongly suggests that the effect of deleting single binding sites for candidate factors that could mediate specific interactions (such as CTCF and cohesin) may not be trivial, and that the full complexity of chromosome folding within TADs (including nonlocal effects due to correlations between distal parts of the fiber) will have to be taken into

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consideration when modeling the experimental results to define the molecular parameters that underlie the structure and dynamics of any particular TAD. SUPPORTING MATERIAL Supporting Materials and Methods, seven figures, and one movie are available at http://www.biophysj.org/biophysj/supplemental/S0006-3495(16) 00146-6.

AUTHOR CONTRIBUTIONS G.T. and L.G. performed research, analyzed the data, and wrote the article; T.P., T.P., and E.H. provided cell lines and performed live-cell imaging; and A.A. and D.H. analyzed live-cell imaging datasets.

ACKNOWLEDGMENTS Work in the E.H. lab was supported by the LABEX Development, Epigenesis, Epigenetics and Life-Time Potential (DEEP grant No. ANR11-LBX-0044), the Initiative d’Excellence of l’Universite´ de Recherche Paris Sciences et Lettres (IDEX PSL grant No. ANR-10-IDEX-0001-02), and the European Research Council Advanced Investigator award (No. 250367). E.H. and D.H. thank the Pierre-Gilles de Gennes Foundation for a collaborative grant. T.P. was funded by the Institut Curie International Ph.D. grant.

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