Physical Modeling of the Heritability and Maintenance of Epigenetic

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Hi-C a for heritability | eoeorganization genome b n nrwJ Spakowitz J. Andrew and , d eateto aeil cec n niern,Safr nvriy tnod CA Stanford, University, Stanford Engineering, and Science Materials of Department b eateto hsc,Safr nvriy tnod A94305; CA Stanford, University, Stanford Physics, of Department f eateto ple hsc,Safr nvriy tnod A94305 CA Stanford, University, Stanford Physics, Applied of Department | is ulse uut1,2020. 10, August published First doi:10.1073/pnas.1920499117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This interest.y the competing wrote no declare A.J.S. authors and per- The S.H.S. A.J.S. and and data; analyzed Q.M., A.J.S. S.H.S., paper. and y research; Q.M., designed S.H.S., research; A.J.S. formed and S.H.S. contributions: Author o ohteipeieihrtneo itn ehlto na silencing gene on of methylation maintenance reliable histone the of and inheritance level nucleosome imprecise the both for epigenetic across varies loci. that and restoration and marks modification strands of daughter kinetics of in the positioning preserved genomic is the modifications that histone find They replication. occupancy modification following replication histone measures after also occupancy that (ChOR-seq), chromatin technique, sequencing a h elcce(3 4.Rever S 24). of (23, during stages cycle subsequent reduced cell during transiently reestablished the gradually are and levels phase methylation histone code epigenetic the maintaining generations. level, across original methyla- its the to returns and density methylation tion counter- nucleo- in strands reduction same daughter initial the the the acts of Methylation maintain unmarked, (21). with to density some filled nucleosomes are synthesized nucleosomes the newly and parental strands, daughter between two repli- the gaps of are the rapidly one nucleosomes on As is parental positioned randomly The DNA genome. (20). replicated chromatin the of into newly of reassembled programming progresses, expression proper machinery and ensure cation identity to cell replication the during cells ter heterochromatic of nature compact 19). (18, and regions isolation physical the HP1α owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To rmwr htcpue outmcaimfrepigenetic for mechanism predictive robust these heritability. a a for captures in behavior that resulting framework observed contributions, experimentally biological the individual on built model Our DNA. is the chromosomal chromosomal within proteins of modifica- control histone posttranslational to tions confer model that that under- those theoretical proteins and well predictive compaction the not a between maintenance currently interplay develop biolog- and are We specific heritability modifications The stood. diseases. the epigenetic mod- of behind of epigenetic range mechanisms in a mul- ical aberrations to which and cells lead types, on ifications identical cell basis genetically various the into differentiate as organisms serves ticellular regulation Epigenetic Significance bfe oe”epaainhsbe rpsdt account to proposed been has explanation model” “buffer A h eut fsal-stp aeigeprmnsso that show experiments labeling stable-isotope of results The daugh- to transferred reliably be must code epigenetic The niae hteieei eeslnigmyb rvnby driven be may silencing gene epigenetic that indicates c,d,e,f,1 PNAS NSlicense.y PNAS | uut2,2020 25, August on-G ´ | . y o.117 vol. mze l 2)developed (25) al. et omez ´ https://www.pnas.org/lookup/suppl/ | o 34 no. c eatetof Department | 20423–20429

BIOPHYSICS AND COMPUTATIONAL BIOLOGY generations. These results qualitatively recapitulate experimen- tal observations that large methylated regions are preserved over generations while small fluctuations are not passed on. Our results demonstrate how chromatin organization, together with methyltransferase activity, could serve as a mechanism for the heritability of the epigenetic code. Thus, our model offers the physical insight needed to draw meaningful biological conclu- sions about the relationship between epigenetic heritability and chromosome organization. Methods Our model for the heritability of the methylation sequence integrates chromosomal compaction and methyl spreading within chromatin. Our model for DNA segregation into heterochromatin and euchromatin is based on the methylation sequence and cooperative HP1 binding (22). We capture the thermodynamics and configuration statistics of this sys- tem using a polymer-based Monte Carlo (MC) simulation. We imple- ment a coarse-graining interaction (37) that allows us to simulate an entire human chromosome with nucleosome-scale discretization. The initial methylation profile is determined from experimental chromatin immunoprecipitation