The fractal globule as a model of chromatin architecturein the cell

Leonid A. Mirny

Published online: 28 January 2011# The Author(s) 2011. This article is published with open access at Springerlink.com

Abstract The fractal globule is a compact polymer statethat emerges during polymer condensation as a result oftopological constraints which prevent one region of thechain from passing across another one. This long-livedintermediate state was introduced in 1988 (Grosberg etal. 1988) and has not been observed in experiments orsimulations until recently (Lieberman-Aiden et al.2009). Recent characterization of human chromatinusing a novel chromosome conformational capturetechnique brought the fractal globule into the spotlightas a structural model of human chromosome on thescale of up to 10 Mb (Lieberman-Aiden et al. 2009).Here, we present the concept of the fractal globule,comparing it to other states of a polymer and focusingon its properties relevant for the biophysics ofchromatin. We then discuss properties of the fractalglobule that make it an attractive model for chromatinorganization inside a cell. Next, we connect the fractalglobule to recent studies that emphasize topologicalconstraints as a primary factor driving formation ofchromosomal territories. We discuss how theoreticalpredictions, made on the basis of the fractal globulemodel, can be tested experimentally. Finally, we discuss

whether fractal globule architecture can be relevant forchromatin packing in other organisms such as yeast andbacteria.

Keywords chromatin . chromosome territories .

conformational capture . fractal globule

AbbreviationsNMR Nuclear magnetic resonance3D 3 Dimensional3C Chromosome conformation capture

methodHi-C The name of a method an extension of 3C

method that is capable of identifyinglong-range interactions in an unbiasedgenome-wide fashion.

H-NS The name of the protein UniProt IDP0ACF8 (HNS_ECOLI)

MukBEF Proteins MukB MukE and MukF,UniProt IDs: P22523 (MUKB_ECOLI),P22524 (MUKE_ECOLI), P60293(MUKF_ECOLI)

CTCF Protein UniProt ID P49711(CTCF_HUMAN)

Introduction

An ensemble of structures

How are meters of DNA packed inside the 5-μm-diameter nucleus of a cell? Recent developments in

Chromosome Res (2011) 19:37–51DOI 10.1007/s10577-010-9177-0

L. A. Mirny (*)Harvard-MIT Division of Health Sciences and Technology,and Department of Physics,Massachusetts Institute of Technology,Cambridge, MA, USAe-mail: leonid@MIT.EDU

imaging (Strickfaden et al. 2010; Müller et al. 2004;Berger et al. 2008; Cremer and Cremer 2010; Yokotaet al. 1995) and chromosome capture techniques(Ohlsson and Göndör 2007; Miele and Dekker 2009;Van Berkum and Dekker 2009; Duan et al. 2010)provided new insights into this problem. Beforelooking at specific observations, however, it is worthasking a question: what kind of DNA structures do weexpect to find in this packing?

Fifty years of research in Structural Biology hasprovided tens of thousands of protein and nucleic acidstructures resolved to a fraction of a nanometer. Such highresolution is possible because billions of copies of aparticular protein or nucleic acid all have precisely thesame shape in the individual cells of a crystal. Moreover,NMR spectroscopy has demonstrated that protein structurein solution largely resembles that in a crystal and, moresurprisingly, that the vast majority of copies of a proteinfreely floating in solution have about the same structure.Several unstructured regions (i.e., regions that havedifferent conformations in individual molecules and/orrapidly interconvert) have recently attracted great attention(Uversky and Dunker 2010; Vendruscolo 2007) in theprotein sciences. Do we expect that most DNA/chromatinhas a stable, well-defined spatial structure analogous tothe situation with proteins? How different are thesestructures in individual cells and how rapidly do theymove around, fold, and unfold?

While some specific loci may have stable con-formations that are the same in all cells, we do notexpect the majority of the chromatin fibers to be foldedin exactly the same way in different cells. The entropiccost of ordering such gigantic molecules as chromo-somal DNA in eukaryotes can run too high to achieveprecise folding. The resulting cell-to-cell variability inchromatin structure is, however, averaged over millionsof cells by methods like chromosome conformationalcapture (Ohlsson and Göndör 2007; Naumova andDekker 2010), which provide detailed informationabout the probability of possible interactions in theensemble. While such variability makes it difficult tobuild precise 3D models, some first models based on5C data have been developed (Baù et al. 2010).Experimental data also allow the study of some generalfeatures of chromatin architecture and principles thatgovern its organization. How can one characterizechromatin folding if no unique structure is attainable?

One productive approach offered by statisticalmechanics and used in structural biology of proteins

(Vendruscolo 2007) is to consider an ensemble ofconformations (not necessarily in equilibrium) foundin different cells and/or at different timepoints duringan experiment. Statistical properties of the ensemblecan tell about the principles that govern DNApacking. Thus, the aim of building a single 3D modelconsistent with the measurements is replaced with thegoal of finding a physical model of folding, whichproduces an ensemble of conformations whose prop-erties resemble that of the ensemble studied experi-mentally by chromatin capture and/or opticaltechniques. In the search for such a model, we turnto the statistical physics of polymers that is concernedwith characterizing states of a polymer that emerge asa result of interactions between the monomers, thesolvent, and surrounding surfaces.

Below, I will describe classical equilibrium statesof the polymer and their biologically relevant andmeasurable statistical properties. Next, I will focus ona non-equilibrium state, the fractal globule, originallyproposed in 1988 (Grosberg et al. 1988) andoriginally named the crumpled globule (here weadopt the former notation). Later, this state wassuggested as a model for DNA folding inside a cell(Grosberg et al. 1993) and recently brought into thespotlight by the discovery that such a state is indeedconsistent with Hi-C data obtained for human cells(Lieberman-Aiden et al. 2009). I will then present asummary of our recent work aimed at characterizingbiophysical properties of the fractal globule, and therelevance of this architecture for a range of biologicalfunctions. Finally, I will discuss our expectationsregarding the possibility of finding the fractal globulearchitecture of chromatin in yeast and bacteria.

Chromatin as a polymer

The approach of statistical physics frequently dealswith a coarse-grained “beads-on-a-string” representa-tion of a polymer (Grosberg and Khokhlov 1994;Gennes 1979; Rubinstein and Colby 2003). Thepower of this approach is that it describes anensemble of polymer conformations that emerges atscales much greater than the size of the individualmonomers and irrespective of their fine structure:whether the monomer is a single chemical group, anamino acid, or a nucleosome.

Several approximations have to be made to modelthe chromatin fiber as a homopolymer, i.e., a polymer

38 L.A. Mirny

with all monomers interacting in the same way,having the same size and uniform flexibility alongthe chain.

