The interplay of chromatin phase separation and lamina interactions in nuclear organization

Abstract

The genetic material of eukaryotes is segregated into transcriptionally active euchromatin and silent heterochromatin compartments. The spatial arrangement of chromatin compartments evolves over the course of cellular life in a process that remains poorly understood. The latest nuclear imaging experiments reveal a number of dynamical signatures of chromatin that are reminiscent of active multiphase liquids. This includes the observations of viscoelastic response, coherent motions, Ostwald ripening, and coalescence of chromatin compartments. There is also growing evidence that liquid-liquid phase separation of protein and nucleic acid components is the underlying mechanism for the dynamical behavior of chromatin. To dissect the organizational and dynamical implications of chromatin’s liquid behavior, we have devised a phenomenological field-theoretic model of the nucleus as a multiphase condensate of liquid chromatin types. Employing the liquid chromatin model of the Drosophila nucleus, we have carried out an extensive set of simulations with an objective to shed light on the dynamics and chromatin patterning observed in the latest nuclear imaging experiments. Our simulations reveal the emergence of experimentally detected mesoscale chromatin channels and spheroidal droplets which arise from the dynamic interplay of chromatin type to type interactions and intermingling of chromosomal territories. We also quantitatively reproduce coherent motions of chromatin domains observed in displacement correlation spectroscopy measurements which are explained within the framework of our model by phase separation of chromatin types operating within constrained intrachromosomal and interchromosomal boundaries. Finally, we illuminate the role of heterochromatin-lamina interactions in the nuclear organization by showing that these interactions enhance the mobility of euchromatin and indirectly introduce correlated motions of heterochromatin droplets.

Significance

The latest imaging experiments have revealed a surprisingly dynamic and stochastic nature of chromatin in eukaryotic nuclei, which is reminiscent of multiphase fluid behavior. To understand and disentangle the complex interplay of forces that contribute to the emergent patterns of organization and dynamics, we have devised a phenomenological field-theoretic model of the nucleus as a multiphase condensate of liquid chromatin types. Armed with a mesoscopic model of nuclear chromatin, we have shed light on the distinct dynamical and structural contributions of chromatin type interactions intermingling of chromosomal territories and lamina binding. We also shed light on the dynamics and coherent motions of chromatin domains which are fully captured by an interplay of microphase separation of chromatin types and lamina binding.

Introduction

The nuclei of eukaryotic organisms have hierarchically organized genomes, the structure, and dynamics of which correlate with the phenotypes and cellular functions (1,2). At the top of the hierarchical order, the nuclear space is partitioned into chromosome territories within which one finds deeper layers of polymeric chromatin order, including microcompartments, nanoscale subcompartments, and loops (3, 4, 5). At the nanometer resolution, chromatin is seen as an amorphous and dynamic organization of polynucleosomal arrays, the nature of which is poorly understood (6,7). At micron scales, however, chromatin shows a relatively simpler picture which is commonly rationalized in terms of two distinct mesoscopic states of chromatin (8): a diffuse genetically active euchromatin (EC) and dense genetically inactive heterochromatin (HC).

The Hi-C experiments reveal the partition of the three-dimensional genome in chromosome compartments of type A and type B, decorated by distinct epigenetic landscapes as revealed by complementary analysis of one-dimensional tracks of histone marks and DNA-protein interactions (9). Polymer simulations using the A/B type decorated copolymeric sequences have shown that one-dimensional sequence of A/B types contain sufficient information for recapitulating both the three-dimensional chromosomal folds and the dynamics at the EC/HC borders (10, 11, 12, 13, 14). It is known that concentrations of EC versus HC in the nuclei are generally time and cell line dependent (15). The HC is known to phase separate into facultative HC (fHC) and constitutive HC (cHC) forms with the former typically residing in the nuclear periphery because of attractive interactions with nuclear lamins whereas the latter is localized in chromocenters (13,16). This spatial organization of EC/HC regions is what is known as the conventional nuclear architecture. A notable exception to the conventional architecture has been found in the nuclei of nocturnal mammals that display inverted architectures (15,17,18) with HC concentrating at the core of the nucleus. Given that the nuclear organization is not random one naturally anticipates a physical connection between EC/HC distribution and its temporal dynamics. The superresolution imaging techniques such as Hi-D (19), reveal a heterogeneous variation of chromatin organization with dynamics that appears to be more consistent with a picture of a multiphase liquid condensate (20, 21, 22, 23).

