Nonequilibrium Theory of Epigenomic Microphase Separation in the Cell Nucleus

Understanding the spatial organization of the genome in the cell nucleus is one of the current grand challenges in biophysics. Certain biochemical—or epigenetic—marks that are deposited along the genome are thought to play an important, yet poorly understood, role in determining genome organization and cell identity. The physical principles underlying the interplay between epigenetic dynamics and genome folding remain elusive. Here we propose and study a theory that assumes a coupling between epigenetic mark and genome densities, and which can be applied at the scale of the whole nucleus. We show that equilibrium models are not compatible with experiments and a qualitative agreement is recovered by accounting for nonequilibrium processes that can stabilize microphase separated epigenomic domains. We finally discuss the potential biophysical origin of these terms.

Figure 1

Phase diagram at the nuclear scale. (a) Sketch of the epigenetic and folding dynamics of magnetic polymers. (b) Equilibrium phase diagram of the free energy density [Eq. (1)]. The three equilibrium phases are UD, uniform ($n=n_0$) and epigenetically disordered ($m^2=0$); UO, uniform ($n=n_0$) and epigenetically ordered ($m^2>0$); and DO, demixed and epigenetically ordered ($n=n_{+}$, $m^2>0$ and $n=n_{-}$,$m^2=0$). The dotted line marks the critical value of the coupling ${\chi }_c(n_0)=a/n_0$, the solid lines identify the boundaries of the coexistence region (binodals) and the dashed lines identify the spinodal region. Insets report representative snapshots from BD simulations performed at monomer density $\rho$ and self-affinity $\epsilon$ (see Supplemental Material [36] and Ref. [17] for details). The value of the parameters used for the snapshots are as follows: $\mathrm{UD}=(\rho {\sigma }^3=0.5,\epsilon /k_{B}T=0.7)$; $\mathrm{UO}=(\rho {\sigma }^3=0.8,\epsilon /k_{B}T=1)$; $\mathrm{DO}=(\rho {\sigma }^3=0.1,\epsilon /k_{B}T=0.9)$. See also Supplemental Material [36], movies M1–M2 and BD M1–M3.

Figure 2

Dynamics and reorganization of epigenomic domains. (a) Snapshots following the quench $Q_1$ from UD to UO ($\chi =0\to \chi =1$ at $n_0=2$). (b) Snapshots following the quench $Q_2$ from UD to DO ($\chi =0\to \chi =3.5$ at $n_0=0.5$). Top rows in (a)–(b) show the evolution of Eq. (2) on a $100\times 100$ grid. Bottom rows show the evolution through BD simulations $(\mathrm{UD}\to \mathrm{UO},\rho {\sigma }^3=0.8,\epsilon /k_{B}T=1;\mathrm{UD}\to \mathrm{DO},\rho {\sigma }^3=0.1,\epsilon /k_{B}T=0.9)$. Ref. [17] gives more details on the BD simulations. (c) Scaling of the mean epigenetic domain (units of lattice size $dl$) computed as the average size of clusters of connected lattice sites with magnetization $|m|>1$ for $Q_{1,2}$, and $|m|>2$ for $Q_3$ ($\chi =0\to \chi =6$ at $n_0=1$). See also Supplemental Material [36], movies M1–M2 and BD M1–M3. The choice of the threshold for $|m|$ is purely for aesthetics and does not change the scaling. The other parameters are set to ${\kappa }_m={\kappa }_n=5$ and ${\mathrm{\Gamma }}_m={\mathrm{\Gamma }}_m=0.1$.

Figure 3

Nonequilibrium chromatin switching creates microphase separated epigenomic domains. (a) The switching rates are set so that any part of the genome can become inactive (at rate ${\sigma }_i$), but an inactive region must reactivate when neutral (at rate ${\sigma }_a$). This choice sets up a current in epigenetic states that breaks detailed balance and time-reversal symmetry. (b) Snapshot from a BD simulation of a melt of polymers with switching rate ${\kappa }_s={10}^{-4}{\tau }_B^{-1}$ (${\tau }_B$ is the diffusion time of a monomer [17]), monomer density $\rho =0.8{\sigma }^{-3}$ and $ϵ/k_B{T}_L=0.9$. The arrows denote a possible loop arising in steady state in BD simulations. (c) Time-dependent snapshots obtained evolving Eqs. (3) with ${\mathrm{\Gamma }}_m={\mathrm{\Gamma }}_n=0.1$, $k_m=k_n=5$, $\chi =6$, $n_0=1$, ${\sigma }_a={\sigma }_i=0.1$. (d) Steady state configurations obtained with the same parameters except for the switching rates, which are given in the figure. (e) Log-log plot of mean epigenetic domain size as a function of time and for different switching rates. The inset shows (again in log-log) the dependence of the average size in steady state against $\sigma ={\sigma }_a={\sigma }_i$. See also Supplemental Material [36], movies M3-M9.