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Proc Natl Acad Sci U S A. 2009 Mar 10; 106(10): 3812–3817.
Published online 2009 Feb 20. doi:  10.1073/pnas.0809501106
PMCID: PMC2656162
Cell Biology

Spatially confined folding of chromatin in the interphase nucleus

Abstract

Genome function in higher eukaryotes involves major changes in the spatial organization of the chromatin fiber. Nevertheless, our understanding of chromatin folding is remarkably limited. Polymer models have been used to describe chromatin folding. However, none of the proposed models gives a satisfactory explanation of experimental data. In particularly, they ignore that each chromosome occupies a confined space, i.e., the chromosome territory. Here, we present a polymer model that is able to describe key properties of chromatin over length scales ranging from 0.5 to 75 Mb. This random loop (RL) model assumes a self-avoiding random walk folding of the polymer backbone and defines a probability P for 2 monomers to interact, creating loops of a broad size range. Model predictions are compared with systematic measurements of chromatin folding of the q-arms of chromosomes 1 and 11. The RL model can explain our observed data and suggests that on the tens-of-megabases length scale P is small, i.e., 10–30 loops per 100 Mb. This is sufficient to enforce folding inside the confined space of a chromosome territory. On the 0.5- to 3-Mb length scale chromatin compaction differs in different subchromosomal domains. This aspect of chromatin structure is incorporated in the RL model by introducing heterogeneity along the fiber contour length due to different local looping probabilities. The RL model creates a quantitative and predictive framework for the identification of nuclear components that are responsible for chromatin–chromatin interactions and determine the 3-dimensional organization of the chromatin fiber.

Keywords: genome organization, polymer model, chromatin folding

The chromatin fiber inside the interphase nucleus of higher eukaryotes is folded and compacted on several length scales. On the smallest scale the basic filament is formed by wrapping double-stranded DNA around a histone protein octamer, forming a nucleosomal unit every ≈200 bp. This beads-on-a-string type filament in turn condenses to a fiber of 30-nm diameter, which detailed organization is still under debate (13). At bigger length scales the spatial organization of chromatin in the interphase nucleus is even more unclear. Imaging techniques do not allow one to directly follow the folding path of the chromatin fiber in the interphase nucleus. Therefore, indirect approaches have been used to obtain information about chromatin folding. One way, pursued in this study, is fluorescence in situ hybridization (FISH) to measure the relationship between the physical distance between genomic sequence elements (in μm) and their genomic distance (in megabases). There have been several attempts to explain the folding of chromatin in the interphase nucleus using polymer models. The strength of polymer models is their ability to make predictions on the structure of chromatin by pointing out the driving forces for observed folding motifs. These predictions can then be tested experimentally. However, a polymer model that is able to explain chromatin folding spanning different length scales is still lacking.

Earlier studies have indicated that the structure of chromatin may be explained by a random walk (RW) model for distances up to 2 Mb, while on a larger scale there is a completely different behavior (4, 5). Folding at larger length scales has been explained using several models. One approach has been to model the fiber as a random walk in a confined geometry (6). Two polymer models have been proposed that introduce loops to explain chromatin folding. One is the random-walk/giant-loop (RWGL) model, which assumes a RW-backbone to which loops of 3 Mb are attached (7). A second model, the multiloop-subcompartment (MLS) model, proposes rosette-like structures consisting of multiple 120-kb loops (5, 8). None of these models is able to describe the folding of chromatin at all relevant length scales. All predict that the physical distance between 2 FISH markers monotonously increases with the genomic distance. Clearly, this is incorrect at bigger length scales, since the chromatin fiber is geometrically confined by the dimensions of the cell nucleus. More so, individual chromosomes have been shown to occupy subnuclear domains that are much smaller than the nucleus itself, i.e., the chromosome territories with sizes in the range of 1 to a few micrometers (9). Evidently, an intrinsic property of the chromatin fiber inside the cell nucleus is that it assumes a compact state that cannot be described by classic polymer models. This raises the fundamental question what physical principles make chromatin to fold in a limited volume.