The reports of ubiquitous condensations of proteins and nucleic acids in the nucleus through liquid-liquid phase separation (24, 25, 26, 27, 28) leave little doubt that multiple aspects of nuclear structure and function are dictated by the liquid behavior and multivalent interactions of nuclear proteins and RNA with the chromatin (29, 30, 31, 32, 33). For instance, the Drosophila variant of HC protein-1 (HP1a), which binds to nucleosomes and plays important roles in HC regulation and maintenance (34, 35, 36), has been shown to undergo liquid-liquid phase separation both in vitro and in vivo (37,38). Recent experiments have also shown that HP1a facilitates the formation of HC domains in early embryos and that partial knockdown of HP1a can lead to major changes in chromatin architectures manifesting in reduced contact frequency within HC regions and increased cross-talk between active and inactive domains (37). The HP1a depletion in differentiated cells, however, does not impact chromatin architecture which suggests distinct roles for HP1a during the embryonic and differentiated stage of nucleus. Interestingly the human analog $HP1\alpha$, has only a weak capacity to form liquid droplets with the chromatin compartmentalization and maintenance taking place largely independent of $HP1\alpha$ (39). Thus, while chromatin appears organized in the nucleus through the formation of liquid droplets containing active and inactive genes, the detailed molecular level nature of chromatin compartmentalization is rather nuanced with dependence on cell line and cell cycle (40, 41, 42, 43).

Abstracting away from the underlying molecular complexity of the nucleus and looking at the mesoscopic scales, chromatin appears to move and display dynamics reminiscent of liquids (30,42,44,45) even though each chromosome is mostly fixed in its territory like a solid body (46,47). The notion of “liquid chromatin” is supported by a number of observations; the viscoelastic response of chromosomal loci (11,47, 48, 49), coherent motions of chromatin domains (49,50), the coalescence and Ostwald ripening of chromatin droplets (51,52), the existence of epigenetic zonations and chains of interlinked ∼200- to 300-nm-wide chromatin domains reminiscent of polymer melts (23). The presence of nonequilibrium, motorized ATP-driven processes is also being reported to modulate chromatin dynamics which manifests in the form of ATP-dependent flows, driven fluctuations, and anomalous diffusion coefficients of chromatin loci (50,53).

To dissect the role of the chromatin’s liquid behavior on the global order and dynamics of nuclear chromatin domains in this work, we develop a field-theoretic description of the nucleus as a liquid condensate of chromatin types. By carrying out simulations with multiphase liquid chromatin model of nucleus, we elucidate a number of recently observed features in nuclear imaging experiments. Namely, we show 1) conditions favoring the segregation of HC droplets versus connected mesoscale chain states of chromatin domains (15,23), 2) the role of fluctuations and heterogeneous diffusive dynamics of chromatin loci (50,52,53), 3) emergence of coherent motions across nuclear compartments (49), and 4) the impact of HC-lamina preferential positioning on intranuclear chromatin order (54). The present model represents a major methodological and conceptual advance relative to a previous study (55), where a simple binary fluid model of nucleus was used as a proof of principle study of the impact of nuclear volume on the long-timescale global ordering of HC occurring on developmental scales without regard to fine dynamics of HC. In the present formulation, we adopt a higher resolution view of HC to study multiphasic stochastic dynamical processes happening on the time scale of 1–10 s during the interphase and compare with nuclear imaging experiments.

A phase field theory for liquid chromatin condensates

In this section, we describe the physical foundation and the mathematical formulation of the field-theoretic model of nucleus as a multiphase condensate of chromatin types. The present model is partially based on the previous mesoscale formulation of nucleus proposed by us (55), which has introduced a single liquid chromatin type for studying chromatin boundary fluctuations and territorial compartmentalization during growth and senescence of Drosophila nucleus (34,35,56). In the present formulation, the nucleus is resolved at a chromatin subtype level corresponding to fHC/cHC and EC forms (Fig. 1). The primary driving forces for emergent nuclear architecture and dynamics are derived from microphase separation of HC subtypes, surface tension of chromatin droplets, and differential affinity for chromatin-lamina interactions. Volume and surface constraints are imposed on chromatin types for capturing chromosomal and nuclear boundaries. Given the dense, active, and heterogeneous nature of nuclear chromatin, it is worth highlighting the advantage of field-theoretic description which manages to avoid the notorious glassy states encountered in the particle-based polymer simulations, thereby facilitating the study of long-timescale chromatin dynamics and patterning at the scale of whole nucleus (57,58).