followed by sequencing (ChIP-seq) data (38, 39). To decide which histone tails are methylated, a cutoff is applied to the ChIP-seq signal such that approximately 50% of the histone tails are methylated. In our model, HP1 can bind and unbind to any histone tail, but it pref- erentially binds to methylated tails based on experimental measurements of the binding affinities (11). To capture the effect of the oligomerization of HP1, we include an experimentally derived energetic benefit for regions Fig. 1. (Top) Images demonstrate the degree of chromosomal com- with higher concentrations of HP1 (11), which then leads to segregation paction in euchromatin (Left) and heterochromatin (Right). Cyan and and compaction. For a given HP1 concentration and methylation profile, purple circles represent histone tails that are unmethylated and methy- our model predicts which genomic regions compact to form heterochro- lated, respectively. (Bottom) Image shows results from Hi-C experiments matin. Fig. 2, Top show the structural progression of the chromosome with (16, 17) (upper-right triangle) compared with our physical model for increasing HP1 concentration (where Gen 0 indicates the initial methyl pro- chromosomal organization (lower-left triangle) (22). Chromosome compart- file determined by ChIP-seq data), demonstrating the role that HP1 plays in ments within the Hi-C contact map indicate segregation into heterochro- modulating the chromosomal state. Our model exhibits a behavior similar matin and euchromatin. to that in previous works (40–42) that treat chromatin as a block copoly- mer. Furthermore, polymer models constructed based on epigenetic data accurately predict many genomic contacts as measured by Hi-C experiments at a genomic scale (23). Theoretical models show that coop- (43, 44). However, our model directly incorporates HP1 binding as an essen- erativity and long-range interactions between nucleosomes are tial player in methylation due to the experimentally established connection required to establish robust bistability (26–29). Models that between HP1 and methyltransferase (45, 46). incorporate physically motivated chromatin connectivity, for Our methylation model captures loop-mediated spreading of methyl instance through experimental measurements of contact prob- marks (36) and predicts which nucleosomes become methylated based on abilities in human and Drosophila cells (30) or a semiflexible the spatial arrangement of the chromatin. Methylation may spread from bead-spring polymer (31–33), agree with experimental results methylated or unmethylated nucleosomes, based on the binding of HP1 to each site (11). We write a kinetic equation for the change in methyla- of the extent of epigenetic spreading and the stable coexistence dp tion probability with respect to time i = k(0)hn i(1 − p ) − k p , where of distinct epigenetic domains. They demonstrate that coupling dt m HP1i i d i pi is the probability that the ith nucleosome is methylated (i.e., trimethy- between three-dimensional (3D) folding and one-dimensional (0) lated in our model), km is the bare methyltransferase rate, and kd is the epigenetic spreading leads to bistability and epigenetic memory. demethylation rate. We define hn i as the average number of HP1-bound HP1i Some models find that barrier elements are necessary to keep tails within a cutoff distance a of the ith nucleosome. The looping radius epigenetic marks confined to a local domain (32–34). a = 15 nm is chosen to approximately correspond to half the resolution of These models lay the groundwork for understanding the con- the MC simulation (similar results are obtained for values between a = 5 nm nection between chromatin architecture and epigenetic heri- and a = 35 nm). We specify the function of the methyltransferase based on tability. We extend this foundation by including in our model experimental measurements of spreading around a nucleosome (45, 46) and α = (0)/ differences in the density of nucleosomes per unit volume in define the parameter d km kd as the relative rate of methylation to heterochromatin and euchromatin, which have been observed demethylation. experimentally. Measurements from orientation-independent We combine the MC simulation for the 3D configurations and the methyl spreading calculation from the kinetic equation for methylation differential interference contrast microscopy indicate that het- by alternating between predicting the chromatin organization and the erochromatin is 5.5 to 7.5 times as dense as euchromatin methylation sequence. We use experimental ChIP-seq data as the methy- (35). The addition of differential densities in our model lation profile for an initial chromatin compaction simulation (identified helps explain the mechanistic factors that cause epigenetic as generation 0). We determine hn i from an average over 26 inde- HP1i