Multiscale Topology of Chromatin Folding

Kevin EmmettDepartments of Physics and

Systems BiologyColumbia UniversityNew York, NY 10032

kje2109@columbia.edu

Benjamin SchweinhartCenter for Mathematical

Sciences and ApplicationsHarvard University

Cambridge, MA 02138bschweinhart@

cmsa.fas.harvard.edu

Raul RabadanDepartments of BiomedicalInformatics and Systems

BiologyColumbia UniversityNew York, NY 10032

rr2579@cumc.columbia.edu

ABSTRACTThe three dimensional structure of DNA in the nucleus (chro-matin) plays an important role in many cellular processes.Recent experimental advances have led to high-throughputmethods of capturing information about chromatin confor-mation on genome-wide scales. New models are needed toquantitatively interpret this data at a global scale. Here weintroduce the use of tools from topological data analysis tostudy chromatin conformation. We use persistent homol-ogy to identify and characterize conserved loops and voidsin contact map data and identify scales of interaction. Wedemonstrate the utility of the approach on simulated dataand then look data from both a bacterial genome and a hu-man cell line. We identify substantial multiscale topology inthese datasets.

Categories and Subject DescriptorsJ.3 [Life and Medical Sciences]: Biology and genetics

Keywordstopological data analysis, chromatin conformation

1. INTRODUCTIONThe 6 billion bases in the human genome would span a lengthof almost two meters if stretched end to end, yet occupya compacted volume inside the nucleus of only a few µm3.Even more remarkably, this million-fold level of compressionis not random, but exhibits a complex hierarchical struc-ture that intimately effects genome function through reg-ulation of gene expression. This multiscale pattern rangesfrom nucleosomes every 150 bases, promoter interactions atthe megabase scale, topologically associated domains at the10 megabase scale, and finally to organization of discretechromosomes [4]. Chromatin conformation is dynamic, andwill change throughout the cellular cycle, under the infuenceof a diverse range of chromatin remodeling proteins, suchas CTCF. Chromatin architecture can further be controlled

epigenetically through post-translational modifications in-cluding methylation and phosphorylation.

Recently developed experimental approaches have providedunprecedented high-throughput access into the three dimen-sional architecture of DNA inside the nucleus [9, 4, 1]. Thesemethods, known as chromosome conformation capture (3C),use next-generation sequencing to probe for enriched physi-cal proximity between nonadjacent genomic loci. Hi-C cou-ples 3C with ultra-deep sequencing to measure genome-wideinteraction patterns in an unbiased manner. However, whilechromatin may fold in three dimensions, Hi-C contact data isonly an indirect representation of these spatial relationships.Several approaches have been developed to use contact mapinformation to generate 3D embeddings of chromatin, how-ever this introduces additional uncertainty in the analysis[1]. Further, the contact map is an average over an ensem-ble of configurations. We would therefore like to directlycharacterize topological properties of the ensemble withoutthe need for such an embedding.

Topological data analysis (TDA) has been applied to severalproblems in genomics [3, 6]. In this brief note we introducethe use of TDA to characterize the complex structure ofchromatin inside the nucleus. Our primary tool is persistenthomology, which extracts global information about geomet-ric and topological invariants in data.

We first demonstrate the approach on data from simulatedpolymer folding. We then consider data from C. crescen-tus, a circular bacterial genome. Finally, we apply our ap-proach to human cell line data, showing how persistent ho-mology can capture complex multiscale folding patterns. Aswe show, tools from topological data analysis may provepowerful at analyzing chromatin interaction data.

2. BACKGROUNDHi-C contact data is generated as follows: First, DNA iscross-linked in formaldehyde, linking segments of chromatinthat are close in spatial proximity. This step links pieces ofchromatin that are in spatial proximity. Second, pieces arefragmented and ligated to form closed loops. Finally, piecesare sheared and sequenced, and the ends of each read aremapped to loci on the genome. The data is summarized asa contact map representing counts of interactions betweennonadjacent loci. For more details, see [4]. From raw fre-quency data, normalization procedures are then applied to

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Figure 1: Three hierarchices of chromatin organi-zation. At the 100 bp scale, DNA chains wraparound protein complexes called nucleosomes. Atthe megabase scale, these chains are compacted intodomains. Lieberman-Aiden et al. proposed closeddomains form a fractal globule structure. At thegenome scale, chromosomes fold into the nucleus inseparate territories. The fractal globule representsdensely packed regions not open for transcription.Fractal globule image from [9]. Reprinted with per-mission from AAAS.

account for biases. Normalization is a difficult problem andseveral such methods have been developed, see [1] for discus-sion. In this work we largely use normalized contact matricesas input. Pearson correlation ρ measures similarity betweenloci, which we convert to a distance as d = 1 − ρ.