Figure 1.

Schematic representation of the mulitphase liquid chromatin model of nucleus. Shown are the key physical interactions that make up the global nuclear free energy functional. Namely, the chromosome territorial interactions (CTs), interactions between cHC and fHC types (cHC-cHC, cHC-fHC, and fHC-fHC) and the lamina interaction which is modeled via surface gradient terms modulated by ${\gamma }_1$ (cHC-lamina) and ${\gamma }_2$ (fHC-lamina). To see this figure in color, go online.

The field theoretic description of nucleus in the present work is best described in terms of experimentally motivated constraint on shapes and sizes of nucleus and chromosomes which are supplemented by the physically motivated interaction terms accounting for polymer intermingling and movement within and between chromosomal territories. To define the shape of the nucleus, we introduce an auxiliary order parameter η that takes 0 inside the nucleus and 1 outside and varies smoothly between these two values through the interfacial region. This region represents the nuclear envelope (NE) whose position is given by the isocontour $\eta =1/2$. The field η is used as an indicator function independent of time to model a fixed nucleus with volume $V_N$. Because the NE is assumed here at the equilibrium state during interphase of the cell cycle, it can be represented by a tanh-like profile that corresponds to the analytical shape of a diffuse planar interface that separates the two bulk phases 0 and 1. Thereby, we simulated the oblate nuclear shape typical of eukaryotic nuclei during interphase by the use of the following expression: $\eta \left(r\right)=\frac{1}{2}\left[1-\text{tanh}\left(r/\left(2\sqrt{2}{\epsilon }_{\eta }\right)\right)\right]$; where $r=\sqrt{\left({\left(x/a\right)}^2+{\left(y/b\right)}^2\right)}$ is the distance from the center of the physical domain $\text{Ω}$, whereas a and b are the semimajor and semiminor axes. The width of the NE is given by the constant $2\sqrt{2}{\epsilon }_{\eta }$.

Within the nucleus, the N-chromosome territories are described by an N-dimensional vector of nonconserved order parameters $\mathit{\phi }\left(\mathbf{r},t\right)=\left\{{\left\{{\phi }_i\right\}}_{i=1,\dots ,N}\right\}$. The microphase separation of A, B, and C chromatin types within chromosome territories is resolved through two additional nonconserved order parameters $\mathit{\psi }\left(\mathbf{r},t\right)=\left\{{\left\{{\psi }_j\right\}}_{j=\mathrm{1,2}}\right\}$, which quantify the epigenetic state of the chromosomal region. Similarly, the phase-field variables ${\phi }_i\left(\mathbf{r},t\right)$, $i=1,\cdots ,N$, and ${\psi }_j\left(\mathbf{r},t\right)$, $j=\mathrm{1,2}$, vary smoothly across their interfaces profile between two values that it is 1 inside its domain and 0 elsewhere.

The dynamics of the chromatin compartmentalization patterns is derived from the free energy functional $\mathcal{F}$ which describes the intranuclear phase separation of chromatin subtypes. The free energy functional employed in the present study for modeling short-timescale nuclear chromatin is a significant extension of our previous model (55), which includes a number of modifications to account for HC phase separation and HC-lamina interactions. The bulk of the present free energy function can be split into two free energy functionals $F_B$ and $F_I$.

The Ginzburg-Landau free energy functional $F_B$ describes the coexistence of two phases associated to each phase-field variables completed by volume constraints terms to ensure the shapes change is given by the following equations:

$$\begin{array}{l}{F}_B[\mathit{\phi },\mathit{\psi }]=\underset{\text{Ω}}{\int }d\text{Ω}[f\left(\mathit{\phi },\mathit{\psi }\right)+\stackrel{N}{\sum _{i=1}}\frac{{\epsilon }_{\phi }^2}{2}{\left(\nabla {\phi }_i\right)}^2\\ +\stackrel{2}{\sum _{j=1}}\frac{{\epsilon }_{\psi }^2}{2}{\left(\nabla {\psi }_j\right)}^2]\\ +a_1{[V_N-\stackrel{N}{\sum _{i=1}}{V}_i\left(t\right)]}^2+a_2\stackrel{N}{\sum _{i=1}}{[V_i\left(t\right)-{\overline{V}}_i\left(t\right)]}^2\\ +a_3\stackrel{N}{\sum _{i=1}}{[v_i\left(t\right)-{\overline{v}}_i\left(t\right)]}^2+a_4\stackrel{N}{\sum _{i=1}}{[w_i\left(t\right)-{\overline{w}}_i\left(t\right)]}^2,\end{array}$$ (1)