As a first approximation, eukaryotic chromatin canbe considered as a polymer fiber formed by DNAwrapped around nucleosomes and separated by link-ers of about 40–60 bp (Routh et al. 2008) This fiberhas a diameter of about 10 nm and a flexibility whichemerges as a result of the flexibility of the linkers andpartial unwrapping of nucleosomal DNA. This permitsestimating its persistence length. Given that thepersistence length of DNA is 150 bp, then about threeto four linkers would provide the flexibilitycorresponding to this persistence length of the fiber.Steric interactions between nucleosomes and possibleoccupancy of linkers by other DNA-binding proteins,however, can make the fiber less flexible, leading tothe estimate that about five to six nucleosomes form apersistence length fragment. Thus, each “bead” is not asingle, but a few neighboring nucleosomes. Thearrangement of neighboring nucleosomes within sucha bead determines its size but is of less concern forlarge-scale architecture: it can be some sort of regularzig-zag pattern or an irregular blob whose fold isdetermined by linker lengths and nucleosome phasing(Routh et al. 2008). If the fiber is modeled as a freelyjointed chain, each segment of the chain shall have alength twice the persistence length, i.e. 10–12 nucleo-somes which corresponds to 2–2.5 kbp of DNA. Thus,a chromosome/region of 10 Mb can be modeled as achain of 4,000–5,000 freely jointed segments. Each“bead” then consists of 10–12 nucleosomes and,depending on their arrangement, will have a volumeexceeding that of its comprising DNA and histones bya factor of three to four, i.e., v � 15� 103nm3,allowing it to be modeled as a sphere of about 20-40 nm in diameter. Alternatively, one can modelchromatin as a homopolymer of the 30-nm fiber thathas been observed in vitro but whose presence in vivois debated (Van Holde and Zlatanova 2007), or aheteropolymer with a polymorphic structure andflexibility, which depends on local nucleosome density(Diesinger et al. 2010). If polymorphisms of the fiberare local, the overall architecture at much greaterlength scales could be independent of a structure ofthe fiber and determined primarily by its polymericnature and long-range interactions.

The nature of interactions between the monomersremains to be discovered. These can include DNA

bridging and packing by specific structural proteins,like cohesin (Nasmyth and Haering 2009), CTCF(Phillips and Corces 2009) or RNA molecules (Ng etal. 2007), and long-range interactions betweenenhancers and promoters mediated by assembly ofthe transcription machinery (Alberts 2008b). Interac-tions between the chromatin fiber and the rest of thenucleus may involve steric confinement by thelamina, anchoring to the nuclear matrix or protein-mediated bridging to the nuclear lamina (Kind andVan Steensel 2010).

More complicated statistical modeling can alsoconsider specific interactions, replacing a homopoly-mer with a heteropolymer of several types of beadsinteracting differently with each other and the lamina,e.g., regions of open and closed chromatin. One canalso take into account how the local density ofnucleosomes influences local flexibility of the chainand size of the “beads” (Alberts 2008a), for example,loss of nucleosomes in a regulatory region can makeit much more flexible. While it may be tempting tomodel bacterial chromatin as naked DNA subject tointeractions mediated by DNA bridging and structuralproteins like H-NS (Fang and Rimsky 2008) andMukBEF (Petrushenko et al. 2010), local supercoilingcan lead to formation of non-trivial DNA packing.The fluctuating filament model recently introduced byWiggins, et al. (2010) models bacterial DNA aspacked into a uniform-density filament of some yet-unknown structure that is likely to include a stack ofplectonemic supercoiled loops. In principle, a poly-mer decorated by supercoiled loops can be modeledas a branched polymer. Chromatin containing lots ofcrosslinks, e.g., mitotic chromosomes (Marko 2008)can be considered as a polymer gel.

In summary, to the first approximation, chromatincan be modeled as a homopolymer formed by DNAwrapped into nucleosomes. Such a polymer isassumed to have a constant diameter, DNA density,and flexibility along the chain, with monomersexperiencing excluded volume and other interactionsas well as spatial confinement. More detailed modelsof a heteropolymer may include heterogeneity ofdensity, interactions, and shapes of the monomers.

Equilibrium states of a single polymer

The properties of a homopolymer under differentconditions are presented in detail in several excellent

The fractal globule as a model 39

books which should satisfy both an expert (Grosbergand Khokhlov 1994; Gennes 1979; Rubinstein andColby 2003) and a novice (Grosberg and Khokhlov1997) in the field. Here, we provide a quick summary,focusing on biologically relevant quantities that aremeasured by optical and chromosome capture experi-ments. Two characteristics of primary interest are (1)the mean spatial distance R(s) between two loci thatare a genomic distance s apart along the chain, aquantity that is measured by fluorescent in situhybridization (FISH); and (2) the probability ofcontact Pc(s) between two loci that are a distance sapart, which can be calculated from the chromatincapture data. Both quantities are averaged over theconformational ensemble in polymer physics and overa population of cells in an experiment.

The random coil

A polymer in which monomers that are far apart alongthe chain do not interact, even when approaching eachother in space, is called an ideal chain. Under certainconditions, the behavior of real chains can be wellapproximated by an ideal chain. Irrespective of thelocal mechanisms of chain flexibility (e.g., a worm-likechain, a freely jointed chain, etc.), the behavior ofsufficiently long fragments of the chain resembles a 3Drandom walk. The characteristic size of the polymer R,which can be defined as either its root-mean-squared

end-to-end distanceffiffiffiffiffiffiffiffiffiffiffiRee2

� �qor its mean radius of

gyration Rg, scale with the polymer length N as

RðNÞ � N 1=2: ð1ÞThe end-to-end distance of a subchain of length s

has the same scaling, i.e. RðsÞ � s1=2. Here andbelow, polymer length is measured in the units ofthe polymer's persistence length ‘p, i.e. N ¼ L=‘p,which depends on the local mechanism of flexibilityand for naked DNA was measured to be ‘p � 150bp.Alternatively, one can use the Kuhn length b which isdefined as a length of a bond in a freely jointed chainthat has the same end-to-end distance. For the worm-like chain model, the Kuhn length is about twice thepersistence length b=2‘p and size of the polymer (theroot-mean-squared end-to-end distance)

RðNÞ ¼ bN1=2; ð2Þwhere N=L/b.

Characteristic for the ideal chain is the power v=1/2 of R(s)~sv. This scaling of the end-to-enddistance with s can be tested by FISH experiments,where two loci, a distance s apart, are labeled andvisualized in individual cells, allowing the measure-ment of spatial distance between them. A recentreview (Emanuel et al. 2009) suggests that signifi-cant cell-to-cell variability, however, makes it hardto obtain reliable estimates of v.

Chromosome capture methods (Miele and Dekker2009; Ohlsson and Göndör 2007; Van Berkum andDekker 2009; Lieberman-Aiden et al. 2009) in turn,can provide data on the probability of contact betweenloci distance s apart along the genome. For the idealchain one can obtain

PcðsÞ � s�3=2: ð3Þ

Note that a polymer in this state is rather expandedand has a low density. For example, a random coilof the Escherichia coli genome has a size of R ¼b

ffiffiffiffiN

p ¼ ffiffiffiffiffiffibL

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi300 � 4:6 � 106

pbp � 12mm, which is

much greater than the size of the E. coli bacterium. Ifexcluded volume interactions between the monomersare taken into account, then the scaling of the polymersize changes to R � N 3=5, a case referred to as the“swollen coil”. The swollen coil has a size even largerthan that of the random coil and is unlikely to be arelevant model for DNA packing.

