A model for the 3D chromatin architecture of pro and eukaryotes

A model for the 3D chromatin architecture of pro and eukaryotes
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  A model for the 3D chromatin architecture of pro and eukaryotes Dieter W. Heermann a,b,c, ⇑ , Hansjoerg Jerabek a , Lei Liu c,d , Yixue Li c,d a Institute for Theoretical Physics, Heidelberg University, Philosophenweg 19, D-69120 Heidelberg, Germany b The Jackson Laboratory, Bar Harbor, ME, USA c Shanghai Institute of Biological Sciences (SIBS), Chinese Academy of Sciences (CAS), 320 Yue Yang Road, Shanghai 200031, PR China d Shanghai Center for Bioinformation Technology (SCBIT), 100 Qinzhou Road, Building 1, Floor12, Shanghai, PR China a r t i c l e i n f o  Article history: Accepted 17 April 2012Available online 5 June 2012 Keywords: ChromosomeFoldingNuclear architectureEntropyPolymerModelling a b s t r a c t Howdochromosomesfoldandhowdoesthisdependontheorganismandtypeof cell?Thisquestionhasbeen addressed by a model and a methodology that uses only very basic constituents to capture the rel-evant features of folding. Key is the dynamic formation of loops within the chromosome. With this andentropy we show that the model is capable to describe the folding of human chromosomes in inter- andmetaphase as well as for the  Escherichia coli  circular chromosomes.   2012 Elsevier Inc. All rights reserved. 1. Introduction The importance of the 3D architecture of chromosomes and thenucleus for the function of the cell has become more and moreapparent in experimental studies (see recent studies related tocancer [1–3] and on expression [4–8]). With this realization came also the insight that loopformationand the possible colocalizationof genes feature prominently in the organization. Of equal impor-tanceistherealizationthatentropy[9–12]drivesmuchofthespa-tial organization.There are several key experiments using various techniques toelucidate the spatial structure. Perhaps the most import experi-mental techniquesarethefluorescenceinsituhybridization(FISH)[13,14] and the chromosome conformation capture (3C) [15–21]that give information on the folding. Chromosome conformationcapture techniques have developed significantly [22] as well asfluorescence in situ hybridization. For example beside one or twocolour FISH now up to 24 colours in Multicolour fluorescenceinsituhybridisation(M-FISH)[23]hasbeendevelopedandtheori-ginal chromosome conformation capture can now be done on agenome wide scale [21].Early studies indicated the existence of loops with a size of about100kb[24]. Ithasnowbecomeclearfromchromosomecon- formation capture studies that loops exist on all scales [21] andthat this is an important aspect of the 3D genome organization,as it implies possible occurrence of the colocalization of genes thatare genomically far apart. However, it is difficult to extract thegeometryfromthesedataastheyonlygivetopologicalinformationof an averaged population and thus may incorporate conflictinginformation.Using multicolour staining of chromosomes it has been estab-lished that chromosomes are separated into distinct chromosometerritories within the nucleus [25]. It has been shown that thismay be a consequence [26] of the fact the chromosomes arelooped.From experiments as well as from modelling a unifying picturefor the 3Dorganization of the genome for prokaryotes and eukary-otes is emerging. More and more evidence is mounting that themajor driving force for the organization is entropy in conjunctionwith the concept of loops [27,26,28–33]. In this paper we showthat a single model based on the idea of dynamic loop formation,i.e., changing co-localization of parts of the chromosome fibre overtime, is able to explain a whole range of experimental results oninter- and metaphase chromosomes for human cells as well as Escherichia coli . We also look at possibilities to describe the exper-iments by different models and show where the shortcomings are. 2. Material and methods The model describes the chromosome fibre on a coarse-grainedspatial level. Hence, details such as individual nucleosomes, meth-ylation and bound protein complexes are all taken into account onan average basis (see below and Appendix A). This is because we 1046-2023/$ - see front matter   2012 