where $f\left(\mathit{\phi },\mathit{\psi }\right)$ is the bulk free energy contribution for multiphase field variables, and gradients terms accounting for the presence of different interfaces in the system and contributing to the interfacial energies. The gradient parameters ${\epsilon }_{\phi }$ and ${\epsilon }_{\psi }$ are controlling the thickness of the interface profile of $\mathit{\phi }$ and $\mathit{\psi }$, respectively. For the bulk free energy density, we use a multiwell potential expressed as: $f\left(\mathit{\phi },\mathit{\psi }\right)={\sum }_{i=1}^N{\phi }_i^2{\left(1-{\phi }_i\right)}^2/4+{\sum }_{i=1}^2{\psi }_i^2{\left(1-{\psi }_i\right)}^2/4$. The terms proportional to $a_i$ account for volume constraints required to enforce the volume of the chromosomal territories at their prescribed values ${\overline{V}}_i$, the fHC at ${\overline{v}}_i$, and the cHC at ${\overline{w}}_i$. The parameters $a_1,a_2,a_3,\text{and}{a}_4$ are positive coefficients that control the thermodynamic driving forces of coarsening processes of different compartments present in the nucleus. The volumes of the i-chromosomal territory $V_i\left(t\right)$, fHC and cHC domains within each chromosome $v_i\left(t\right)$ and $w_i\left(t\right)$ are defined as the spatial integral over the physical domain $\text{Ω}$ of their interface profiles given by the associated phase-field variables ${\phi }_i\left(\mathbf{r},t\right)$, ${\psi }_1\left(\mathbf{r},t\right)$, and ${\psi }_2\left(\mathbf{r},t\right)$. Using the usual phase-field approximation of the volume could change the coexistence phases values defined by the phase-field variables in 0 and 1. We thus used an interpolation function, defined as $h\left({\phi }_i\right)={\phi }_i^3\left(10-15{\phi }_i+6{\phi }_i^2\right)$, for approximating the volumes of different compartments of the nucleus while keeping the position of the local free energy minima at the coexistence phase values. Thus, the volumes of these domains are approximated by the following expressions: $V_i\left(t\right)=\underset{\text{Ω}}{\int }d\text{Ω }h\left({\phi }_i\right);v_i\left(t\right)=\underset{\text{Ω}}{\int }d\text{Ω }h\left({\psi }_1\right)h\left({\phi }_i\right);\text{and}{w}_i\left(t\right)=\underset{\text{Ω}}{\int }d\text{Ω }h\left({\psi }_2\right)h\left({\phi }_i\right)$.

Next, we define the free energy functional $F_I$ which accounts for the geometrical constraints on the nucleus, excluded volume interactions between different domains within the nucleus, and the interaction between heterochromatic subtypes with the NE. The functional $F_I$ is expressed as:

$$F_I\left[\eta ,\mathit{\phi },\mathit{\psi }\right]={\beta }_0\stackrel{N}{\sum _{i=1}}\underset{\text{Ω}}{\int }d\text{Ω}h\left(\eta \right)\left(1-h\left(\eta \right)\right)h\left({\phi }_i\right)+{\beta }_{\phi }\stackrel{N}{\sum _{i\ne j}}\underset{\text{Ω}}{\int }d\text{Ω}h\left({\phi }_i\right)h\left({\phi }_j\right)+{\beta }_{{\psi }_1,{\psi }_2}\underset{\text{Ω}}{\int }d\text{Ω}h\left({\psi }_1\right)h\left({\psi }_2\right)+\underset{\text{Ω}}{\int }d\text{Ω}\left[1-\stackrel{N}{\sum _{i=1}}h\left({\phi }_i\right)\right]\left[{\beta }_{{\psi }_1}h\left({\psi }_1\right)+{\beta }_{{\psi }_2}h\left({\psi }_2\right)\right]+\underset{\text{Ω}}{\int }d\text{Ω}g\left(\nabla \eta ,\nabla \mathit{\psi }\right).$$ (2)