How can be explained that a polymer folds such that, irrespective of the length of the polymer, its physical extend does not increase? We have shown that this can be achieved by bringing parts of the polymer together that are nonadjacent along the contour of the polymer, thus forming loops on all length scales (10). There is extensive experimental evidence that chromatin loops exist in the interphase nucleus. Various studies have indicated that the chromatin fiber forms loops that at their bases may be attached to a still poorly defined structure that is called nuclear scaffold/matrix (11). Recent investigations indicate that the formation of chromatin loops involves specific proteins, including SatB1 (12), CTCF and other insulator binding proteins (13). Other studies show long-range chromatin-chromatin interactions due to transcription factories in which transcriptionally active genes at different positions on a chromosome and from different chromosomes, come together (14).

The random loop (RL) polymer model offers a unified description of chromatin folding at different length scales (10). We show that the RL model adequately describes a large set of experimental data that systematically measure the in situ 3D distances between pairs of FISH probes that mark specific points on the chromatin fiber of the q arms of chromosomes 1 and 11 in human primary fibroblast. We show that the RL model presents a simple explanation of the spatial confinement of the chromatin fiber in chromosome territories. A heterogeneous extension of the model with respect to local transcriptional activity is presented showing a good correlation with short distance measurements in different regions. Our results suggest that the formation of loops of a broad size range is a key determinant of chromatin folding at genomic length scales between 0.5 Mb and 75 Mb.

Results

Random Loop Polymer Model.

Chromatin polymer models predict the relationship between mean square physical distance and genomic distance between 2 FISH markers on the same chromosome. Although parameters such as gene activity and epigenetic state probably influence local chromatin properties, we assume here as a first approximation that chromatin can be modeled as a homogeneous polymer. This simplifying assumption has been made for all polymer models on chromatin so far (5, 7, 15). However, below we extend the model to incorporate heterogeneity of the polymer. For a polymer with N monomers, classical models predict that the mean square displacement between the end points of the polymer scales like

equation image

in which ν depends on the type of polymer model (see below). Unavoidably, Eq. 1 is in conflict with the confined geometry of chromosomes inside the interphase nucleus. The recently developed random loop (RL) polymer model overcomes this problem, as the mean square displacement becomes independent of the chain length at bigger length scales (10).

The RL model assumes that the polymer consists of a Gaussian chain backbone with N monomers (numbered by indices 1 to N), the spatial positions denoted by r1rN. Loops are introduced by assigning each pair of monomers {i, j},|ij| > 1 a probability P to interact and form a loop, i.e., 2 monomers that are not adjacent along the backbone interact with a probability P. As a consequence loops on all length scales are generated randomly as illustrated in Fig. 1A. Obviously, assuming random loop formation as we do in the RL model is an approximation, since in the living cell chromatin-chromatin fiber interactions will most likely depend on physical interactions between specific regulatory elements.

Fig. 1.
The random loop polymer model. (A) The diagram schematically shows a small part of the polymer, which is build up of loops with a broad range of sizes. The attachment points are marked by colored circles. (B) Molecular dynamics simulations of a polymer ...

The RL model introduces 2 important features that have not been addressed by polymer models for chromatin up to now. First, it takes into account that intrapolymer interactions, i.e., loop-attachment points, vary from cell to cell and therefore measurements are an average over the ensemble that is represented in the model by assigning a probability for looping (disorder average). Second, it does not assume a fixed loop size, in contrast to the RWGL and MLS models. In the RWGL model, for example, the assumption of loops of a fixed size leads to a random walk behavior on a scale larger than the loop size, with the loops playing the role of “effective monomers.”

