Active Hydrodynamic Theory of Euchromatin and Heterochromatin

Abstract

The genome contains genetic information essential for cell's life. The genome's spatial organization inside the cell nucleus is critical for its proper function including gene regulation. The two major genomic compartments -- euchromatin and heterochromatin -- contain largely transcriptionally active and silenced genes, respectively, and exhibit distinct dynamics. In this work, we present a hydrodynamic framework that describes the large-scale behavior of euchromatin and heterochromatin, and accounts for the interplay of mechanical forces, active processes, and nuclear confinement. Our model shows contractile stresses from cross-linking proteins lead to the formation of heterochromatin droplets via mechanically driven phase separation. These droplets grow, coalesce, and in nuclear confinement, wet the boundary. Active processes, such as gene transcription in euchromatin, introduce non-equilibrium fluctuations that drive long-range, coherent motions of chromatin as well as the nucleoplasm, and thus alter the genome's spatial organization. These fluctuations also indirectly deform heterochromatin droplets, by continuously changing their shape. Taken together, our findings reveal how active forces, mechanical stresses and hydrodynamic flows contribute to the genome's organization at large scales and provide a physical framework for understanding chromatin organization and dynamics in live cells.

Figures (9)

• Figure 1: Illustration of the cell nucleus, with chromatin compartmentalized into euchromatin (cyan) and heterochromatin (pink). We define the local density of euchromatin and heterochromatin by $\rho_E$ and $\rho_H$, respectively, and their corresponding velocity fields $\boldsymbol{u}_E$ and $\boldsymbol{u}_H$. At a microscopic level, euchromatin is loosely packed and transcriptionally active chromatin. Here, bound molecular motors to the euchromatin exert extensile stresses onto the neighboring aqueous environment, resulting in solvent flows (see inset). In contrast, heterochromatin consists mostly of transcriptionally silent genes, being densely packed by cross-linking molecules, which exert contractile stresses on the surrounding fluid, also resulting in solvent flows as shown in the inset.
• Figure 2: (a) Growth rates $\lambda_C$ of density perturbations for each Fourier mode $k$, perturbed around the homogeneous densities $\bar{\rho}_E$ and $\bar{\rho}_H$; here, we choose $\bar{\rho}_E=\bar{\rho}_H$. (b) Linearly unstable (red) and stable (green) regions, where the stability boundary is shown by the dashed curve. Contour lines in the unstable region indicate the largest unstable mode $k_\star$, while the three points correspond to those in (a) using the same color convention. (c) Density snapshots of the heterochromatin in a three-dimensional periodic domain of equal lengths $L$, by initially applying small density perturbations about the base state $\bar{\rho}_{E,H}=10$. Only high-density regions are shown (see color bar); low-density regions are rendered transparent, with opacity indicated by the grayscale bar. The instability leads to the formation of small heterochromatic droplets, which subsequently coalesce into a single spherical droplet at long times. The red-highlighted slice indicates the two-dimensional plane used in panel (f), and the overlaid arrows indicate the $x$ and $y$ directions. All coordinates are nondimensionalized by the box size $L$, which corresponds to the nuclear diameter. (d) Total interfacial area $A$ of droplets, rescaled by the effective area $A_0 =\sqrt[3]{36\pi }\,V_0^{2/3}$, where $V_0$ is the volume of all droplets at time $t$. The blue stars correspond to the snapshots in (c). At long times, $A/A_0$ relaxes to one; that is, a single spherical droplet. (e) Two-point correlations of density fluctuations from their mean, normalized by their standard deviations, at a radial separation distance $r$, where the average is taken over all possible points separated by $r$ and all snapshots with $t/T\geq{0.20}$. The size of shaded regions corresponds to the standard deviation of the sample. (f) Densities $\rho_E$ and $\rho_H$, and total density $\rho_T = \rho_E+\rho_H$, shown on the two-dimensional slice highlighted in red in panel (c). Streamlines represent the projected velocity fields $\boldsymbol{u}_S$, $\boldsymbol{u}_E$, and $\boldsymbol{u}_H$ on this slice. The streamlines are colored by the magnitude of the velocity, as shown by the insets, whereas the background color indicates the density. Herein, we use $\eta=0.01$, $\bar{\rho}_C = 100$, and $\Gamma=25$.