The first term in $F_I$ corresponds to the energy penalty reflecting a geometrical constraint on the nuclear volume that is required to restraint nuclear components’ motion inside the nucleus. The other terms represent the excluded volume interactions between chromosome territories, fHC-cHC regions interactions and HC/EC regions mixing affinity, with the interactions’ strengths described by the parameters ${\beta }_{\phi },{\beta }_{{\psi }_1,{\psi }_2},{\beta }_{{\psi }_1},\text{ and }{\beta }_{{\psi }_2}$, respectively. The last term represents the interaction of the fHC and cHC with the NE though a g-function. This function represents the local Lamina-interaction energy contribution to the free energy functional, and it is expressed as the following equation:

$$g\left(\nabla \eta ,\nabla \mathit{\psi }\right)=\nabla h\left(\eta \right)\times \left({\gamma }_1\nabla h\left({\psi }_1\right)+{\gamma }_2\nabla h\left({\psi }_2\right)\right),$$

where ${\gamma }_1$ and ${\gamma }_2$ are two positive parameters controlling the binding affinity of HC types to the nuclear lamina.

The full free energy functional of the nuclear chromatin $\mathcal{F}[\eta ,\mathit{\phi },\mathit{\psi }]$ that we minimize to get the evolution equations of the nuclear structures is equal to the sum of the two energy functional contributions $F_B$ and $F_I$ explicitly presented in Eqs. 1 and 2. The phase-field parameter η that describes nuclear geometry is assumed constant in the total free energy functional. For the structural chromatin types, there are only two independent states described by ${\psi }_1$ and ${\psi }_2$ due to the following constraint on the set of the phase field variables ${\psi }_j$ given by: ${\sum }_{j=1}^3{\psi }_j=1$. The dynamics of nuclear chromatin is governed by the Allen-Cahn evolution equations of the phase-field variables $\left\{\mathit{\phi },\mathit{\psi }\right\}$:

$$\begin{array}{l}\frac{\partial {\phi }_i}{\partial t}=-L_{\phi }\frac{\delta \mathcal{F}}{\delta {\phi }_i},i=1,\cdots ,N\\ \frac{\partial {\psi }_1}{\partial t}=-L_{{\psi }_1}\frac{\delta \mathcal{F}}{\delta {\psi }_1}+{\zeta }_{{\psi }_1}\left(\mathbf{r},t\right),\\ \frac{\partial {\psi }_2}{\partial t}=-L_{{\psi }_2}\frac{\delta \mathcal{F}}{\delta {\psi }_2}+{\zeta }_{{\psi }_2}\left(\mathbf{r},t\right),\end{array}$$ (3)

where $L_{\phi }$, $L_{{\psi }_1}$, and $L_{{\psi }_2}$ are mobility coefficients that are proportional to the relaxation time of different phase-field variables. We have chosen the mobility coefficients to match the magnitudes of in vivo measurements of EC and HC diffusion coefficients (59). The terms ${\zeta }_{{\psi }_j}\{j=\mathrm{1,2}\}$ in Eq. 3 account for the fluctuations at the boundaries of EC/fHC and EC/cHC islands due to the finite size nature of the droplets. The fluctuations are modeled as white noise: $⟨{\zeta }_{{\psi }_j}\left(\mathbf{r},t\right){\zeta }_{{\psi }_j}\left({\mathbf{r}}^{\text{′}},t^{\prime }\right)⟩=A_{{\psi }_j}\delta \left(\mathbf{r}-{\mathbf{r}}^{\text{′}}\right)\delta \left(t-t^{\prime }\right)$, where the amplitude of noise is given by: $A_{{\psi }_j}=2k_{B}T{L}_{{\psi }_j}$. The amplitude of noise $A_p$ sets the “effective temperature” $T_{eff}$ of the nucleus (60), which can be taken as a measure of ATP activity in comparisons with the experiment (50,61).

