Polymer model with Epigenetic Recoloring Reveals a Pathway for the de novo Establishment and 3D Organization of Chromatin Domains

One of the most important problems in development is how epigenetic domains can first be established, and then maintained, within cells. To address this question, we propose a framework that couples three-dimensional chromatin folding dynamics to a “recoloring” process modeling the writing of epigenetic marks. Because many intrachromatin interactions are mediated by bridging proteins, we consider a “two-state” model with self-attractive interactions between two epigenetic marks that are alike (either active or inactive). This model displays a first-order-like transition between a swollen, epigenetically disordered phase and a compact, epigenetically coherent chromatin globule. If the self-attraction strength exceeds a threshold, the chromatin dynamics becomes glassy, and the corresponding interaction network freezes. By modifying the epigenetic read-write process according to more biologically inspired assumptions, our polymer model with recoloring recapitulates the ultrasensitive response of epigenetic switches to perturbations and accounts for long-lived multidomain conformations, strikingly similar to the topologically associating domains observed in eukaryotic chromosomes.

Popular Summary

Epigenetic modifications are biochemical marks that form patterns along chromosomes and regulate gene expression. The epigenetic patterning of chromosomes enables cellular differentiation, while leaving the underlying DNA sequence unchanged. How these patterns are established and then maintained across cell division is far from understood, however. Here, we show that a polymer model with “recolorable” segments can explain epigenetic memory and the self-organization of long-lived epigenetic domains.

We theoretically consider a model of epigenetic switches that consists of a chromatin fiber modeled as a semiflexible chain made of beads (roughly 30 nm) connected by a harmonic potential. In our model, we assume that two similar epigenetic marks experience self-attractive interactions. We investigate how the folding dynamics of the chromatin in three dimensions is coupled to the creation of epigenetic modifications and how the model transitions between an epigenetically disordered phase and an epigenetically ordered phase. We are able to explain the physically observed phenomenon of “epigenetic memory,” in which genes that are switched off are only infrequently reactivated. In addition, we show that by breaking detailed balance, our model is able to display epigenetic domains with long-lived boundaries that last throughout the simulation runtime of a few hours and that are reminiscent of so-called “topologically associated domains,” recently observed in vivo. In conclusion, we argue that epigenetic modifications are likely modulated by a positive feedback mechanism and that their ATP-mediated regulation is key to control topologically associated domains and, ultimately, gene expression.

We expect that our findings will motivate future studies of how epigenetic domains and, consequently, gene expression are coupled with chromatin folding dynamics.

Figure 1

A 3D polymer model with “recoloring” for the propagation of epigenetic marks. (a)–(c) Multivalent binding proteins, or readers (shaded spheres), bind to specific histone modifications and bridge between similarly marked segments (distinguished here via their “color”). Histone-modifying enzymes, or writers (solid squares), are assumed here to be chaperoned by the bridge proteins. The writing (or recoloring) activity is a consequence of 3D contiguity (perhaps through facilitated diffusion [35]), which is modeled here as a Potts-like interaction between spatially proximate monomers [41] (a). The positive-feedback mechanism and competition between different epigenetic marks results in a regulated spreading of the modifications (b), which, in turn, drives the overall folding of the polymer (c). A sketch of a biological reading-writing machinery is shown in (d). Heterochromatin binding protein HP1 is known to recruit methyltransferase proteins (e.g., SUV39H1), which, in turn, trimethylates lysine 9 on histone 3 (H3K9me3) [29, 39, 42]. Similarly, the Polycomb Repressive Complex (PRC2) is known to comprise the histone H3 Lys 27 (H3K27) methyltransferase enzyme EZH2 [12, 39, 43] while binding the same mark through the interaction with JARID2 [43, 44].

Figure 2

The two-state model above the critical point evolves into an epigenetically coherent state via a symmetry-breaking mechanism. Top row: Typical snapshots of 3D configurations adopted by the polymers as a function of time for two choices of $\alpha =\epsilon /k_B{T}_L$ below and above the critical point ${\alpha }_c\simeq 0.9$ (for $M=2000$; see Ref. [54]). Middle row: Time evolution of the total number of beads of type $q$, $N_b(q,t)$, for four independent trajectories (the dashed one corresponds to the trajectory from which the snapshots are taken). Bottom row: Time evolution of the color of each polymer bead, viewed as a “kymograph” [57]. By tuning $\alpha >{\alpha }_c$, the whole polymer is taken over by one of the two self-attracting states via a symmetry-breaking mechanism. (see also Movies M1 and M2 in Ref. [54]).