• Figure 3: (a) Growth rates $\lambda_I$ associated with perturbations in the nematic order parameter $\boldsymbol{Q}$ about the isotropic state $\frac{1}{3}\boldsymbol{I}$ for different values of the activity strength $\alpha$. (b) Two-point correlation of the solvent velocity $\boldsymbol{u}_S$ at a separation $r$, as a function of $\alpha$, normalized by the variance, and averaged over all snapshots with $t/T\geq0.25$. Correlation length $\xi_S$ in units of the system size $L$ is shown in the inset plot. (c) Two-point correlation of the barycentric velocity of the chromatin $\boldsymbol{u}_C = (\rho_E \boldsymbol{u}_E+\rho_H \boldsymbol{u}_H)/\rho_T$ at a separation distance $r$, normalized by the variance, and averaged over all snapshots with $t/T\geq{0.20}$. Its associated correlation length $\xi_C$ (in units of $L$) is shown in the inset. In both (b) and (c), the size of shaded regions corresponds to the standard deviation of the sample. (d) Net interface area $A$ in terms of the spherical area $A_0$ as in Fig. \ref{['fig:figures/figure_2']}(d) for different $\alpha$. For nonzero activity, the ratio $A/A_0$ fluctuates about some mean value which increases as we increase $\alpha$, since at large activity $\alpha$ the shape of the droplets is continuously deformed and occasionally split into smaller droplets. (e) Heterochromatin density for different values of $\alpha$ at snapshots indicated by the stars in (d). The color and opacity scheme and axis conventions are the same as in Fig. 2(c). Herein, $\eta=0.01$, $\bar{\rho}_C = 100$, $\Gamma=25$, and $\bar{\rho}_{E,H}=10$.
• Figure 4: (a) Two-point correlation of fluctuations from the mean of the scalar order parameter $q$, at a radial separation $r$, normalized by the variance. Inset shows the correlation length $\xi_q$ in the scalar order parameter $q$ at increasing values of $\alpha$. (b) Three-dimensional plot of $q$ for different $\alpha$, where only the high values (${q}\geq0.96$) and low values (${q}\leq0.36$) are opaque, while the rest is transparent; see color bar and the corresponding opacity levels. (c) Two-point correlation in fluctuations of the director field $\boldsymbol{n}$ around its mean, at a radial separation $r$, and normalized by the variance. Inset shows the associated correlation length $\xi_n$ of the director field. (d) The highly aligned regions in (b), shown in yellow, are instead colored by the orientation of the director field $\boldsymbol{n}$, as shown by the inset legend. By interpolating between three colors (cyan, magenta, and yellow), antipodal points on the unit sphere are associated with the same color (e.g. pure yellow corresponds to both north and south pole). The dark gray display the high density regions of heterochromatin, as in Fig. \ref{['fig:figures/figure_3']}(e); see color bar of $\rho_H$ and opacity levels. In both (b) and (d), the same axis convention as in Fig. 3(e) is used. Here, we use $\eta=0.01$, $\bar{\rho}_C = 100$, $\Gamma=25$, and $\bar{\rho}_{E,H}=10$.
• Figure 5: (a) Total interface area $A$ of heterochromatic droplets in a spherical nucleus, rescaled by $A_0 =\sqrt[3]{36\pi }\,V^{2/3}_0$, where $V_0$ is the volume of all heterochromatic droplets at time $t/T$. Note that $A/A_0$ does not approach one, even in the absence of activity ($\alpha=0$). Herein, $\eta=0.01$, $\bar{\rho}_C=100$, $\Gamma=25$, $\bar{\rho}_E =10$, and $\bar{\rho}_H =40$. (b) Snapshots of the heterochromatin density $\rho_H$ (see color bar) and its associated projected velocity $\boldsymbol{u}_H$ onto the surface of a spherical cut, with $\alpha=0$. As before, we observe coarsening of the initial droplets; however, at long times, the material is completely localized near the nuclear boundary, wetting the entire surface and forming a spherical shell. The shell thickness is determined by the initial amount of heterochromatin. (c) Time evolution of the averaged radial position of heterochromatin, $\langle r\rangle_H = N_H^{-1}\!\int r\space\rho_H(r,\theta,\varphi)\,\mathrm{d}V$, rescaled by the nucleus radius $R$. (d) Snapshots of $\rho_H$ for nonzero activity strength ($\alpha={250}$), using the same color bar as in (b), and the corresponding projected velocity fields of heterochromatin $\boldsymbol{u}_H$ onto the spherical cuts.