The equilibrium globule

If attraction between the monomers dominates overexcluded volume repulsion, or if the polymer isconfined to a sufficiently small volume, the polymerundergoes a coil-globule transition into an equilibriumglobule. The size of the equilibrium globule scales withthe polymer length as

R � N1=3: ð4Þ

Hence, the volume occupied by the polymer scaleslinearly with polymer length: V~R3~N, i.e., monomersfill a fixed fraction of the volume, and the density ofmonomers r � N=V � const is independent of thepolymer length and is uniform inside the globule.This uniform-density contrasts with that of an idealchain where the volume populated by the polymerV � R3 � N3=2 and the density decrease with N as:r � N=V � N�1=2, resulting in monomers occupying

40 L.A. Mirny

a tiny fraction of the volume of the coil. Relevant forFISH experiments is the scaling of the end-to-enddistance of a subchain with its length s. This scalingin the globule differs from that of the whole chain. Infact, according to the Flory theorem (Grosberg andKhokhlov 1994), interactions of a chain in a densemelt are screened by other chains, making that chainbehave almost like an ideal chain (i.e., a randomwalk). In other words, a chain inside a globulebehaves like a random walk (RðsÞ � s1=2), until the“walker” hits the boundary of the confining volume(or the boundary of the globule). After such a“collision” the walker starts a new random walk fromthe boundary inside the globule. After several suchcollisions, the volume of the globule becomes filledwith random walks that are uncorrelated with eachother. Since the volume is filled by the chainuniformly, the walker that experienced several colli-sions (i.e., s1=2 > RðSÞ � N1=3) is equally likely to befound anywhere within the volume. The end-to-enddistance of a subchain then scales as

RðsÞ � s1=2 for s � N 2=3

const for s > N 2=3:

(ð5Þ

Note that a similar ideal regime of chains isobserved in other dense polymer systems such asmelts of many individual polymers. The contactprobability for a subchain of an equilibrium globulescales (Lua et al. 2004) approximately as

PcðsÞ �s�3=2 for s � N2=3

const for s > N 2=3:

(ð6Þ

Figure 1 shows these scaling behaviors for simu-lated equilibrium globules. Interestingly, FISH datafor yeast chromosomes labeled at the centromere andtelomere show a similar roll-over into a plateau, acharacteristic feature of the equilibrium globule (seefigures in Therizols et al. 2010; Emanuel et al. 2009).

Another important property of the equilibriumglobule is its entanglement. Computer simulations(Virnau et al. 2005; Lieberman-Aiden et al. 2009) andtheoretical calculations (Metzler et al. 2002; Grosberg2000) have demonstrated that a long polymer foldedinto an equilibrium globule is highly knotted. Suchknots can hamper folding and unfolding processes(Bölinger et al. 2010) making knotted conformationsrare among naturally occurring protein structures

(Virnau et al. 2006; Lua and Grosberg 2006). Becauseof the high degree of entanglement of the globule,folding into such knotted conformations requires apolymer to thread its ends through different loopsmany times. Since its slithering motion is rather slowand diffusive (polymer ends move equally forward andbackward), formation of the entangled equilibriumglobule is a very slow process (with equilibration time,~N3 (Grosberg et al. 1988)).

The fractal (crumpled) globule

According to de Gennes, polymer collapse proceedsby the formation of crumples of increasing sizes: first,small crumples are folded, leading to formation of aneffectively thicker polymer-of-crumples, which nextforms large crumples itself, etc. Grosberg et al. (1988)demonstrated that this process should lead to forma-tion of a long-lived state that they called a crumpledglobule (recently referred to as a fractal globule).They also conjectured that such a globule is charac-terized by a hierarchy of crumples thus forming a self-similar structure (Grosberg et al. 1988). Thesecrumples emerge due to topological constraints: everysufficiently long chain experiences such constraintsimposed by other parts of the polymer and collapsesinto a crumple subject to these confining interactions(Khokhlov and Nechaev 1985).

Since the fractal globule is space-filling, its volumescales linearly with polymer length the same way theequilibrium globule does:

RðNÞ � N1=3: ð7ÞAccording to the conjecture by Grosberg et al (1988),

the fractal globule consists of globules (crumples) formedon all scales (Figs. 2 and 3), the scaling of the size of asubchain of length s should follow the same law:

RðsÞ � s1=3: ð8Þ

This applies to sufficiently long subchain, i.e. s>N*, where N* is the minimal length of the polymerthat can form a spontaneous knotted structure and isbelieved to be about 10–20 Kuhn's lengths (Grosberget al. 1988). The self-similar conformation of thefractal globule resembles a statistical fractal with afractal dimension of 3 (for comparison, the Gaussiancoil formed by an ideal chain has a fractal dimensionof 2). Comparison of these equations with the scaling

The fractal globule as a model 41

for the equilibrium globule (Eq. 5) reveals two majordifferences: (1) the scaling of the end-to-end distancehas a power of 1/3 for the fractal globule, rather than1/2 for the equilibrium globule; and (2) the plot of R(s) r s for the fractal globule does not have a plateaulike that present for the equilibrium globule (seeFig. 1a). Such differences could be detected by high-resolution DNA FISH experiments with averagingover a sufficiently large number of cells (e.g., Yokotaet al. 1995) but sufficient cell-to-cell variability canmake it hard to distinguish the powers of 1/2 and 1/3(Emanuel et al. 2009).

The contact probability for the fractal globule wasnot computed in the original (Grosberg et al. 1988)contribution and was difficult to compute analyticallywithout making drastic simplifications. Our groupused simulations to obtain the scaling of Pc(s) in thefractal globule. We use traditional Monte Carlosimulations of a polymer freely jointed chain modeledas spherical impenetrable beads with diameter b(Lieberman-Aiden et al. 2009; Imakaev and Mirny2010). The simulations took care to not violatetopological constraints as tested by computingAlexander polynomials on reduced chains (Virnau etal. 2006). Simulating collapse from a polymer coil byapplying a confining spherical cage or via pairwiseinteractions, we obtained fractal globules for chains as

A B

Fractal globule Equilibrium globule

C D

Fig. 2 Conformations of the fractal (a) and equilibrium (b)globules. The chain is colored from red to blue in rainbowcolors as shown on the top. The fractal globule has a strikingterritorial organization, which strongly contrasts with themixing observed in the equilibrium globule. Territorial organi-zation of the fractal globule (c) is evident when two chains of1,000 monomers each are outlined. The equilibrium globule(d), in contrast, has two chains mixed together in space

A B

-1

-3/2

<R

>

P c

distance, s (monomers) distance, s (monomers)

Fig. 1 a Root-mean squared end-to-end distance R(s) as afunction of the genomic distance s between the ends of asubchain (in the units of ‘) for globules of N=32,000monomers. Blue, equilibrium globule; green, fractal globule.At small s, both globules show scaling characteristic of the self-avoiding random walk (3/5), followed by 1/2 of the ideal coil.

Notice there is a plateau for the equilibrium globule. b Theprobability of a contact as a function of genomic distance s forthe equilibrium globule (blue) and the fractal globule (green).Notice the robust scaling of −1 which spans two orders ofmagnitude for the fractal globule

42 L.A. Mirny

long as N=500,000 monomers (Imakaev and Mirny2010). The resultant fractal globules show a robustscaling

PcðsÞ � s�1 ð9Þ

over a broad range of polymer lengths (N=4,000–500,000) and subchain lengths (s=101–105). Compar-ison with the contact probability for the correspondingequilibrium globule (6) shows a significant differencein the exponent (−1 vs −3/2) and the lack of a plateaufor large s which is present in the equilibrium globule(Fig. 1b). These features have been used in the recentanalysis of the human Hi-C data (Lieberman-Aiden etal. 2009).

The fractal globule in human chromatin architecture

Recently, chromosomal contacts in human cells havebeen characterized by the Hi-C experiments (Lieberman-Aiden et al. 2009). Among several important observa-tions brought to light by this study is the dependence ofthe contact probability Pexp

c ðsÞ on genomic distance s:

Pexpc ðsÞ � sa; a � �1; ð10Þ

for s in the range from 0.5 Mb to about 6 Mb. Theoriginal paper made this statement based on a linearfitting of logPexp

c ðsÞ vs log s. Our more recent analysisusing a maximum-likelihood estimator and systematicmodel selection has (1) confirmed that Hi-C data

contact probability Pexpc ðsÞ is best fit by a power-law

with α very close to −1, and (2) that such a fit couldextend beyond the 5–10-Mb range (Fudenberg andMirny 2010).