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Institute for Theoretical Physics, Heidelberg Univer-sity, Philosophenweg 19, D-69120 Heidelberg, Germany. E-mail address: (D.W. Heermann).Methods 58 (2012) 307–314 Contents lists available at SciVerse ScienceDirect Methods journal homepage:  firstly are lacking the detailed knowledge of all of these and theircorrelation. Secondly, since the information on the folding thatweobtainfromtheexperimentsareaverages wecanonlydescribethe folding on an average basis. Indeed there is a large cell-to-cellvariation in the conformation of individual chromosomes. Thirdly,to describe the conformations on the scale of say 0.5Mb, the de-tailsonthebaseorevenkilobasepairlevelsenteronlyonanaver-age basis. This can be seen from Fig. 1. The small scale details areeliminated in favour of the overall structure. This is shown for ametaphase model conformation. Eventually only a backbone is re-tained yielding the structural information above a certain scale.Dependingonthequestionaddressedmoredetails canbeincorpo-rated, hence the route back is always open.This backbone is the basis for the dynamic loop model [32]. Toincorporate the interaction between proteins and protein com-plexes with the chromatin fibre and the induced colocalizationthe backbone is decorated with sites such that if two of these sitesmeet, they stay colocalized for a specific amount of time. Thismimicsthefactthatproteinandproteincomplexassociationisdy-namic and induces, along with the local conformational changes,global properties of the fibre. Such a specific relation is, for exam-ple, the genomic distance between two fluorescent marks alongthe fibre and their spatial separation [31].We will use this idea, i.e., a backbone that eventually formsloops which themselves form loops so we have loops within loopswithin loops, etc. (cf. Fig. 2) to describe experiments on meta- andinterphase human chromosomes and nuclei as well as for  E. coli where the model is adapted by starting out from a circularbackbone.The effect of the loop formation can be clearly seen in Fig. 3.Shown is the effect that the looping has on the structure of thechromosome. In panel A looping is only present on a small scale,i.e., no long range looping is allowed. In panel B the looping is al-lowed to extend further. The effect is clearly a compaction into asausage-like structure. Loops emanate from a backbone. If on theother hand loops are allowed to form on all scales (panel C) thena clearly confined structure forms. 3. Results Models of the folding are based on the idea that the chromo-some is nothing but a linear polymer (macromolecule) and thusfolding properties could be deduced by considering the statisticsof such polymers mimicking the cell-to-cell variability due to den-sityvariationandotherdynamicstructureswithinthecellnucleus.They differ, however, in the degree of taking the environment intoaccount. van den Engh et al. [34] assume the chromosome to be alinearpolymerinconfinement.Inasimilardirectiongoestheworkby Emanuel et al. [35]. They propose that the chromatin organiza-tion can be explained by a globular state polymer model, i.e., thepolymer has essentially collapsed to form a globule.A step forward is the model by Sachs [36], the random-walk/giant-loop model, where for the first time the interaction of the Fig. 1.  The coarse-graining process is depicted. To establish large scale properties such as the folding the details enter only on an average basis. Shown is a part of aconformationofametaphasechromosome(leftpanel).Theoverall conformationpropertycanbeobtainedbysystematicallyeliminatingsmaller detailssuchasnucleosomesor small scale chromatinloops, as shownin the middle andfinallyin the right panel. Thered ‘‘backbone’’ presentsthe gross structure of the part of thechromosome that canbe used as a model for the overall conformation of a chromosome on a particular scale. Fig. 2.  