In a first approach the RL model assumed that the probability P for 2 monomers to interact is the same for any pair of monomers (10). Such model allows a semianalytical calculation of the mean square displacement, which rapidly becomes independent of polymer length. The RL model ignored excluded volume interactions for reasons of mathematical tractability. Because this may have a major impact on the behavior of the model, we have analyzed how the predictions of the model change if we lift this limitation. We have used molecular dynamics (MD) simulations to obtain chain conformations and to introduce excluded volume interactions in the model. Because 2 averaging processes have to be performed, i.e., over the thermal disorder and over the ensemble of loop configurations, simulations are very time-consuming. Since here we are only interested in large-scale behavior, a coarse-graining approach can be used. In our simulations we equilibrate polymers of length N = 300 (for details on the MD simulations see SI Appendix). Fig. 1B shows the results of simulations for different looping probabilities P. In contrast to classical polymer models, the mean square displacement becomes independent of the contour length at intermediate length scales, resulting in a spatially confined polymer structure. Interestingly, already a small number of loops results in an almost complete independency of the mean square displacement of the genomic distance, without any additional assumptions. It is stressed that loops on all length scales are necessary to make the mean the square displacement independent of contour length (see also ref. 10). Looping probabilities P in Fig. 1B range from 3 × 10−4 to 3 × 10−3, corresponding to 13 up to 133 loops per N = 300 polymer. As expected, the plateau value of 〈R2〉 rapidly decreases, because the number of loops increases and therefore the polymer becomes more compact. For P smaller than 10−4 leveling-off becomes less pronounced, becoming a normal SAW model as P approaches zero. Notably, qualitatively the same behavior is observed for the RL model ignoring excluded volume interactions (10). We therefore conclude that at bigger length scales excluded volume interactions contribute only to a limited extend to the behavior of the RL model.

Experimental Data to Test the Model.

To explore whether the RL model is able to explain experimental data on chromatin folding in the interphase nucleus we have performed systematic measurements that relate the physical distance between 2 pairs of genomic sequence elements and their genomic distance. To do so, we applied the FISH technique on primary human fibroblasts under conditions that preserve 3-dimensional (3D) nuclear structure, in combination with semiautomated 3D confocal microscopy and 3D image processing and analysis (16). We have concentrated on the q-arms of chromosomes 1 and 11, because the human transcriptome map shows that these chromosome arms contain pronounced gene dense and transcriptionally highly active regions, and gene-poor low activity areas [Fig. 2A and (17)]. Such domains have been named ridges (regions of increased gene expression) and anti-ridges, respectively. Approximately 60 bacterial artificial chromosomes (BACs) were selected that recognize approximately evenly spaced genomic sequences, together spanning a large part of the q-arm of chromosome 1 and essentially the complete q-arm of chromosome 11 (see Table S1). For most 3D distance measurements 30–50 nuclei were imaged and quantitatively evaluated, resulting in practice in 45–75 measurements for each pair of BAC probes, i.e., each genomic distance, allowing statistical analysis of the datasets. The distribution of measured distances represents cell-to-cell variation, which relates to the conformational ensemble that the polymer model averages over. We have analyzed exclusively cells in G1 to reduce cell cycle effects on chromatin folding. Fig. 2A shows the transcriptome map of the 1q and 11q areas. The starting points of the arrows above the maps indicate the positions of the reference FISH probes. The arrowheads marks the locations of the FISH probe that has the largest genomic distance to the reference probe. All physical distances have been determined with respect to the reference probe. Green arrows and green data points refer to ridges, red ones to anti-ridges. Black arrows in Fig. 2A indicate long distance measurements beyond ridge and anti-ridge domains. Physical distances were measured in 3D space between the centers of gravity of the 3D FISH signals of the individual BAC probes.

Fig. 2.
Experimental data. (A) Domains of different transcriptional activity and gene density (ridges and anti-ridges) are shown on the human transcriptome map of the q-arms of chromosomes 1 and 11. Each vertical line in the map represents a specific gene. The ...