The scaling of s−1 is easy to intuit. First, it meansthat loci twofold farther apart are twofold less likelyto interact. Second, if contacts are interpreted aschromatin loops, then there is no mean or character-istic loop length: loops of all length are present andthe mean is not well defined for s−1 scaling. Thisobservation contrasts with the earlier loop models ofchromatin packing (Münkel et al. 1999; Sachs et al.1995).

Another important feature of Pexpc ðsÞ obtained by

Hi-C experiments is the lack of a clear plateau at larges, that would be indicative of the equilibrium globule(see Fig. 1b). While some rise in the slope is observedfor sa50� 100Mb, it is not statistically significantdue to the fact that some chromosomes are shorterthan this range and due to the lack of dynamic rangewhich would allow extremely low-frequency inter-actions to be resolved at such distances.

Another recently introduced polymer model ofchromatin is a Random Loop model (Mateos-Langerak et al. 2009; Bohn et al. 2007). In thismodel, any pair of monomers has a fixed probabilityof having an attractive interaction. Such attractiveinteractions lead to polymer collapse. Since theinteractions are uniformly distributed, the final con-figurations show all the features of the equilibrium

A

B

Fractal globule

Equilibrium globule

Fig. 3 The fractal globule(a) consist of dense globulesformed on all scales. Sub-chains of 100, 300, 1,000,and 3,000 monomers (left toright) are shown by ared tube in a globule ofN=32,000 monomers. Forcomparison, same regionsof the equilibrium globule(b) are diffuse inside theglobule

The fractal globule as a model 43

globule, most prominently, the saturation in the sizeof fragments: R(s)=const for large s (Mateos-Langerak et al. 2009). Such a plateau in R(s) isconsistent with FISH data for a range beyond 25 Mb(for chromosome 11). The plateau in R(s), however,will inevitably lead to a plateau in the contactprobability (i.e., Pc(s)=const for large s), which is instriking disagreement with Hi-C data for human cells.

The fractal globule, in turn, is the only ensemble ofconformations that is consistent with (a) the 1/sscaling of the contact probability for distances in therange of up to 10 Mb, and (b) the lack of plateau inPc(s). Therefore, over this megabase range, the Hi-Cresults exclude several models for higher orderchromatin structure, including the equilibrium glob-ule, the random loop as well as the swollen or idealcoils, and any regular arrangement of open loops.While available FISH data is generally lackingsufficient precision to discriminate between exponentsof 1/2 and 1/3 for R(s) (Emanuel et al. 2009), data forlarge distances s>10 Mb in human chromosome 4(Yokota et al. 1995) are best fit by RðsÞ � s0:32

(Münkel et al. 1999) which is consistent with thefractal globule's 1/3.

It is presently difficult to propose an adequatemodel for chromatin packing for s>25 Mb, due to asignificant scatter in the FISH data and a smallfrequency of contacts at this range in the Hi-C data.However, forthcoming high-resolution chromosomecapture data combined with optical methods couldchange this.

Folding, unfolding, and loop opening

Beyond being the only model that fits Hi-C data, thefractal globule has several important properties thatmake it an attractive way of organizing chromatin ina cell. The fractal globule is easy to form: as weshowed by simulations (Lieberman-Aiden et al. 2009;Imakaev and Mirny 2010) a non-specific collapse of apolymer naturally leads to a fractal globule confor-mation, provided that topological constraints are inplace, i.e., the chain cannot cross itself. For achromatin fiber, such a collapse of ~5–10-Mb loopscould be induced by DNA-binding condensing/linkingproteins like cohesin (Nasmyth and Haering 2009),CTCF (Phillips and Corces 2009; Ohlsson et al. 2010),or structural RNAs (Ng et al. 2007) and can span largechromosomal domains.

The fractal globule is unentangled, i.e., it containsno knots since it maintains the topology of an openstate. Dynamics of chromatin opening from theunentangled fractal conformation are very differentfrom that of the knotted conformation of the equilib-rium globule, as we demonstrated by simulations.Figure 4A shows opening of a region of about 1 Mbin two types of globules of 8 Mb each. While in afractal globule, it can easily unfold if molecularcrosslinks (i.e., attractive interactions) that keep itcondensed are removed, a similar region of theequilibrium globule does not fully open up as itremains trapped by multiple entanglements (Fig. 4B).

Such ability to rapidly unfold can be of greatimportance for gene activation, which has been shownto cause decondensation of large (0.5–2 Mb) genomic

A B

Fig. 4 Opening of a loop that is a part of the fractal globule(a), and the equilibrium globule (b). Globules of 32,000monomers were folded by pairwise attractive interactions. Thefractal globule was formed by Molecular Dynamics whichkeeps track of topological constrains, while the equilibriumglobule was equilibrated folded by Monte Carlo simulations(Reith and Virnau 2010) that violate topological constrainsleading to significant entanglement. On the next step ofmolecular dynamics simulation, attractive interactions for aregion of 3,000 monomers were removed allowing the region toopen up due to the chain entropy. In the fractal globule, theregion opened up forming a large loop (a). The same regionfailed to open from the equilibrium globule (b) due chainentanglements in this state

44 L.A. Mirny

regions (Hubner and Spector 2010; Müller et al. 2001).Our model suggests that such displacement/modifica-tion of crosslinking proteins/RNAs in a spatially smallarea of condensed loop is sufficient to trigger its large-scale decondensation. Modification or displacement ofcrosslinking proteins can be accomplished by somecomponents of transcription machinery or polymerasecomplex that are recruited to the activated locus. Thiscan explain why decondensation depends on thepresence of transcription factor activation domains(Carpenter et al. 2005) or polymerase activity (Mülleret al. 2001). In simulations, the unfolded loop canrapidly move around allowing it to sample space.Such dynamics can allow chromatin loops to searchnuclear environment for transcription factories (Cope etal. 2010; Hubner and Spector 2010). Since theunfolded loop has a size of RðsÞ � s1=2 as comparedto size of the folded loop RðsÞ � s1=3, the foldedregion can exceed in size a much longer domain foldedinto a compact fractal globule (Fig. 4). This argumentis consistent with experiments of Müller et al. (2001)which demonstrated that a 0.5-μm spot decondensesinto a 1–10-μm spot.

Thus, the fractal globule architecture allows rapidand large-scale opening of genomic loci as well astheir spatial motion in the unfolded state. Importantly,all these events happen spontaneously in response tolocal removal/modification of crosslinking proteinsand are driven by the entropy of the polymer chain.

The fractal globule, topological constraints,and chromosomal territories

Our recent computer simulations of polymer folding(Lieberman-Aiden et al. 2009; Imakaev and Mirny2010) demonstrated that when a chain is folded into afractal globule, each sequential region of the chainoccupies a distinct spatial region (see Figs. 2 and 3).This segregation of subchains is akin to the segregationof polymer rings that occurs due to topologicalconstraints and was suggested as a mechanism thatleads to formation of chromosomal territories (Rosaand Everaers 2008; Dorier and Stasiak 2009; DeNooijer et al. 2009; Grosberg et al. 1988; Vettorel etal. 2009). In contrast to chromosomal territories thatseparate chromosomes into spatially distinct regions,spatial segregation in the fractal globule occurs on allscales (Fig. 3). This suggests the presence of genomicterritories where a continuous genomic region is

spatially compact, and different regions occupy differ-ent locations (Fig. 2). The fractal globule suggests thepresence of genomic territories in a broad range ofscales: from tens of kilobases to tens of megabases.While subchromosomal domains have been visualizedas non-overlapping spatial entities (Visser and Aten1999) more systematic study of genomic territories cantest predictions made by the fractal globule model.