Thekeyideasthat underlietheDynamicLoopModelare shown. Panel AandBshowthe two fundamental chromosome types: linear and circular. Panel Cshowstheconceptofthedependenceofthespatialdistance R  asafunctionofthegenomicdistance  g  . The looping as shown in panel D influences this distance. As more andmore loops are built the basic difference between linear and circular chromosomesstart to disappear, i.e., the end effects are dominated by the loops.308  D.W. Heermann et al./Methods 58 (2012) 307–314  proteins and protein complexes was taken into account. However,the loops were modelled to be of a specific size so that the overallpropertyofthechainwasstillthatofalinearchain.Alongthesameline is the model by Muenkel and Langowski [37].Grosberg [38] and recently Lieberman-Aiden and co-workers[21] proposed that chromatin is organized as a fractal globule. Incontrasttotheequilibriumglobularstatemodel,suchamodelrep-resents a long-lived, non-equilibrium knotless conformation. Theaverage distance  R  between two loci  g   (see Fig. 2) along the chro-matin fibre is proportional to  g  1 = 3 for a fractal globule, inconsistentwith the levelling-off observed on the scale of the wholechromosomes.TodescribekeypropertiesonthescaleofseveraltensofMb,weassume a homogeneouslooping probability P   between eachpair of monomers along the contour of the polymer. Note that this proba-bility mimics the attachment of protein and protein complexessuch as for example CTCF or cohesin that will enable the loopformation.To explain differences in chromatin compaction on the scale of several genes and linking to their expression we introduce differ-ent local looping probabilities:   The backbone of the chain is given by a simple self-avoidingwalk or random walk polymer.   Two parts of the polymer form loops with a certain probability P  .   Loops are not static but can change in the course of time; theirsize and position are chosen from a broad range.   Averaging is done over the ensemble of possible conformationsas well as over the ensemble of loops.Thus, in contrast to the above discussed models we have intro-duced loops on all scales.Looped polymers show a levelling-off of the mean square spa-tial distance  h R 2 ð  g   ¼ j i   j jÞi ¼ ð 1 = N  Þ P M k ¼ 1 ð r  ð k Þ i   r  ð k Þ  j  Þ 2 , where  M   isthe number of samples and the superscript  k  labels the samplesas a function of the genomic distance  g   between two loci  i  and  j (MSD). This means that the MSD reaches an almost constant pla-teaulevelaboveacertaingenomicdistance.Hencethevolumethatthe chromosome occupies is constraint. For chromatin fibres ineukaryotic cell nuclei, this is observed in FISH experiments [31].For genomic distances above roughly 10 mega base pairs (Mb)the MSD shows a levelling-off in human cells, which indicates thateach chromosome is confined to a certain volume much smallerthan the nuclear volume.This hints to the fact that chromatin can be described as alooped polymer. In the model the degree of confinement, i.e., theplateau level of the MSD, can largely be varied by changing theprobability for loop formation (see Fig. 4). Hence, also the experi-mentally observed differences in chromatin compaction betweentranscriptionally highly and lowly active genomic regions mightresult from variations in chromatin looping. Fig. 3.  Shownareloopedstructureswithvaryingdegreesofloops.Inpanel A,loopsareonlyformeduptoasmallscale. Thisresultsinathinlinearchromosomalstructure.Inpanel B the range of looping is increased, while in C loops on all scales are allowed, resulting in a strongly confined chromosomal structure. P = 0.07P = 0.09P = 0.12P = 0.28 Fig. 4.  Shown is the mean square spatial distance (MSD) as a function of the genomic distance. Above a certain genomic distance the MSD remains almost constant. Theheight of this plateau level is a measure for the confinement of the chromosome and strongly depends on the looping probability of the fibre. Here, the different graphsrepresentchromatinfibreswithdifferentloopingprobabilities: P   ¼  0 : 07(dashedblue),0 : 09(dottedcyan),0 : 12(continuousgreen)and0 : 28(dashed-dottedred).Wesee,thatthe higher the degree of looping is, the lower the plateau level and subsequently the stronger the confinement is. D.W. Heermann et al./Methods 58 (2012) 307–314  309  Alteringtheloopformationprobabilitybyaconstantvaluedoesnot lead to a simple shift of the loop distribution as a function of the genomic distance, but changes the characteristics of the distri-bution. For the distribution of contacts  P  c  ð  g  Þ  between sites alongthe fibre one obtains P  c  ð  g  Þ ¼  ag  a ;  ð 1 Þ where  a  is a constant and  a  a characteristic exponent. This power-law behaviour of the loop probability can also be observed in chro-mosome conformation capture experiments (3C), like 4C, 5C and Fig. 5.  