Plots of the mean square distance as a function of the genomic distance, covering a large part of the q-arm of chromosome 1 (27 Mb) and essentially the complete q-arm of chromosome 11 (75 Mb), are shown in Fig. 2C. Results show that the average physical distance to the reference probe does not increase at genomic distances beyond 3–10 Mb. The maximal distances are in the 1.5- to 2.5-μm range, similar to the size-range of chromosome territories and well below the diameter of the cell nucleus. The observed leveling off is most probably related to the limited space that chromosomes occupy in interphase, i.e., the chromosome territories (9). Fig. 2B shows how the mean square physical distance to the reference FISH probe depends on the genomic distance for the ridge and anti-ridge domains on chromosome 1q and the ridge and on chromosome 11q. Above ≈3 Mb genomic distance the measured physical distances level off, similar as seen for long genomic distances (Fig. 2C). Average physical distances for anti-ridges are smaller than observed for ridges, reflecting their different degrees of compaction, agreeing with earlier measurements (16, 18).

All measurements show considerable cell-to-cell variation for the physical distances. This is not due to errors in 3D measurements, since their precision is better than 100 nm (see Materials and Methods). Also, differences between cells due to different cell cycle stages are unlikely, because all analyzed nuclei were in G1. Apparently, cell-to-cell variation is an intrinsic property of chromatin folding, reflecting the thermal and conformational ensemble. These experimental results show that there are at least 2 regimes for chromatin folding: one at short genomic distances up to ≈2 Mb, at which the mean square distance increases with the genomic distance, and another at large genomic distances, beyond 10 Mb, where the mean square distance is independent of genomic length. Below we integrate these experimental results in the RL polymer model introduced above.

Integration of Short- and Long-Length Scale Experimental Data by the RL Model.

The RL model proposes that large-scale chromatin folding is driven by chromatin looping. The prediction of a leveling-off in the mean square displacement is in agreement with the experimental data. How can we bring theory and experiment together? The simulations use a polymer with a length N = 300. By mapping a coarse-grained monomer to 500-kb chromatin we obtain a chain of an effective length of 150 Mb, i.e., the size range of a human chromosome. In the model the mean square displacement is a complex function of the chain length N, separation between monomers Nm and looping probability P: 〈R2〉 = fN(Nm,P). In this context the single variable parameter is P, because N is fixed to 300. To compare our simulation results to the experimental data we have to introduce a scaling factor for the 〈R2〉 axis. This factor is somewhat arbitrary and on this level of coarsening strongly depends on monomer geometry and does not reflect biological parameters in a simple manner (10). In Fig. 1C we have scaled the results of the simulations in Fig. 1B to the experimental data, using 320 nm per coarse-grained monomer. This number has been determined such that the model fits to the plateau level of the experimental data. Fig. 1C shows that the RL model is able to qualitatively describe the large-scale genomic distance data quite well. This is remarkable because we do not include information about at what positions along the chromatin fiber loops are formed.

At shorter genomic distances, i.e., on the length scale of ridges and anti-ridges (0.5–2 Mb), another folding regime dominates, because the measured mean square distances at this scale increase with genomic distances and as a first approximation Eq. 1 applies. To see whether one of the basic polymer models for which Eq. 1 holds true (the random walk, self-avoiding walk or globular state (19)), applies to our data, we conducted a sensitive comparison between these polymer models and the experimental dataset by dividing out the leading order term Nm of Eq. 1 and analyzed the ratio 〈R2〉/Nm as a function of the contour length for the measurements shown in Fig. 2B. The value of 〈R2〉/Nm should be independent of the contour length. We use data up to genomic distances of 2 Mb to keep away from distances at which leveling off begins. Figs. 3 A and B show that neither the RW, nor the SAW model fulfills this criterion. Fig. 3C indicates that a scaling with ν = 1/3, as defined for the globular state (GS) model, is more consistent with the experimental data, indicating a considerably more compact state than predicted by the RW and SAW models. We have incorporated data of Yokota et al. (20) in Fig. 3 (blue data points) in support of this conclusion.