While scaling of the contact probability suggestedthe presence of fractal globules for genomic regionsof up to 5–10-Mb long, it is possible that fullchromosomes and their relative packing follow thesame principle of fractal globule architecture. Severalrecent studies have suggested that topological con-straints can lead to the emergence of chromosomalterritories (Rosa and Everaers 2008; Dorier andStasiak 2009; De Nooijer et al. 2009; Visser andAten 1999; Vettorel et al. 2009). By simulatingchromosomes as polymer chains or rings of variouslengths that are either confined to a small volume(Dorier and Stasiak 2009; Visser and Aten 1999; DeNooijer et al. 2009) or equilibrated in a melt of otherchromosomes (Vettorel et al. 2009) these studies haveobserved spatial segregation of chains. Such segrega-tion closely resembles chromosomal segregationobserved by optical microscopy (Cremer and Cremer2010). Moreover, Vettorel et al. (2009) demonstratedthat polymer rings equilibrated in a high-density melthave statistical properties resembling that of thefractal globule. Dorier and Stasiak (2009) have shownthat topological constraints are more important thanexcluded volume in inducing spatial segregation ofrings. However, unrealistically short rings uponextreme confinement used in Dorier and Stasiak(2009) necessitate further studies of this phenomenon.Rosa and Everaers (2008) examined the equilibriumand kinetics of polymer rings and chains. They reportobserving robust RðsÞ � s1=3 scaling for equilibratedrings and as a transient, long-lived intermediate ofconfined polymer chains. They note that this scalingis consistent with RðsÞ � s0:32 obtained for humanchromosome 4 using FISH techniques (Yokota et al.1995). It was also noted (Rosa and Everaers 2008)that an equilibrium conformation of a compactpolymer does not exhibit such self-similar behavior.

In summary, these studies demonstrated thattopological constraints, the same ones that lead toformation of the fractal globule, lead to spatialsegregation of chromosomes. Segregation in the

The fractal globule as a model 45

fractal globule, leads to emergence of “genomicterritories” on all scales above some N*. Biologically,this means that any region of the genome folded intoa fractal globule is spatially compact, rather thanspatially spread. Decondensation and spreading couldbe caused by either active displacement of cross-linking proteins/RNAs (e.g., during gene activation)or by violation of the topological constraints (e.g., bytopoisomerase II or unrepaired double-stranded DNAbreaks, see below).

Mixing and crosstalk

Despite the territorial organization created by topo-logical constraints, there is a great deal of interactionbetween individual regions of the fractal globule.Figure 5 presents two neighboring crumples inside afractal globule, showing a great deal of interdigitationof the two crumples. In fact, our study demonstrated(Lieberman-Aiden et al. 2009) that the number ofinteractions M(s) a region of length s has with the restof the fractal globule scales as

MðsÞ � s � R3ðsÞ � V ðsÞ; ð11Þi.e., linearly with its volume, rather than its surfacearea. As we showed analytically, this scaling followsdirectly from PðsÞ � s�1 scaling of the contactprobability. This means that individual regions deeplypenetrate into each other's volumes (see 5), rather thantouch each other on the surface, as spheres, polyhe-dra, or other squishy but impenetrable objects woulddo. In other words, a crumple of a fractal globule hasa fixed (independent of its size) fraction of its volumethat is involved in interactions.

Moreover, in the fractal globule the number ofcontacts between two crumples of lengths s1 and s2(s1;2 N ) that are separated by a distance l along thechain scales as

M1;2ðlÞ � s1s2l

� V1V2

l: ð12Þ

Thus, the number of interactions is proportional tothe product of the crumples' volumes. Such penetra-tion means a great deal of possible crosstalk betweenindividual regions of all sizes (loci, chromosomalarms, etc.) despite their spatial segregations. Thus,the fractal globule simultaneously provides twoseemingly contradictory features: spatial segregationof genomic regions on all scales and their extensivecrosstalk.

Stability of the fractal globule

While providing a number of advantages, the fractalglobule is a long-lived intermediate on the way tobecoming an equilibrium globule. What are thefactors that determine its metastability? How cancells maintain the fractal globule organization ofchromatin for a long time?

The original theory of the fractal globule (Grosberget al. 1988) suggested that (1) the lifetime of thefractal globule was determined by a time (~N3)required to thread the ends of the polymer throughthe whole globule, allowing the formation of suffi-ciently knotted state; (2) a chain with attached ends(e.g., a loop or a polymer ring) should remain in thefractal globule state. We tested these conjectures bysimulations demonstrating that equilibration of thefractal globule is indeed a very slow process (seeFig. 6) with the time exceeding ~N3. Rosa andEveraers estimate that it would take more than500 years for a chromosomal fiber to equilibrate(Rosa and Everaers 2008). We also found that,contrary to the second conjecture, a chain withconfined ends nevertheless slowly interconverts intoan equilibrium globule, while remaining unentangled(Imakaev and Mirny 2010).

The lifetime of the fractal globule naturallydepends on the stringency of the topological con-straints. Such constraints can be violated in the cell byDNA topoisomerase II enzyme (topo II). Topo II cutsboth strands of one DNA double helix, passes anotherunbroken DNA helix through it, and then religates the

Fig. 5 Despite having an organized territorial architecture,spatially neighboring regions of the fractal globule (shown inred and blue) have a large number of interactions betweenthem, deeply penetrating into each other's volumes. Thenumber of interactions of crumples has scales linear with itsvolumes (see Eq. 12). Thus a fixed fraction of crumples volume(rather than its surface) is involved in interactions.

46 L.A. Mirny

cut DNA. In doing so, it can knot and unknot DNA(Vologodskii 2009). To test the role of topo II, weperformed simulations where occasional strand pass-ing was allowed. These simulations show rapidequilibration of the fractal globule into an equilibriumone. This result suggests that active topo II during theinterphase could destroy the fractal globule architec-ture and chromosomal territories requiring some othermechanisms for their stabilization.

It was proposed that formation and maintenance ofchromosomal territories requires topological con-straints (Dorier and Stasiak 2009). Such constraintswill be kept if topo II were unable to act on nucleosomedchromatin fibers. Recent experiments however demon-strated that at least in vitro topo II is able to act onnucleosomed DNA as efficiently as on naked DNA,reducing its positive supercoiling (Salceda et al. 2006).However, the ability of topo II to facilitate the passageof two chromatin fibers through each other in vivo, aswell as the activity of topo II enzyme during theinterphase remain to be studied.

Stabilization of the fractal globule could involveanchoring as well as reversible and irreversiblecrosslinking of DNA by proteins or RNA molecules.Simulations show that while reversible crosslinkingcannot prevent eventual equilibration, it can signifi-cantly slow it down (Imakaev and Mirny 2010).

To manifest in the cell, fractal globule andtopological territories should not necessarily be stableindefinitely. They should persist at least for theduration of single cell cycle, as chromosomal archi-tecture is re-established upon mitosis. Recent photo-activation experiments beautifully demonstrated thatchromosomal architecture is maintained for 10–15 hand is completely reset upon mitosis (Strickfaden etal. 2010). Mechanisms that suppress stand passing

and otherwise stabilize the fractal globule as well asthe rest of chromatin architecture during the inter-phase are yet to be established.