domains to phase-segregate and thereby preserve the epigenetic pendent structures to obtain a representative nucleosome connectivity sequence. map. This map serves as the input to the methyl spreading master equa- We develop a theoretical model for the reestablishment tion, which we solve for the steady-state methylation profile. We then of epigenetic modifications following DNA replication. Our convert the steady-state profile into a specific methylation sequence by theory is based on our current understanding of heterochro- selecting the methylation status of each nucleosome randomly based on the probabilities pi from the master equation. Next, we use the resulting matin/euchromatin segregation and methylation spreading (22, methylation sequence as the input to another chromatin compaction simu- 36). We capture both the relationship between methylation lation. We repeat this process to observe compaction and reestablishment state and HP1 binding as well as the cooperative interaction over multiple generations, where a generation is one cycle through the between HP1 and methyltransferase. Our model reproduces the MC simulation and reamplification calculation. The specific steps we take robustness of the epigenetic code over the course of multiple are as follows:

20424 | www.pnas.org/cgi/doi/10.1073/pnas.1920499117 Sandholtz et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 ueoe ra ag fcnetain oilsrt h ulsetu of spectrum full the illustrate to concentrations of range broad a over dure 3. step methyla- to the Return on 7) based sequence methylation the probabilities. methylation determine the Randomly for equation 6) master the Solve 5) (This nucleosomes. obtain of 50% to demethylate simulation randomly MC replication, are mimic an tails To run histone 4) sequence, of methylation 50% the that on so Based signal 3) ChIP-seq the to cutoffs corresponding Apply sections 200-bp 2) into 39 ref. from data ChIP-seq the Divide 1) from the concentrations HP1 shows free to Plot con- of one range methylation (Bottom) a of at respectively. for generation methylated ferral versus tails, nucleosomes tails, of both methylated nucle- histone of concentrations designate at fraction both colors three methylated at purple and for and unmethylated tail, 4 gold, are and Cyan, that nm.) 2, osomes 250 0, bars, (Scale generations HP1. at structure matin 2. Fig. adot tal. et Sandholtz inprobabilities. tion equation master the of condition.). equilibrium initial since the 6 of Fig. independent is for necessary only is step binding. HP1 and positions nucleosome equilibrium methylated. nucleosomes. to [ ic h nvv ocnrto fH1i nnw,w efr hsproce- this perform we unknown, is HP1 of concentration vivo in the Since Fraction methylated [HP1] = 0.193 M [HP1] = 0.700 M [HP1] = 1.649 M = HP1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 012345 H1 .8 M [HP1] =0.082 mgsso ersnaiecngrtoso h chro- the of configurations representative show Images (Top) 1.649 Gen 0 Gen 2 Gen 4 Gen 2 Gen Gen 0 (red). µM H1 .0 M [HP1] =0.700 Generation [ = HP1] H1 .4 M [HP1] =1.649 0.082 (blue) µM osbeeieei eair eslc h value the select We behavior. epigenetic possible a raiaini on nteSaoizlbrtr iHbrepository GitHub laboratory Spakowitz the in ( found large-scale is organization their mal Availability. conventional in Data both and differences Materials for major valid the remains despite heterochro- present nuclei organization. within we inverted compartmentalization, compaction local that and as on mechanism primarily well depends the model How- as our heterochromatin (50). Since into domains matin. nuclei compartmentalization chro- euchromatin inverted exhibit large-scale types and and cell the conventional both between ever, in differences organization major mosomal suggests experiments nucleosome range binding. in HP1 fluctuations a and local represent positions for to account heterochro- and us within structures allows chromosomal connectivity nucleosomes of leads average neighboring which an of nucleosomes, Considering number matin. of enhanced condensation an local here to the present on we effects primarily the depend rearrangement, influ- chromosomal will that large-scale dynamics ence model chromosomal of While model a our (22). as segregation build well as heterochromatin experi- (45) we the methyltransferase of on Accordingly, function based observed 46). mentally methylation and (45, condensation chromosomal proximity captures close in osomes euchromatin. chromosomal on methyla- and and heterochromatin efforts H3K9 dynamics into for our methylation segregation achieve focus of to descriptions therefore aim accurate We for attaining we precision). that inaccessible single-nucleosome detail remains (i.e., of scales tion level time facets the these with all over models and dynamics kinetics chromosomal methylation rear- of dynamic capturing considerable Simultaneously undergoes DNA rangement. chromosomal which over days, than faster much are dynamics dynamics. looping methylation DNA that dif- suggest and rate coefficient methylation fusion the for approxi- estimates these Together, is (49). 