Shown is the probability  P  c   that two loci that are a genomic distance  g   apart come in contact with each other (see Eq. (1)). The different graphs represent chromatinfibres with different looping probabilities  P  . A linear fit of   P  c   in the grey region reveals that the characteristic exponent  a  strongly depends on  P  : for  P   ¼  0 : 07 (dashed blue) a  ¼  1 : 35 ; P   ¼  0 : 09(dottedcyan) a  ¼  1 : 27 ; P   ¼  0 : 12(continuousgreen) a  ¼  1 : 07and P   ¼  0 : 28(dashed-dottedred) a  ¼  0 : 96.Asthegraphswouldbehighlyoverlapping,they were vertically shifted by a constant value in order to better illustrate the differences in the distributions. Fig. 6.  Shown is the probability  P  c   that two loci that are a genomic distance  g   apart come in contact with each other for the case of a self-avoiding polymer in a sphericalconfinement. Depending on the bond length cut-off   L  and the density  q  inside the sphere, the characteristic exponent  a  (see Eq. (1)) of the distribution, determined from alinearfit inthegreyregion, changes: for  L  =0.35 l m, q  ¼  4(dashedblue) a  ¼  1 : 28, for L  =0.2 l m,  q  ¼  8(dottedcyan) a  ¼  1 : 24, for L  ¼  0 : 4 l m, q  ¼  16(continuous green) a  ¼  1 : 06 and for  L  ¼  0 : 45 l m,  q  ¼  24 (dashed-dotted red)  a  ¼  0 : 90.310  D.W. Heermann et al./Methods 58 (2012) 307–314  Hi-C [21]. For human cells for example, the characteristic exponentis around   1 for short genomic distances between roughly 2–10Mb. In the model, the value of   a  depends on the loop formationprobability. The stronger the looping, the higher the value of   a  (c.f.Fig. 5).However, simulations of un-looped self-avoiding walk (SAW)polymers in confinement hint to the fact that the characteristicsof the loop distribution might solely be a result of polymer com-paction. This fact was obtained by confining a SAW polymer in asphere and calculating the bonds between non-adjacent mono-mers by defining a bond length cut-off: all monomers that are clo-ser to each other than the cut-off were declared to be bonded. Thisin-silico procedure mimics 3C-like experiments where formalde-hyde is added to the chromatin in order to cross-link parts of thegenomic fibre that are in spatial proximity. The results show thatthe bond distribution’s characteristic exponent  a  strongly dependson the polymer compaction as well as the bond length cut-off defining the interactions between non-adjacent monomers (c.f.Fig. 6).In Fig. 7 we show a phase diagram exhibiting the characteristicexponent  a  of the contact distribution as a function of the bondlengthcut-off and the monomer density in the sphere the polymeris confined to. Even though chromatin is known to not behave likea SAW polymer, the calculated contact distributions are in agree-mentwiththeresultsfrom3C-likeexperiments.Since a mainlyde-pends on the density and the bond length cut-off, we propose thatthecontactdistributionsobtainedin3C-likeexperimentsprimarilyreflectthechromatindensityinsidethenucleus.Hence,thecharac-teristicexponent a  doesnotallowtomakeconclusionsaboutwhattype of polymer model has to be favoured when describing chro-matin fibres.Finally, the model can also be used to study  E. coli  with its cir-cular chromosome [33]. The results from FISH experiments couldbe successfully be reproduced. 4. Conclusions While various models based solely on linear polymers can de-scribe aspects of some experiments, only models that incorporateloopsarecapableofdescribingmultiplepropertiesof chromatinfi-bres. Among these, the Dynamic RandomLoop Model is capable of  Fig. 7.  Shown is a phase diagram of the characteristic exponent  a  (colour-coded) from Eq. (1) as a function of the chromatin density  q  and the bond length cut-off   L .  a becomes less negative if either the density or the bond length cut-off increases. Fig. 8.  Illustration of the bond fluctuation model. Each monomer sits in the centerof a unit cube (large sphere), rendering all of its eight vertices occupied (smallspheres). Bond vectors are allowed to fluctuate with a maximum distance of  b  ¼  ffiffiffiffiffiffi  10 p   . D.W. Heermann et al./Methods 58 (2012) 307–314  311
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