Other organisms

The relevance of the fractal globule architecture tochromosome organization in other organisms dependson the space available for chromosomes in the nucleusduring the interface, i.e., due to DNA density. The DNAdensity in a diploid human cell is approximately6,000 Mb packed in the nuclear volume of � 300mm3,i.e., 20Mb=mm3. Baker's yeast, in contrast has a densityof about 12Mb=3mm3 ¼ 4Mb=mm3. At least a fivefolddifference in density could entail very differentchromatin architectures. While a significant fraction ofthe yeast nucleus is occupied by nucleoli that areinaccessible to the chromatin (Therizols et al. 2010) theremaining volume may be almost sufficient for a loosecoil formed by an ideal or swollen coil. For example, along yeast chromosomal arm of N=0.5 Mb correspondsto about 200b (with Kuhn length b=30 nm) and has acharacteristic size RðNÞ � b

ffiffiffiffiN

por � bN3=5 � 0:4�

0:7mm which easily fits inside the available volume ofthe yeast nucleus and matches high-resolution FISHmeasurements (Therizols et al. 2010). Chromosomecapture (Ohlsson and Göndör 2007; Van Berkum andDekker 2009; Lieberman-Aiden et al. 2009; Duan et al.2010) measurements in yeast (Duan et al. 2010) willprovide critical information about the scaling of thecontact probability Pc(s) revealing whether chromatinin yeast is packed into a fractal globule or not.Topological constraints in yeast, may nevertheless leadto segregation of chromosomes into less pronouncedchromosomal territories (Haber and Leung 1996;Berger et al. 2008; Therizols et al. 2010).

t=1 10 100 1000

Fig. 6 Equilibration of the fractal globule. A series of snap-shots obtained at four logarithmically spaced timepoints of longequilibration simulations. Notice gradual loss of the territorialorganization, characteristic of the fractal globule, and increas-ing mixing, leading to formation of the equilibrium globule.

Since the ends of the globule remain attached to the surfacewhile being able to slide on it, the structure remainsunentangled. This equilibration is very slow. The details ofthese simulations will be published elsewhere

The fractal globule as a model 47

Recent experiments have shed light on the organi-zation of the bacterial chromosome (Wiggins et al.2010; Toro and Shapiro 2010). Wiggins, et al. (2010)have demonstrated linear organization of the bacterialchromosome in E. coli. The origin of replication wasfound to be positioned close to cell center, while thetwo “arms” extend symmetrically. This study demon-strated that the spatial distance between any locus andthe origin goes precisely linearly with the genomicdistance between the two. This organization andfluctuations in loci positions are explained by amechanical Fluctuating Spring model, which repre-sents the whole DNA-filled nucleoid as a fluctuatingelastic filament, rather than resolving how DNA isorganized inside the nucleoid. Another statisticalmodels of DNA packing in bacteria (Jun and Wright2010) is concerned with a potential mechanism ofDNA segregation upon cell division, suggesting thatchain entropy is sufficient for spontaneous segrega-tion of two DNA chromosomes (Jun and Wright2010). A statistical polymer model that can explainthe observed linear scaling of spatial and genomicdistance has yet to be developed. We conjecture that afractal globule confined to the elongated geometry ofthe E. coli nucleoid can exhibit such linear scalingdue to segregation of subchains. Again, chromosomecapture can provide data complementary to opticalmeasurements, yielding a clearer understandingof the principles that govern folding of bacterialchromosome.

The fractal globule, topological constraints,and cancer

There are a few interesting connections between theconcept of the fractal globule and cancer. From thehistoric work of Boveri (Boveri 1914) to recentcharacterization of cancer genomes (InternationalCancer Genome Consortium 2010), it has beenknown that cancer cells carry numerous genomicrearrangements. Chromatin structure could play a rolein molecular mechanisms involved in formation ofgenomic rearrangements and influence the distribu-tion of rearrangements observed in cancer.

Recent characterization of somatic copy-numberalteration across many human cancers (Beroukhim etal. 2010) have provided a high-resolution map of suchevents and revealed two classes of rearrangements:global, such as deletions or amplifications of a

complete chromosomal arm; and focal which occuron much smaller scales. The abundance of suchevents and significant sample-to-sample differencesin the patterns of observed alterations suggest that thevast majority of these events are passenger mutations,i.e., random genetic events. Strikingly, the frequencyof an alteration (insertion or deletion) of a genomicregion of length s scales as

f ðsÞ � s�1 ð13Þ

for the range of 0:1 � s � 5Mb. This resembles thescaling of the probability of contact between two locidistance s apart obtained by Hi-C for humanchromosomes (Lieberman-Aiden et al. 2009). Weconjecture that these two scaling laws are connected:if two loci form a spatial contact, they are more likelyto be subject to a recombination/repair event thatleads to deletion or amplification of the formed loop.This way, the 1/s scaling in the contact probabilityleads to the same scaling in the frequency of genomicalterations. Such connection between chromatin struc-ture and the frequency of chromosomal alterationshave not been reported earlier and constitutes anotherexperimentally testable hypothesis that stems from thefractal globule model.

Another interesting connection between the fractalglobule and cancer stems from the fact that double-stranded DNA breaks can lead to strand passing andhence to violation of topological constraints. Double-stranded breaks are widespread in certain forms ofcancer and are produced by deficiencies of repair andrecombination machineries (Weinberg 2007). Topo-logical constraints, on the other hand, are central forthe maintenance of the fractal globule and chromo-somal territories. Abundant double-stranded breaksare likely to cause partial opening of domains foldedinto fractal globules leading to some degree ofchromosome decondensation. Note that if the equi-librium globule were the state of the chromatin,double-stranded breaks would have little effect sincea highly knotted conformation of the equilibriumglobule constrains motion of the fiber. Consistent withthese conjectures are experimental findings of localchromatin decondensation at the sites of double-stranded breaks (Kruhlak et al. 2006) and globalchromatin decondensation upon malignant transfor-mation (Ye et al. 2001). Such decondensation, in turncan help cancer cells to reverse chromatin condensa-

48 L.A. Mirny

tion and gene silencing associated with cell differen-tiation (Weinberg 2007). One provocative hypothesisis that cancer cells may use double-stranded breaks tosupport the dedifferentiation process.

Double-stranded breaks can also lead to fasterequilibration of the globule and melting of theboundaries of chromosomal territories. Note that theHi-C data (Lieberman-Aiden et al. 2009) discussedabove were obtained for two cancer cell lines(GM06990 and K562), both showing contact proba-bilities characteristic for the fractal globule. Furtherchromosome capture and fluorescence microscopyexperiments on cells subject to different levels ofdouble-stranded break induction treatment could testthese predictions.

Summary and outlook

Introduced about 20 years ago and proposed then as amodel for DNA packing (Grosberg et al. 1988;Grosberg et al. 1993), the concept of the fractalglobule is an attractive model of chromatin organiza-tion during interphase in human cells. It is the onlystatistical polymer model that is consistent with bothchromosome conformational capture data and FISHscaling: It delivers experimentally observed PðsÞ �s�1 scaling (Lieberman-Aiden et al. 2009); andprovides scaling of the end-to-end distance close tothe FISH scaling of RðsÞ � s0:32 (Rosa and Everaers2008; Yokota et al. 1995). The span of genomiclengths over which the fractal globule persists has yetto be established, as chromosome capture data fit thefractal globule for 0:1|s|10Mb, while FISH datahas close to 1/3 scaling on longer scales sa10Mb(Rosa and Everaers 2008). High-resolution single-molecule single-cell microscopy methods may be ableto overcome current limitations of the FISH methodcaused, in part, by significant cell-to-cell variability ofspatial distances.