Fig. 3.
Comparison of the short distance (0.5–2 Mb) experimental data with the random walk, self-avoiding walk and globular state polymer models. The panels show the experimental short distance data for the ridge (green) and anti-ridge (red) on the q-arm ...

Although the exponent ν = 1/3 (Eq. 1) is true for the globular state polymer model, one should be aware of the fact that the model is only valid for end-to-end distances of a polymer, whereas we here deal with intrachain distances. Fitting the RL model to our experimental data shows that such value of ν is only valid in a narrow range of genomic distances before a plateau level is reached. Finally, for other loci even higher levels of compaction with scaling exponent ν ≈ 0.1–0.2 have been observed (15). Thus, the interpretation of the data in terms of one of the classical polymer model would be an extreme oversimplification. In contrast to the RW and SAW models the RL model is based on intrachain attractive forces, i.e., chromatin loops. On short length scales the RL model also shows a power-law dependence of the mean-square displacement in relation to genomic distance. Fig. 4A shows that practically any value for the exponent ν (Eq. 1) <0.5 (RW model) can be obtained by choosing different looping probabilities P.

Fig. 4.
Incorporation of chromatin fiber heterogeneity into the RL model by assuming different looping probabilities. (A) Qualitative behavior of the random loop model. The relationship between the mean square displacement between 2 monomers and their contour ...

Therefore, we extended the original RL model, which assumes the same looping probability P for all pairs of monomers, to incorporate local differences in P values (thus making the polymer heterogeneous). We assign different looping probabilities for different regions based on the distribution of ridges and anti-ridges in the human transcriptome map as shown in Fig. 2A (17). As a first approximation we divide the polymer in ridge and anti-ridge regions and define 3 different looping probabilities, i.e., PR, defining loop formation in ridge regions, PAR for anti-ridges and Pinter for the interaction between such regions. Fig. 4B shows the result of a simulation for PR = 3 × 10−5, PAR = 7 × 10−5 and Pinter = 1 × 10−5. The RL model with these values describes the folding of the ridge and anti-ridge of chromosome 11 remarkably well. Details on the implementation of the RL model with different P values can be found in SI Appendix. A fit of the RL model for the same set of P values to ridge and anti-ridge data of chromosome 1 is shown in Fig. S1.

An alternative way to introduce heterogeneity in looping probability into the RL model is to assume that loops on short length scales are more abundant than loops on large scales. For the original RL model the loop-size distribution is s(l) ∼ 1/(N − l) (10). Heterogeneity in looping probability can be implemented by assuming that the probability p for a pair of monomers {i, j} to interact depends on their genomic distance l = |ij|. This can be achieved by a power-law distribution p(l) = al−b + c. The reason for assuming such kind of distribution is that a power-law behavior arises naturally in the distribution of random contacts in random or self-avoiding walks. Fig. 4C shows that this does not change the qualitative behavior of the RL model, i.e., it still shows leveling-off, provided that there is a significant probability to form large loops. This indicates that the qualitative behavior of the RL model is not very sensitive to the distribution of loop sizes along the length of the polymer.

Discussion

In this study we present a polymer model that qualitatively explains the folding of a chromosome in a limited volume, e.g., a chromosome territory (9). This random loop (RL) model predicts that loop formation is the major driving force for chromatin compaction (10). The RL model assumes that the measured observables, e.g., the mean square displacement, are derived from an ensemble of loop configurations formed by interactions between different parts of the polymer with a certain probability P. A major characteristic of the RL model is that the mean square displacement becomes independent of the contour length at longer distances.