Hi-C experiments have identified topologically associateddomains. Existing computational analyses have focused onidentifying significant off-diagonal contacts and associatingthem with specific genomic interactions. Here we focus onthe global scales of chromatin folding.

We use persistent homology to analyze Hi-C contact maps.Persistent homology captures information about loops andvoids in a dataset using homology. Homology informationis tracked across a scale parameter ε via a series of nestedsimplicial complexes (see [2] for more details). Invariantsare summarized in a barcode diagram indexed by homologydimension Hd. Each bar in the diagram, indexed as PHi, isannotated with a birth time, bi, and a death time, di. H1

gives information about looping between loci, and H2 givesinformation about voids. Following [10], we define the sizeof a PH class as

xi =bi + di

2. (1)

The distribution of PH class sizes reflects the scales of foldingobserved. We use Dionysus to compute persistent homology[11].

2.1 Long-Range Chromatin InteractionsLong-range chromatin interactions can manifest in a num-ber of different biological consequences at megabase scales.In Figure 2 we show a cartoon of two possible types of inter-action. On the left we see a single interaction mediated bya binding protein that brings two nonadjacent loci into con-tact. This could reflect a promoter-enhancer interaction, for

A B

Figure 2: Two examples of long range chromatin in-teractions resulting in a topological loop. (A) A pro-tein mediated (red) point-interaction between an en-hancer (green) and promoter (yellow) sequence. (B)A transcription factory consists of dense RNA poly-merase (green) around a structural core in whichadjacent genomic loci (colored segments) will becotranscribed. Transcription factors (purple) areshown.

example. We call this interaction a one-jump loop. On theright, we see a more complex interaction representing mul-tiple nonadjacent loci surrounding a dense compartment ofpolymerase proteins. This phenomenon is known as a tran-scription factory, and genes adjacent to a given factory willhave correlated levels of expression. We call this interactiona multi-jump loop, because the minimal hole generated bythe filtration will span multiple nonadajacent loci.

2.2 Minimal Cycle AlgorithmIt is important to be able to localize a cycle in order toannotate particular loops an interactions. To define a notionof minimal cycle corresponding to a PH class, we first usethe contact map to locate an “essential edge” which a cyclemust contain. To do this, the values of the heatmap areperturbed so that they are unique and there is a well-definedmap from PH birth times to pairs of chromatin segments.That is, we can associate to each PH class (bi, di) a unique“essential edge” that enters the filtration at the birth timebi. We define a minimal cycle corresponding to (bi, di) to beone containing the essential edge that traverses the shortestlength along the genome, and is homologically independentfrom the minimal cycles of all classes born before bi [13].This does not uniquely specify a cycle, and we break ties bypreferring cycles with shorter jumps. Specifically, if x y ... zis homologous to x x+1 y ... z (where x > y) then the latteris considered better. We use a breadth-first search startingwith the essential edge to locate a minimal cycle for a givenPH class. Then, we shorten any jumps if possible.

3. POLYMER SIMULATIONSTo explore the use of topological methods for analyzing chro-matin data, we used code from [5] to simulate equilibriumfolded polymer conformations. The model uses a MonteCarlo approach to simulate chromatin as a one-dimensionalpolymer chain confined to a volume and allowed to cometo an equilibrium conformation. After equilibration, the 3Ddistance between monomers can be used as a measure of thecontact frequency.

In Figure 3 we show the output of one such a simulation.Here, we simulated a 50 megabase chromatin segment as a

H0

H1

A B

Figure 3: Polymer Simulation. (A) 50 Mb polymerwith 10 fixed loops is allowed to reach an equilbriumconformation. (B) PH identifies 10 H1 loops.

chain of 1,000 15 nm monomers. Each monomer correspondsto approximately 6 nucleosomes, or 1200 bp. We inserted10 fixed loops into the chain at random positions on the in-terior of the chain. These loops represent recurrent protein-mediated interactions and mimic chromatin folding patternsobserved in real data. An ensemble of 5,000 conformationswere generated and then averaged to yield the contact mapdepicted on the left. On the right, we show the output ofpersistent homology on the average contact map. Persis-tent homology recovers 10 H1 intervals, consistent with thesimulation and showing that topological information can beextracted from Hi-C-like contact maps.