Parametrization of the Drosophila nucleus

To solve numerically the set of evolution equations of the phase-field variables resulting from the free energy minimization (3), we use the finite element method combined with the preconditioned Jacobian Free Newton Krylov technique. The model has been implemented in MOOSE finite element C⁺⁺ library (62,63), which is built using high-performing computational libraries MPI, LibMesh, and PETSc needed for solving nonlinear partial differential equations (63). Once all the equations in Eq. 3 are transformed in the weak form to extract the residual vectors, we compute the solution of the phase-field variables by using the different kernels implemented in the Moose phase-field module. More technical details of the numerical implementation of the phase-field module in MOOSE are given in (63,64). By introducing two parameters corresponding to the characteristics scale of time $\tilde{\tau }$ and length l, we have written and solved the coupled equations system (Eq. 3) of interfaces motions in dimensionless form. The total genome length of Drosophila is around $150\mathrm{M}\mathrm{b}$, and the nucleus diameter is estimated to $5\mu \mathrm{m}$. The model is resolving chromatin at $\approx Mb$-scale resolution discriminating between chromatin compartments which give rise to cHC, fHC, and EC domains in the nucleus. Thereby, the length scale and characteristic time were fixed to $l=1\mu \mathrm{m}$ and $\tilde{\tau }=0.005\mathrm{s}$. In this work, all simulations were performed on a rectangular mesh domain of dimension $6\mu \mathrm{m}\times 9\mu \mathrm{m}$. The same domain mesh as the one in (55) is used here: a quadrilateral elements are chosen to generate the fine mesh with 640,000 elements. The time-step used for time integration is set to 0.04 in dimensionless units, which is chosen to insure numerical stability of all simulations. The initial conditions used to solve the chromatin dynamics equations system are generated by a tanh-like function: $1/2\left[1-\text{tanh}\left(r/2\sqrt{2}{\epsilon }_{.}\right)\right]$, where r represents the signed distance function. An elliptical shape is chosen as the initial shape for different domains inside the nucleus described by the set of phase-field variables $\left\{\eta ,\mathit{\phi },\mathit{\psi }\right\}$. The nuclear shape is maintained at the center of the computational domain in an elliptical shape, in which the semimajor axes are fixed to $a=2.5\mu \mathrm{m}$ and the semiminor axis to $b=4\mu \mathrm{m}$; thus the nuclear volume is: $V_N=\pi ab=31.416\mu {\mathrm{m}}^2$ consistent with empirical measurements of Drosophila nucleus during interphase (65). The initial coordinate of the eighth chromosomal territories $\left(N=8\right)$ within the nucleus is given by ${\mathbf{X}}_{\mathbf{i}}$ = {(3,1.2); (1.2,2.9); (2.7,3.25); (4.5,2.75); (3.15,5.5); (4.8,6); (3,8.25); (1.35,6)}, with the major and minor axis are set to $0.5$ and $0.1\mu \mathrm{m}$, respectively. The HC domains (cCH-fCH) are generated at the center of the chromosomal territories, with the major and minor axis are set to 0.28–$0.26$ and 0.08–$0.06\mu \mathrm{m}$, respectively.

For simplicity, the mobility of chromosomal and HC compartments are set to be equal $L=L_{{\phi }_i}=L_{{\psi }_i}$. This parameter fixes the domain’s interface relaxation time: $\tau \propto L^{-1}$. The interface relaxation time τ is used to set the unit timescale for the kinetics of phase transitions. To investigate effects of diffusion coefficient of chromatin on nuclear morphology, given by $D_{{\psi }_1}=L{\left(l{\epsilon }_{{\psi }_1}\right)}^2/\tilde{\tau }$ for the HC and $D_{{\psi }_2}=L{\left(l{\epsilon }_{{\psi }_2}\right)}^2/\tilde{\tau }$ for chromocenter, simulation were performed by varying the interfacial parameters ${\epsilon }_{{\psi }_1}$ and ${\epsilon }_{{\psi }_2}$. The diffusion coefficient values taken for A/B types of HC are 4, 12, and 20 μm²/s. The diffusion coefficient of chromosomes $D_{\phi }$ is fixed to 1μm²/s. These values of diffusion coefficients are motivated by experimental measurements of EC/HC mobility in live cells (59). The interaction coefficient between chromosomal territories and NE is chosen to be strong enough to maintain them inside the nucleus by setting ${\beta }_0=16.67$. To ensure a well separated chromosome territories, we set the chromosome-chromosome interaction parameter with strong interaction same as the one in (55) ${\beta }_{\phi }=40$. In our previous work (55), we have shown that strong affinity between EC-HC leads to mixed states for HC droplets whereas weaker mixing affinity leads to their fusion. We thus set the HC-HC interaction parameter ${\beta }_{{\psi }_2}$ to 0.1 for all simulations, and chromo-chromocenter and hetero-chromocenter interaction parameters ${\beta }_{{\psi }_1}$ and ${\beta }_{{\psi }_1,{\psi }_2}$ are varied. The interaction parameters between A/B types of HC and nuclear lamina, ${\gamma }_1$ and ${\gamma }_2$, are varied to evaluate the effects of competition between binding energies and chromatin compartment interactions. The remaining model’s parameters including the growing domain kinetic are set as $a_1=0.16$ and $a_2=a_3=a_4=2$. The fluctuation amplitude of EC-HC $A_1$ is fixed at 15 and two different values are considered for EC-chromocenter interface $A_2$ to 2.5 and 15. We note that although the nuclear shape, size, chromosome numbers, and diffusion coefficients are calibrated after Drosophila nucleus the model is sufficiently general for drawing broader conclusions about the chromatin structure and dynamics in eukaryotic nuclei. The parameters involved in the model discussed above are summarized in Table 1.