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HeLa in for the ments chromosome that is for using than suggest timescale In shorter measurements the system. much that the is assume in dynamics we timescales connectivity, relevant averaged numerically the an of the understanding on current based analytically methy- the solved Then, contacts. is 50%. looping of determined equation randomly probability kinetic a are with lation sequence demethylated be methylation of to function ChIP-seq–derived selected a original, nucleosomes as methylated the profile, probability methylation from methylation initial the the determine for To time. equation master the ing unbound of value concentration same moderate (0.700 a the HP1 for approximately generation to successive each returns after nucleosomes methylated of tion ua eragmn n ehlsraig i.2sosthe shows 2 Fig. spreading. methyl and struc- of rearrangement the generations tural over and reestablishment methylation HP1, study we with bound are condensed. fully nucleosomes of is chromosome all concentrations the high regions almost At some increases, HP1, uncondensed. with of HP1 others transition, and fraction of phase condensed the concentration a chromosome undergoes the low, the chromosome and As is low, uncondensed. also HP1 is is of HP1 same with concentration bound the observe nucleosomes the We has concentrations). when 0 HP1 that generation all for that sequence (note methylation structure 2, chromatin Fig. on in illustrate We Discussion and Results Spakowitz the on found is (http://www.stanford.edu/∼ajspakow/ ). dynamics website methylation group for solution equation ter n oedcmnainfrtemas- the for code/documentation and https://github.com/SpakowitzLab) utemr,eprmna vdneo oprmnaiaini Hi-C in compartmentalization of evidence experimental Furthermore, nucle- to confined be to experimentally shown is H3K9 of Methylation many of order the on occur reestablishment and replication of cycles Cell our and data experimental by informed is methodology of choice Our solv- by replication after kinetics reestablishment the investigate also We olwn h rcdr ecie ntepeiu section, previous the in described procedure the Following µM). 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY progression of chromatin structure and the overall fraction of lying fluctuations in the methylation sequence. The fact that methylated nucleosomes over multiple generations. The aver- window averaging is sufficient to capture the local state sug- age methylation level exhibits either amplification or reduction gests that this mechanism for methylation is robust against local with generation, depending on whether the initial chromatin fluctuations in the methylation state. The shape of the methy- state is uncondensed (low HP1 concentration) or condensed lation profile closely resembles that of the number of neigh- (high HP1 concentration). An intermediate HP1 concentration bors, demonstrating a clear connection between chromosomal of [HP1] = 0.700 µM (purple) results in a fraction methylated organization and epigenetic state. that is approximately constant over the generations of reestab- Experimental evidence shows that the epigenetic state is lishment. At this optimal concentration, the structure maintains maintained at genomic length scales despite fluctuations at a “core” and “corona” with different degrees of nucleosome den- individual nucleosomes (23). As Fig. 3 demonstrates, the mech- sity. This difference in nucleosome density results in a disparate anism in our model allows for the overall preservation of number of neighbors that participate in the conferral of methyl condensed/methylated (i.e., transcriptionally repressed) regions marks, causing the condensed core to reestablish its methylation and uncondensed/unmethylated (i.e., transcriptionally active) and the uncondensed corona to maintain its unmethylated state. regions even as small fluctuations are not passed on. Our results At low concentrations of HP1, the entire chromosome is uncon- reflect a mechanism of imprecise inheritance and thus exhibit a densed, resulting in the entire chromosome losing methylation. behavior similar to experimental observations. Conversely, the entire chromosome is condensed at high HP1 While there is some small loss and gain of methylation, overall concentrations, leading to the entire chromosome reestablish- the domain boundaries are remarkably stable. The boundaries ing to a higher fraction methylated than in generation 0. This in that do shift are those that border regions too small to maintain turn leads