Here, we extend the original RL model beyond the limitations of its original formulation (10). We performed extensive MD simulations to establish the effect of excluded volume on the behavior of the RL model. It turns out that the introduction of excluded volume does not alter the model's main properties. We also explored the effect of heterogeneity along the contour length of the fiber, creating polymers with domains of different local looping probability and therefore different compaction. This for instance mimics the distribution of ridges and anti-ridges on chromosomes (Fig. 2A). Introducing such heterogeneity improves the prediction of the model with respect to the folding of ridges and anti-ridges at short length scales (≈1 Mb) (Fig. 4B), but does not alter the overall behavior of the model at bigger length scales (see Fig. S2).

We have performed systematic 3D-FISH measurements to validate the model. At genomic length scales >10 Mb, distances between pairs of FISH probes are shown to be independent of genomic distance for the q-arms of chromosomes 1 and 11 in G1 nuclei of human primary fibroblasts. This property is most likely due to the confinement of interphase chromosomes in chromosome territories. Measurements of Trask and coworkers (6, 7, 20, 21) did not show such leveling off of physical distances at large genomic distances. Rather, they reported a monotonous increase with increasing genomic distance up to 180 Mb and interpret this as evidence that chromatin folding reflects a RW polymer model. At least in part, this discrepancy can be explained by the fact that these authors used different cell fixation and FISH labeling methods, which preserve the structure of the nucleus less well than those used here. Also, most measurements have been carried out 2-dimensionally. Together, this is likely to result in systematic distortions of their datasets. At short distances (<2 Mb) our experimental results are similar to those obtained by others (4); however, their interpretation in terms of a RW differs from ours. Shopland and coworker also determined distances on the short length scale (22), suggesting probabilistic 3D folding states of chromatin. These probabilistic folding states can be explained by the probabilistic chromatin-chromatin interactions in the RL model.

The leveling-off of physical distances at large genomic distances that we observe (Fig. 2) is in good agreement with the RL model (Figs. 1C and and44B). This leveling-off is due to the presence of loops on all length scales and the averaging procedure over the ensemble of loop configurations. Although polymer models involving loops have been proposed before to explain chromatin folding (7, 8), these models cannot explain experimental results that show that the mean square displacement becomes independent of genomic distance above a few megabases. The RWGL model, which assumes fixed-size loops, results in 2 folding regimes at different length scales, in both of which the mean square displacement increases monotonically with genomic distance (7). The MLS model assumes rosette-like structures with multiple loops of fixed size (120 kb) and results in a power-law dependence of the mean-square displacement on genomic distance, similar to Eq. 1 (8).

We have extended the RL model to take into account local differences in chromatin compaction, as for instance found in ridges and anti-ridges along the q-arms of chromosomes 1 and 11 (Fig. 2), by locally assigning different looping probabilities to the polymer. Although still highly simplifying, this explains remarkably well the difference in compaction of ridges and anti-ridges, assuming a 2.5-fold difference in looping probability for the studied region on human chromosome 11 (Fig. 4B). There is abundant experimental evidence for heterogeneous chromatin looping along the chromatin fiber. For instance, loops with sizes in the 10-kb range have been observed in the beta-globin locus, where gene activity is correlated with loop formation that brings together different regulatory elements of the locus (23). Another example are loops between promoter and enhancer sequences, which span a broad genomic length scale in the 1- to 1,000-kb range (24). Even larger loops are associated with transcription factories, which bring together transcriptionally active genes from different parts of a chromosome, and from different chromosomes (25).

Thus, the RL model allows a unified description of the folding of the chromatin fiber inside the interphase nucleus over different length scales and explains different levels of compaction by assuming different looping probabilities, related for instance to local differences in transcription level and gene density. The RL model creates a basis for explaining the formation of chromosome territories, not requiring a scaffold or other physical confinement. While there is a lot of evidence that chromatin-chromatin interactions play a crucial role in genome function (e.g., see refs. 23 and 25), our study proposes that it also plays an important role in chromatin organization inside the interphase nucleus on the scale of the whole chromosome (tens of megabases) and on that of subchromosomal domains in the size range of a few megabases. Importantly, various aspects of the RL model can be experimentally verified, e.g., by perturbing chromatin-chromatin interactions and analyzing its effect on chromatin folding. Although experimental data on loop distributions are not yet available, experimental techniques such as the 4C technology (26, 27) will allow the measuring of looping probabilities and loop size distribution along the length of complete chromosomes. These and other experimental parameters can be incorporated into the RL model, moving toward a stepwise more realistic polymer model for chromatin folding in higher eukaryotes.