Figure 3

The two-state model displays a discontinuous transition at the critical point marked by coexistence. The plot shows the joint probability $P(R_g,\tilde{m})$ for a chain of $M=50$ beads, obtained from 100 independent simulations of duration $10^6{\tau }_{\mathrm{Br}}$ each (1000 recoloring steps) at $\alpha =1.15$ (the critical point for $M=50$). Single trajectories are shown in Ref. [54]. One can readily appreciate that the system displays coexistence at the critical point, therefore suggesting it is a discontinuous, first-order-like transition [see Ref. [54], Fig. S3 for plots of $P(R_g,\tilde{m})$ at other values of $\alpha$].

Figure 4

Within the two-state model, epigenetic dynamics slows down with increasing $\alpha$. (a,b) These panels show the kymographs and the number of beads in state $q$, $N_b(q,t)$, for two values of $\alpha$ above the critical point (${\alpha }_c\simeq 0.9$ for $M=2000$). Counterintuitively, the symmetry breaking of the chain towards an epigenetically coherent state slows down with increasing interaction strengths (compare also with Fig. 2). (c) This panel shows the time evolution of the squared gyration radius $R_g^2$ of the polymer from the moment the collapse starts. (d) This panel (see also Movie 3 in Ref. [54]) shows the behavior of the epigenetic magnetization [defined in Eq. (3)] as a function of time. As expected, larger values of $\alpha$ therefore lead to a faster polymer collapse dynamics (faster decay of $R_g$); surprisingly, however, this is accompanied by a slower recoloring dynamics towards the epigenetically coherent state [slower growth of $m(t)$]. The longevity of the epigenetic domains thereby formed can be quantified by looking at the growth of the epigenetic magnetization. For $\alpha =5$, $m(t)$ can be extrapolated to reach, say, 0.5 at about 3 $10^7$ ${\tau }_{\mathrm{Br}}$, which corresponds to 5000 minutes of physical time according to our time mapping (see Sec. 2).

Figure 5

The network of interactions is short-ranged for fast-collapsing coils. We show a snapshot of the network of bead-bead contacts taken at $t=10^6{\tau }_{\mathrm{Br}}$ for two simulations with (left diagram) $\epsilon =1k_B{T}_L$ and (right diagram) $\epsilon =5k_B{T}_L$. For clarity of visualization, each node of the network coarse grains 10 beads along the chain. The node size and color intensity encode the number of interactions within the coarse-grained monomers. Edges are only drawn between nodes that contain interacting monomers, and their thickness is proportional to the (normalized) number of contacts. To improve the visualization, only edges corresponding to contact probabilities between monomers in the top 30% are displayed. Snapshots of the respective 3D conformations are also shown. It is important to notice that higher values of $\alpha$ lead to short-ranged networks, which translates into fewer edges but larger nodes in this coarse-grained representation.

Figure 6

The two-state–with–intermediate-state model displays ultrasensitive response to external signals such as replication or chromosome inactivation. We show the time evolution of the system starting from a mixed metastable state (MMS) and for $\epsilon =k_B{T}_L$. At $t=0$, a localized perturbation of the MMS is externally imposed by recoloring a segment of 200 beads (10% of the polymer length). This perturbation triggers the collapse of the whole chain into an epigenetically coherent state that is reached within about 4 $10^5$ Brownian times. At $t=4$ $10^5$ ${\tau }_{\mathrm{Br}}$, we next simulate semiconservative replication of the collapsed chromatin fiber. This is achieved by assigning a random color to 50% of the beads all along the polymer. Following this extensive (i.e., nonlocal) color perturbation, the polymer returns to the epigenetically ordered phase. These results show that the epigenetically coherent phase is robust and stable with respect to extensive perturbations, in stark contrast to the much more sensitive MMS. Movie M4 in Ref. [54] shows the whole dynamics. Contact maps are shown in Ref. [54], Fig. S11.

Figure 7

Breaking detailed balance leads to the formation of TAD-like structures. Simulations correspond to $M=2000$, $T_{\text{Rec}}=0.1\epsilon /k_B$, $T_L=2\epsilon /k_B$ (i.e., $\alpha =\epsilon /k_B{T}_L=0.5$, see Ref. [54] for other cases). (a) Plot of the number of red (and blue) colored beads $N_b(q,t)$ as a function of time. Notice that these curves do not seem to diverge within the simulation runtime, opposite to the ones reported in the previous sections. (b) The kymograph of the system showing the presence of long-lived boundaries between distinct epigenetic domains. (c) A contact map averaged over the last 2 $10^5$ Brownian times: The upper half shows the contact probability between beads; the lower half is color-coded to separately show the probability of red-red, blue-blue, and mixed contacts. (d) A snapshot of the 3D configuration. The visible TAD-like structures in the snapshot and in the contact map are enumerated as in the kymograph, to ease comparison. Note that the TAD-like structures are long-lived but metastable, while they coarsen on very long time scales. More details are given in the text and Ref. [54], and other values of $T_L$ are given in Figs. S14 and S15, as well as different initial conditions in Fig. S16. See also Movies.