to further compaction and methylation in subsequent their original epigenetic state, whether methylated or unmethy- generations. lated. We note that unlike experimental ChIP-seq profiles, which The methylation level of a region of ∼100 nucleosomes is a show heterogeneous methylation levels within domains, our better predictor of local compaction state (i.e., located in hete- methylation profile becomes increasingly homogeneous in both rochromatin or euchromatin) than an individual nucleosome’s euchromatic and heterochromatic domains with each passing methylation state (22). We determine the fraction of methy- generation. One possible explanation for this difference is that lated nucleosomes in sliding windows of 101 nucleosomes, with we apply a cutoff to multicell ChIP-seq data to approximate a window centered around each nucleosome. Throughout our a starting methylation profile for a single-cell simulation. The current work, we focus our analysis on the fraction of nucleo- subsequent observed change in part represents relaxation away somes methylated per window rather than the methylation state from the inherently inexact initial condition. In our determi- of individual nucleosomes. nation of the methylation sequence, we applied a cutoff such Fig. 3 shows the methylation profile and number of neighbors that 50% of histone tails would be methylated, but this per- for each successive generation for a subsection of the chromo- centage is not well established and could be different from the some for the optimal HP1 concentration [HP1] = 0.700 µM (see number we prescribed. Furthermore, our model does not cap- SI Appendix, Fig. S1 for methylation profiles over a broader ture all processes in the cell, so it is possible that processes chromosomal region). The height of the methylation profile cor- beyond the scope of our model are contributing to hetero- responds to the fraction of nucleosomes methylated in the sur- geneity in the epigenetic profile. Another, perhaps less likely, rounding window of 101 nucleosomes. The color represents the explanation is that the actual methylation profile is more homo- compaction state of the nucleosome as predicted by the window geneous than the inherently stochastic data produced by the fraction methylated and cutoffs established from Fig. 5 (cutoff ChIP-seq protocol. selection is described later in this section when Fig. 5 is intro- In our model, the increasing homogeneity occurs because of duced). Cyan, gold, and purple colors designate nucleosomes entropic effects. Once a heterochromatic domain has enough that are in euchromatin, the boundary, and heterochromatin, methylation to maintain compaction and an elevated methyla- respectively. Regions with more neighbors tend to maintain their tion level, then the lowest energy and highest entropy state will methylation over time and lie predominantly in heterochromatin occur when methyl marks are more evenly spread among nucleo- or on the boundary. Notably, the window-averaged plots in Fig. 3 somes. Small euchromatic and heterochromatic domains cannot show the local state of the chromosome and do not reveal under- maintain their methylation level when surrounded by a much larger region of the opposite compaction state. Gradually these small domains are consumed by the larger ones, with entropy driving a homogenizing redistribution of methyl marks. That is, small domains that were initially euchromatic and heterochro- matic gain or lose methylation, respectively, and as a result the overall methylation level within the larger domain equalizes. We define the correlation coefficient χ (i.e., the Pearson cor- relation coefficient) for the current methylation state to the generation-0 methylation state

(j ) (0) (j ) (0) hmi mi i − hmi ihmi i χ = (j ) (0) , [1] σm σm

(j ) where mi is the window-averaged methylation (window size of Fig. 3. The evolution of the methylation profile and number of neighbors 101 nucleosomes) centered at the ith nucleosome for the current for a subsection of the chromosome at the optimal concentration of HP1. state at the end of the methylation process for the j th genera- The height of the methylation profile corresponds to the fraction of methy- (0) tion (i.e., mi corresponds to the window-averaged methylation lated nucleosomes in the surrounding window of 101 nucleosomes. The (j ) (0) color represents the compaction state of the nucleosome as predicted by the for generation 0). The methylation SDs σm and σm give the window fraction methylated and cutoffs established from Fig. 5. Cyan, gold, SD of the methylation sequence for the current generation and and purple colors designate nucleosomes that are in euchromatin, boundary generation 0, respectively. Fig. 4 shows the correlation coeffi- between euchromatin/heterochromatin, and heterochromatin, respectively. cient χ versus the concentration of HP1 for reestablishment over