Materials and Methods

Cell Culture and Fluorescence in Situ Hybridization.

Human primary female fibroblasts (04–147) were cultured in DMEM containing 10% FCS, 20 mM glutamine, 60 μg/mL penicillin and 100 μg/mL streptomycin. Cells were used up to passage 25 to avoid effects related to senescence. BACs were selected from the BAC clones available in the RP11-collection at the Sanger Institute (Table S1). Genomic distances were defined as the distance between centers of the BACs. BAC DNA was isolated using the Qiagen REAL prep 96 kit (Qiagen) and DOP-PCR amplified (16). Nick-translation was used to label the probes, either with digoxigenin or biotin (Roche Molecular Biochemicals). FISH was carried out as described in ref. 16.

Confocal Laser-Scanning Microscopy.

For each experiment >45–75 nuclei were imaged. Twelve-bit 3D images were recorded in the multitrack mode to avoid cross-talk, using a LSM 510 confocal laser-scanning microscope (Carl Zeiss) equipped with a 63x/1.4 NA Apochromat objective, using an Ar-ion laser at 364 nm, an Ar laser at 488 nm and a He/Ne laser at 543 nm to excite DAPI, FITC and Cy3, respectively. Fluorescence was detected with the following bandpass filters: 385–470 nm (DAPI), 505–530 nm (FITC) and 560–615 nm (Cy3). Images were scanned with a voxel size of 50 × 50 × 100 nm.

Image Processing and Data Evaluation.

Automated image analysis was carried out on raw datasets with the ARGOS software to identify nuclear sites labeled by BACs and to compute their 3D position in the nucleus as described in ref. 16. In short, chromatic aberration was measured via Tetraspeck Microspheres (Molecular Probes) and corrected for in the analysis. After background subtraction, images were treated with a bandpass filter to remove noise. Subsequently, images were segmented and ensembles of interconnected voxels were regarded as the site labeled by a BAC. The center of mass was calculated for each labeled site at subvoxel resolution and 3D distances between BACS were measured. To estimate the systematic measuring error we hybridized cells with a mixture of the same BAC marked with 2 different fluorophores and measured the distances between the 2 signals. Measurements resulted in accuracy better than 50 nm in all 3 dimensions: x = 7 ± 9 nm; y = 40 ± 11 nm; z = 22 ± 12 nm.

Random Loop Model.

The chromatin fiber is modeled as a polymer consisting of N coarse-grained monomers. In a general approach the Hamiltonian can be written as

equation image

where the position vectors of the monomers are denoted as r1, …, rN. The first term assures the connectivity of the chain, the second term accounts for excluded volume interactions. The third term accounts for the formation of loops and its disorder. The interaction constants κij are random variables with a specific probability distribution. Simulations were carried out using the ESPResSo package within the NVT-Ensemble and Langevin thermostat (28). Simulated chains have a length of N = 300 monomers. Details on the MD simulations can be found in SI Appendix. Simulations were performed on the HELICS2-cluster at the Interdisciplinary Center for Scientific Computing (IWR) in Heidelberg.

Supplementary Material

Supporting Information:

Acknowledgments.

We thank the Sanger Institute and Eric Schoenmakers (University Nijmegen, Nijmegen, The Netherlands) for providing BACs. We thank Jens Odenheimer for helpful comments concerning data analysis. This work was supported by European Commission (as part of the 3DGENOME program) Contract LSHG-CT-2003-503441. M.B. thanks the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences for partial support and the research training group “Simulational Methods in Physics” for funding.

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0809501106/DCSupplemental.

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