Several biophysical properties of the fractal glob-ule make it a particularly appealing model ofchromatin organization.

& The fractal globule is formed spontaneously dueto topological constraints by chromatin condensa-tion and is able to maintain its topological state fora long time.

& By virtue of being largely unknotted, any region ofthe fractal globule can easily and rapidly unfold and

translocate (Fig. 4), becoming accessible to tran-scriptional and other protein machinery of the cell.

& Folding into the fractal globule leads to formationof genomic territories (Fig. 2), i.e., a conformationwhere any specific genomic locus is folded intocompact crumples (Fig. 3), and distinct locioccupy distinct spatial locations. Despite thisterritorial organization, folded loci form a verylarge number of interactions with each other(Fig. 5, with the number of interactions propor-tional to the volumes of interacting crumples).When expanded to the scale of whole chromo-somes, these features of the fractal globulecorrespond to chromosomal territories and suggestextensive crosstalk between the chromosomes.

& The fractal globule is a long-lived intermediatethat gradually converts into an equilibrium glob-ule (Fig. 6), which lacks many of the properties ofthe fractal globule and is not consistent with theexperimental data. Activity of the topo II enzymesignificantly accelerates this process, while cross-linking between remote chromosomal loci slows itdown. Mechanisms that help to maintain thefractal globule state are yet to be found.

Many of these properties, predicted theoreticallyand observed in simulations, can be tested experi-mentally and can help to better characterize the stateof the chromatin inside a cell. For example, genomicterritorial organization can be tested using high-resolution optical microscopy by methods like PALMor STORM (Betzig et al. 2006; Rust et al. 2006).

The role of topo II enzyme in the organization ofthe interface of chromosomes is intriguing. Its abilityto facilitate passage of nucleosomed chromosomalfibers, thus violating topological constraints, can befurther studied experimentally. Similarly, stability ofchromosomal/genomic territories despite the activityof topo II enzyme in vivo can be assayed by inducedtopo II overexpression. Chromatin-pulling experi-ments (Marko 2008) can help to test the degree ofDNA entanglement and to characterize contributionsof topological constraints and crosslinking by proteinsto the folded state. These questions are central tounderstanding the role of topological constraints inthe formation and support of chromosomal organiza-tion during the interphase (Rosa and Everaers 2008;Dorier and Stasiak 2009; De Nooijer et al. 2009;Vettorel et al. 2009; Lieberman-Aiden et al. 2009).

The fractal globule as a model 49

Chromosome-capture methods (Dekker 2008) canreveal how chromatin is organized in differentorganisms, different tissues, at different stages of cellcycle, and to observe the evolution of its structuralstate upon differentiation or malignant transformation.

Solving a precise structure of chromatin akin to thestructure of a folded protein may not be feasible aschromatin structures can differ significantly from cell-to-cell. However, approaches based on the statisticalphysics of polymers and high-quality experimentalmeasurements can help characterize the state of thechromatin as a conformational ensemble, revealingbasic organizing principles behind chromatin foldingand dynamics.

Acknowledgments I am grateful to Maxim Imakaev, GeoffryFudenberg, and Daniel Reich for many productive conversa-tions and providing their results prior to publication; toAlexander Grosbger, Job Dekker, Erez Lieberman-Aiden,Natalia Naumova, Jon-Matther Belton, and Gaddy Getz forproductive discussions; and to Albert Liau for meticulousproofreading the manuscript. LM is supported by PhysicalScience in Oncology Center at MIT funded by the NationalCancer Institute.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

References

Alberts B (2008a) Molecular biology of the cell. GarlandScience, New York

Alberts B (2008b) Molecular biology of the cell: chapter 7.Garland Science, New York

Baù D, Sanyal A, Lajoie BR et al (2010) The three-dimensionalfolding of the α-globin gene domain reveals formation ofchromatin globules Nature Structural and MolecularBiology

Berger AB, Cabal GG, Fabre E et al (2008) High-resolutionstatistical mapping reveals gene territories in live yeast.Nat Methods 5:1031–1037

Beroukhim R, Mermel CH, Porter D et al (2010) The landscapeof somatic copy-number alteration across human cancers.Nature 463:899–905

Betzig E, Patterson GH, Sougrat R et al (2006) Imagingintracellular fluorescent proteins at nanometer resolution.Science 313:1642–1645

Bohn M, Heermann DW, Van Driel R (2007) Random loopmodel for long polymers. Phys Rev E 76:051805

Bölinger D, Sułkowska JI, Hsu H-P et al (2010) A stevedore'sprotein knot. PLoS Comput Biol 6:e1000731

Boveri T (1914). Zur frage der entstehung maligner tumoren(concerning the origin of malignant tumours). TheCompany of Biologists.

Carpenter AE, Memedula S, Plutz MJ, Belmont AS (2005)Common effects of acidic activators on large-scalechromatin structure and transcription. Mol Cell Biol25:958–968

Cope NF, Fraser P, Eskiw CH (2010) The yin and yang ofchromatin spatial organization. Genome Biol 11:204

Cremer T, Cremer M (2010) Chromosome territories. ColdSpring Harb Perspect Biol 2:a003889

De Nooijer S, Wellink J, Mulder B, Bisseling T (2009) Non-specific interactions are sufficient to explain the positionof heterochromatic chromocenters and nucleoli in inter-phase nuclei. Nucleic Acids Res 37:3558–3568

Dekker J (2008) Gene regulation in the third dimension.Science 319:1793–1794

Diesinger PM, Kunkel S, Langowski J, Heermann DW (2010)Histone depletion facilitates chromatin loops on thekilobasepair scale. Biophys J 99:2995–3001

Dorier J, Stasiak A (2009) Topological origins of chromosomalterritories. Nucleic Acids Res 37:6316–6322

Duan Z, AndronescuM, Schutz K et al (2010) A three-dimensionalmodel of the yeast genome. Nature 465:363–367

Emanuel M, Radja NH, Henriksson A, Schiessel H (2009) Thephysics behind the larger scale organization of DNA ineukaryotes. Phys Biol 6:025008

Fang FC, Rimsky S (2008) New insights into transcriptionalregulation by h-ns. Curr Opin Microbiol 11:113–120

Fudenberg G, Mirny LA (2010) Statistical properties of thechromosome capture data for human and yeast cells

Gennes PGD (1979) Scaling concepts in polymer physics.Cornell University Press, Ithaca

Grosberg (2000) Critical exponents for random knots. Phys RevLett 85:3858–3861

Grosberg AY, Khokhlov AR (1994) Statistical physics ofmacromolecules. AIP, New York

Grosberg AIU, Khokhlov AR (1997) Giant molecules: here,and there, and everywhere-. Academic, San Diego

Grosberg AY, Nechaev SK, Shakhnovich EI (1988) The role oftopological constraints in the kinetics of collapse ofmacromolecules. J Phys 49:2095–2100

Grosberg A, Rabin Y, Havlin S, Neer A (1993) Crumpledglobule model of the three-dimensional structure of DNAEPL. Europhys Lett 23:373

Haber JE, Leung WY (1996) Lack of chromosome territorialityin yeast: promiscuous rejoining of broken chromosomeends. Proc Natl Acad Sci USA 93:13949–13954