Table 1.

Parameters used in the MELON model

Parameter	Description	Value
$L_{.}$	Mobility coefficients $L_{\phi }=L_{{\psi }_1}=L_{{\psi }_2}$ = L	1
Τ	Relaxation time of phase-field variables $\tau =L^{-1}$	1
${\epsilon }_{.}$	Gradient parameters setting the width of the interface and the surface tension
	${\epsilon }_{\phi }^2$ is fixed	0.005
	${\epsilon }_{\psi }^2$ is varied	$0.02,0.06,0.1$
$a_{.}$	Are penalty coefficients $a_1$, $a_2=a_3=a_4$
	$a_1$	0.16
	$a_2$	2
${\beta }_0$	Geometry parameter controlling the intensity of the energetic belt around the NE	16.67
${\beta }_{\phi }$	Strength of the excluded volume interaction between chromosomal territories	40
${\beta }_{{\psi }_1}$	Strength of cHC selfinteraction (varied)	0.1–4.5
${\beta }_{{\psi }_2}$	Strength of fHC self-interaction	0.1
${\beta }_{{\psi }_1,{\psi }_2}$	Strength of interaction between cHC-fHC (varied)	0.1–0.5
${\gamma }_1$	Strength of interaction between cHC and the NE (varied)	0–1.1
${\gamma }_2$	Strength of interaction between fHC and the NE (varied)	0–1.1
$\tilde{\tau }$	Characteristic time	0.005 s
L	Characteristic length scale	$1\mu \mathrm{m}$
$D_{.}$	Diffusion coefficients $D_{.}=L{\left(l{\epsilon }_{.}\right)}^2/\tilde{\tau }$
	$D_{\phi }$ is fixed	$1\mu {\mathrm{m}}^2/\mathrm{s}$
	$D_{{\psi }_1}$; $D_{{\psi }_2}$ are varied	$\mathrm{4,12,20}\mu {\mathrm{m}}^2/\mathrm{s}$

Results

Chromatin compartmentalization patterns emerge from phase separation of HC types

To dissect the impact of the phase separation of HC types on spatial organization of chromatin in the nucleus, we first carry out simulations by varying the free energy terms controlling the interaction of cHC-cHC, fHC-fHC, and cHC-fHC in the absence of lamina-HC anchoring terms. The interactions between CTs and the thickness of chromatin intermingling regions have a significant impact on the interchromosomal interactions by coordinating chromatin domain formation. Therefore, we first dissect the contribution of each of these interactions on the chromatin organization. Subsequently we will use the distinct classes of nuclear architectures as steppingstone for more detailed investigation of phase separation kinetics, HC patterning and the role of lamina-HC interactions in the later sections. We note that the concept of chromosome territories and the degree of chromatin intermingling in the current multiphase liquid chromatin model is essentially controlled through a single parameter ${\epsilon }_{\phi }$.