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BIOPHYSICS AND COMPUTATIONAL BIOLOGY 1.0 final methylation state. The average h. . .imeth indicates an aver- age over nucleosomes that are categorized as high-methylation (purple), intermediate-methylation (gold), and low-methylation

 0.8 (cyan). The predicted behavior suggests that degree of methy- lation is positively correlated with the kinetics of reamplifi- cation of the signal, resulting in regions of high methylation reaching their final methylation state faster than regions of 0.6 low methylation. Although experimental data for H3K9me3 are lacking, the kinetics for H3K27me3, another repressive trimethylation modification, are available (25). Experimental 0.4 ChOR-seq measurements find that highly methylated regions return to their original methylation level more quickly than less methylated regions (25), which is the same qualitative trend

Fraction reamplication 0.2 as our results. Conclusion 0.0 Our work demonstrates that methyl spreading and chromatin 0.0 0.5 1.0 1.5 2.0 compaction may act jointly as a mechanism for preserving Time the epigenetic code. Essentially, the proposed mechanism uti- lizes the phase transition associated with heterochromatin com- Fig. 6. Plot of the fraction reamplification γ versus time for nucleosomes paction as a template for the methylation sequence in the categorized as high- (purple), intermediate- (gold), and low-methylation next generation. The preferential binding of HP1 to methylated (cyan). nucleosomes results in cooperative compaction of chromoso- mal regions. HP1 molecules bind more prevalently in regions the right region is primarily purple, indicating that nucleosomes that are highly methylated, which causes those regions to con- that are originally in heterochromatin have a high number of dense into heterochromatin with more nucleosomes in close neighbors and thus remain in heterochromatin. As a result, spatial proximity. The increase in neighboring nucleosomes then nucleosomes will have roughly the same number of neighbors for causes those highly methylated regions to maintain their methy- the transfer of methyl marks and will retain their original methy- lation in the subsequent generation due to the cooperative lation state, whether unmethylated or methylated. Below the interaction between locally bound HP1 and methyltransferase. optimal concentration of HP1, nucleosomes in heterochromatin In contrast, fewer HP1 molecules bind in regions that are less move into euchromatin, losing both neighbors and methylation, methylated, which causes those regions to decondense, driving as inferred from the presence of gold and purple in the left nucleosomes farther apart. Those less-methylated regions main- peak of Fig. 5, Top. For concentrations above the optimal value tain their unmethylated state due to the decrease in neighboring (bottom plot), the presence of gold and cyan in the right peak nucleosomes. By creating such a feedback loop, HP1 may serve suggests that nucleosomes in euchromatin move into heterochro- as a global regulator for the overall methylation level in the matin, gaining both neighbors and methylation. In both cases, chromosome and enable epigenetic domains to persist over gen- the methylation sequence is not maintained. These results pre- erations at the optimal concentration of HP1. This proposed dict that it is the cooperative nature of chromatin compaction mechanism represents a robust strategy for maintaining the epi- and methyl spreading that ensures the stability of the epigenetic genetic code that is not sensitive to large-scale chromosomal sequence over generations. dynamics and cell-to-cell variability in chromosomal organiza- In addition to recapitulating the long-term conservation of tion. Our work illustrates how large methylated domains could epigenetic domains over multiple cell cycles, our model also cap- be passed successfully from one generation to the next, while tures the dynamics of the restoration of methylation after a single small fluctuations in the methylation profile would not be main- replication event. We define the fraction reamplification: tained in the next generation, thus reaffirming a trend seen in experiments. Our model proposes that physical mechanisms * + that operate globally and cooperatively contribute to long-term mi (t) − mi (t = 0) γ = (j ) , [2] epigenetic heritability. mi − mi (t = 0) meth ACKNOWLEDGMENTS. Financial support for this work is provided by the NSF Physics of Living Systems Program (PHY-1707751). S.H.S. and Q.M. where mi (t) is the window-averaged methylation of the ith acknowledge funding support from the NSF Graduate Fellowship program (j ) nucleosome during reamplification, and mi = mi (t → ∞) is the (DGE-1656518).

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