Hubner MR, Spector DL (2010) Chromatin dynamics. AnnuRev Biophys 39:471–489

Imakaev M, Mirny LA (2010) Fractal globule as model ofchromatin architectute: biophysical properties

International Cancer Genome Consortium (2010) Interna-tional network of cancer genome projects. Nature464:993–998

Jun S, Wright A (2010) Entropy as the driver of chromosomesegregation. Nat Rev Microbiol 8:600–607

Khokhlov AR, Nechaev SK (1985) Polymer chain in an arrayof obstacles. Phys Lett A 112:156–160

Kind J, Van Steensel B (2010) Genome-nuclear laminainteractions and gene regulation. Curr Opin Cell Biol, May

50 L.A. Mirny

Kruhlak MJ, Celeste A, Dellaire G et al (2006) Changes inchromatin structure and mobility in living cells at sites ofDNA double-strand breaks. J Cell Biol 172:823–834

Lieberman-Aiden E, Van Berkum NL, Williams L et al (2009)Comprehensive mapping of long-range interactionsreveals folding principles of the human genome. Science326:289–293

Lua RC, Grosberg AY (2006) Statistics of knots, geometry ofconformations, and evolution of proteins. PLoS ComputBiol 2:e45

Lua R, Borovinskiy AL, Grosberg AY (2004) Fractal andstatistical properties of large compact polymers: a compu-tational study. Polymer 45:717–731

Marko JF (2008) Micromechanical studies of mitotic chromo-somes. Chromosome Res 16:469–497

Mateos-Langerak J, Bohn M, De Leeuw W et al (2009)Spatially confined folding of chromatin in the interphasenucleus. Proc Natl Acad Sci USA 106:3812–3817

Metzler R, Hanke A, Dommersnes PG, Kantor Y, Kardar M(2002) Equilibrium shapes of flat knots. Phys Rev Lett88:188101

Miele A, Dekker J (2009) Mapping cis- and trans- chromatininteraction networks using chromosome conformationcapture (3c). Methods Mol Biol 464:105–121

Müller WG, Walker D, Hager GL, Mcnally JG (2001) Large-scale chromatin decondensation and recondensation regu-lated by transcription from a natural promoter. J Cell Biol154:33–48

Müller WG, Rieder D, Kreth G et al (2004) Generic features oftertiary chromatin structure as detected in natural chromo-somes. Mol Cell Biol 24:9359–9370

Münkel C, Eils R, Dietzel S et al (1999) Compartmentalizationof interphase chromosomes observed in simulation andexperiment. J Mol Biol 285:1053–1065

Nasmyth K, Haering CH (2009) Cohesin: its roles andmechanisms. Annu Rev Genet 43:525–558

Naumova N, Dekker J (2010) Integrating one-dimensional andthree-dimensional maps of genomes. J Cell Sci 123:1979–1988

Ng K, Pullirsch D, Leeb M, Wutz A (2007) Xist and the orderof silencing. EMBO Rep 8:34–39

Ohlsson R, Göndör A (2007) The 4c technique: the’rosetta stone’for genome biology in 3d? Curr Opin Cell Biol 19:321–325

Ohlsson R, Bartkuhn M, Renkawitz R (2010) Ctcf shapeschromatin by multiple mechanisms: the impact of 20 yearsof ctcf research on understanding the workings ofchromatin. Chromosoma, Feb

Petrushenko ZM, Cui Y, She W, Rybenkov VV (2010)Mechanics of DNA bridging by bacterial condensinmukbef in vitro and in singulo. EMBO J 29:1126–1135

Phillips JE, Corces VG (2009) Ctcf: master weaver of thegenome. Cell 137:1194–1211

Reith D, Virnau P (2010) Implementation and performanceanalysis of bridging monte carlo moves for off-latticesingle chain polymers in globular states. Comput PhysCommun 181:800–805

Rosa A, Everaers R (2008) Structure and dynamics ofinterphase chromosomes. PLoS Comput Biol 4:e1000153

Routh A, Sandin S, Rhodes D (2008) Nucleosome repeat lengthand linker histone stoichiometry determine chromatin fiberstructure. Proc Natl Acad Sci USA 105:8872–8877

Rubinstein M, Colby RH (2003) Polymer physics. OxfordUniversity Press, Oxford

Rust MJ, Bates M, Zhuang X (2006) Sub-diffraction-limitimaging by stochastic optical reconstruction microscopy(storm). Nat Methods 3:793–795

Sachs RK, Van Den Engh G, Trask B, Yokota H, Hearst JE(1995) A random-walk/giant-loop model for interphasechromosomes. Proc Natl Acad Sci USA 92:2710–2714

Salceda J, Fernández X, Roca J (2006) Topoisomerase ii, nottopoisomerase i, is the proficient relaxase of nucleosomalDNA. EMBO J 25:2575–2583

Strickfaden H, Zunhammer A, Van Koningsbruggen S, KöhlerD, Cremer T (2010) 4d chromatin dynamics in cyclingcells. Nucleus 1:1–14

Therizols P, Duong T, Dujon B, Zimmer C, Fabre E (2010)Chromosome arm length and nuclear constraints deter-mine the dynamic relationship of yeast subtelomeres. ProcNatl Acad Sci USA 107:2025–2030

Toro E, Shapiro L (2010) Bacterial chromosome organizationand segregation. Cold Spring Harb Perspect Biol 2:a000349

Uversky VN, Dunker AK (2010) Understanding protein non-folding. Biochim Biophys Acta 1804:1231–1264

Van Berkum NL, Dekker J (2009) Determining spatialchromatin organization of large genomic regions using5c technology. Methods Mol Biol 567:189–213

Van Holde K, Zlatanova J (2007) Chromatin fiber structure:where is the problem now? Semin Cell Dev Biol 18:651–658

Vendruscolo M (2007) Determination of conformationallyheterogeneous states of proteins. Curr Opin Struct Biol17:15–20

Vettorel T, Grosberg AY, Kremer K (2009) Statistics of polymerrings in the melt: a numerical simulation study. Phys Biol6:025013

Virnau P, Kantor Y, Kardar M (2005) Knots in globule and coilphases of a model polyethylene. J Am Chem Soc127:15102–15106

Virnau P, Mirny LA, Kardar M (2006) Intricate knots inproteins: function and evolution. PLoS Comput Biol2:e122

Visser AE, Aten JA (1999) Chromosomes as well as chromo-somal subdomains constitute distinct units in interphasenuclei. J Cell Sci 112(Pt 19):3353–3360

Vologodskii A (2009) Theoretical models of DNA topologysimplification by type iia DNA topoisomerases. NucleicAcids Res 37:3125–3133

Weinberg RA (2007) The biology of cancer. Garland Science,New York

Wiggins PA, Cheveralls KC, Martin JS, Lintner R, Kondev J(2010) Strong intranucleoid interactions organize theEscherichia coli chromosome into a nucleoid filament.Proc Natl Acad Sci USA 107:4991–4995

Ye Q, Hu YF, Zhong H et al (2001) Brca1-induced large-scalechromatin unfolding and allele-specific effects of cancer-predisposing mutations. J Cell Biol 155:911–921

Yokota H, Van Den Engh G, Hearst JE, Sachs RK, Trask BJ(1995) Evidence for the organization of chromatin inmegabase pair-sized loops arranged along a random walkpath in the human g0/g1 interphase nucleus. J Cell Biol130:1239–1249

The fractal globule as a model 51