4. CAULOBACTER DATAWe examined interaction data from Caulobacter crescen-tus as published in [8]. C. crescentus has a 4MB circu-lar genome. In that paper, chromatin interaction domains(CIDs) were identified at scales between 30 to 420 kb. Theauthors proposed a structural model consisting of brush-likeplectonemes arranged along the circular fiber.

Here we look at look at sample GSM1120446, a wildtypeCaulobacter cell. In Figure 4A we show the contact mapdata binned at 10 kb resolution. Clearly identifiable are thestrong interactions along the diagonal, as well as the circularoff-diagonal interactions. In Figure 4B is the barcode dia-gram computed from this contact map. Finally, in Figure 4Cwe see that the size of H1 invariants is strongly bimodallydistributed.

We used the minimal-cycle algorithm to determine a repre-sentative basis for each H1 loop. Figure 5 shows the setof minimal cycles arranged along the genomic axis. Wedivide the loops between small- and large-scale loops asidentified in Figure 4C. On the left, we see that the small-scale loops cover small genomic scales and are regularly dis-tributed along the genome. These small loops may associateto small nucleoid-associated proteins or structural mainte-nance complexes (see [14]), or may simply reflect stochasticfolding. On the right, we see the large-scale loops coverbroader genomic regions (average size 100kb). These loopsdo not associate with the CIDs identified in [8], but ratherreflect larger scale folding patterns.

5. HUMAN DATAWe examined one of the original human Hi-C data sets aspublished in [9]. In that paper, the authors proposed a two-

H0

H1

C

BA

Figure 4: (A) Contact map and (B) barcode dia-gram for Caulobacter. (C) Distribution of H1 barsizes for Caulobacter shows a bimodal scale of fold-ing patterns.

Figure 5: Minimal cycles projected linearly forCaulobacter. Left: small loops distribute uniformlyacross the genome. Right: pattern of large loops,which segregate into two chromosomal domains.

compartment model of chromosomal organization associatedwith open and closed chromatin states and correlated withgene expression patterns. At the megabase scale, they pro-posed a fractal globule model in which nearby loci along thepolymer are spatially proximate in 3D.

We looked at data from GM06690, a healthy human lym-phoblastoid cell line. In Figure 6 we show an example fromchromosome 1 measured at 1 MB resolution. On the leftis the observed contact map. The gray band in the mid-dle represents the position of the centromeres. On the rightis the barcode diagram computed persistent homology. Weobserve substantial structure in both H1 and H2.

In Figure 7 we show the distribution of H1 bar sizes. We ob-serve a strong bimodal structure, representative of two scalesof chromatin folding. This is consistent with the results in[9], which identified topologically associated domains at the10MB scale.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

H0

H1

H2

Figure 6: Hi-C data chromosome 1 from GM06690human cell line data, from [9]. Left: Contact maprepresentation. Right: PH identifies complex mul-tiscale topology.

Figure 7: Distribution of H1 bar sizes for GM06690human cell line data shows a bimodal scale of foldingpatterns.

Because the contact map is at 1 MB resolution, it is toocoarse to capture nucleosome-level folding patterns (200bp).More recent work has yielded Hi-C datasets at kilobase res-olution [7, 12], however at this resolution the contact mapis too large for an efficient persistent homology computationacross the entire genome (or even an entire chromosome). Itis possible that the linear nature of the chain may make aheuristic homology computation feasible.

6. CONCLUSIONSPatterns of chromatin conformation inside the nucleus ex-hibit complex, multiscale structures that are intimately tiedto genome function. Here we have used methods from TDAto characterize the scale and conformation of these struc-tures. TDA is a natural framework to study data of this typebecause there is a clear definition of the ambient embeddingspace. Using simulation we showed that persistent homologycaptures recurrent loops. In real data, we observed multi-scale structures reflecting hierarchical patterns of chromatinorganization. In the present work we have examined only in-trachromosomal interactions. Interchromsomal interactionshave also been reported, however the resulting contact mapsare too large for homology computations at sufficient reso-lution. Future work will focus on identifying heuristics toimprove these calculations.

7. ACKNOWLEDGMENTSKE and RR are supported by NIH grants U54-CA193313-01and R01-GM117591-01. BS is supported by the Center ofMathematical Sciences and Applications at Harvard Univer-sity. The authors thank Robert MacPherson, Arnold Levine,and Nils Baas for useful discussions.

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