The evolution of nuclear chromatin patterns driven by the attractive interactions between the two HC types cHC and fHC $\left({\beta }_{{\psi }_1,{\psi }_2}=0.1\right)$ is shown in Fig. 2 A and with repulsive interaction $\left({\beta }_{{\psi }_1,{\psi }_2}=0.5\right)$ in Fig. 2 B. The simulated nuclear patterns are generated for two extreme cases of cHC mixing affinity that is set by the parameter ${\beta }_{{\psi }_1}$. We note that the mixing affinity parameters ${\beta }_{{\psi }_1}$ and ${\beta }_{{\psi }_2}$ represent the degree of the attractive interaction between different HC regions from neighborhood chromosomes. The simulation results with a stronger and weaker mixing affinity are shown in the top and the down panels of the Fig. 2, A and B, respectively. Depending on the strength of chromosomal territorial interaction ${\beta }_{\phi }$, the decrease of mixing affinity parameters leads to enhanced mobility of HC droplets to move through chromosome territories. Thus, it will be easy for similar-type HC from different chromosomes to merge and initiate the phase separation. This connection between similar-type HC from neighboring chromosomes is shown in the Fig. 2. We observe that for an attractive interaction between cHC-fHC, with a stronger mixing affinity ${\beta }_{{\psi }_1}=4.5$, the fHC droplets located initially on neighboring chromosome centers move toward periphery of their territories and coalesce forming a larger droplet. Meanwhile the cHC droplets remain in the center (Fig. 2 A, top panel). For a similar mixing affinity of HC types, which are set to ensure a weaker mixing within a chromosome ${\beta }_{{\psi }_1}={\beta }_{{\psi }_2}=0.1$, both of the HC droplets are moving through the chromosomal periphery in the direction of the nuclear center resulting in a single droplet (Fig. 2 A, bottom panel). In contrast, a strong interaction between cHC and fHC HC (Fig. 2 B) leads to phase-separated states of chromatin compartments irrespective of the mixing affinity of HC types. One can also observe an increase in the distance between HC droplets when the cHC-fHC interaction is made stronger.

Figure 2.

Compartmentalization patterns of nuclear chromatin. The snapshots show the emergence of chromatin compartmentalization at different time steps initiated with attractive and repulsive interactions between the cHC and fHC HC types. (A) Simulated structures obtained for attractive interaction between cHC-fHC with a weaker mixing affinity of cHC (top panel) and stronger mixing affinity (bottom panel). (B) Simulated structures obtained with repulsive interaction between cHC-fHC with a weaker mixing affinity of cHC HC (top panel) and stronger mixing affinity of cHC (bottom panel). The green color indicates the chromosome territories, while the blue and red colors indicate the HC types cHC and fHC regions, respectively. (C) Relaxation dynamics of chromosomal volume V, the cHC volume v, and fHC volume w. (D) Heterochromatin droplet heterogeneity quantified as area distribution as a function of simulated time. To see this figure in color, go online.

To fix the total time for simulating interphase nuclear dynamics as well as to assess the temporal convergence compartmentalization patterns, we have also analyzed the temporal evolution of the chromosome territories volume V, the fHC and cHC HC volumes v and w, respectively (Fig. 2 C) and the distribution of fHC droplet areas (Fig. 2 D). The different volumes of the nucleus’ components are relaxed to reach their prescribed values and thus occupy the whole volume of the cell nucleus. Likewise, the relative fraction of the two HC types whose we have evaluated by $v/V$ and $w/V$ are well around $25\text{\%}$.

Thermodynamics of chromatin phase separation dictates coherent motions of chromatin domains

Here, we study the role of differential mobility of chromatin types on the kinetics of nuclear compartmentalization. To vary the mobility we first express the diffusion coefficients of chromatin types in terms of mobility coefficients $L_{{\psi }_j}$that allows fixing both the relaxation times of HC fields ${\psi }_j$ and the diffusive interface thickness ${\epsilon }_j$, such that $D_{{\psi }_j}=L_{{\psi }_j}{\left(l{\epsilon }_{{\psi }_j}\right)}^2/\tilde{\tau },j=\mathrm{1,2}$. With this formulation, we can vary these two parameters independently for setting the diffusion coefficients. Simulations were performed with different values of diffusion coefficients associated with the cHC and fHC HC domains by varying their diffusive interface thickness ${\epsilon }_{{\psi }_j}$. In these simulations, we consider only the case of repulsive cHC-fHC interactions $\left({\beta }_{{\psi }_1,{\psi }_2}=0.5\right)$ with an identical mixing affinity for cHC and fHC types $\left({\beta }_{{\psi }_1}={\beta }_{{\psi }_2}=0.1\right)$.

We introduce and use local displacement vector fields of chromatin to analyze the chromatin domains’ motion within the nucleus and the resulting compartmentalization patterns. The local displacement fields of chromatin domains are correlated to their interface’s motion; therefore, they can be determined by the interface’s normal velocity. We use the following expression to characterize the velocity fields of HC droplets (66,67):

$${\mathbf{v}}_{{\psi }_{\mathbf{j}}}=-\frac{\partial {\psi }_j}{\partial t}\frac{\nabla {\psi }_j}{\left|\right|\nabla {\psi }_j|{|}^2},j=\mathrm{1,2}$$

The velocity fields of HC domains are proportional to the diffusion coefficients via the term $\partial {\psi }_j/\partial t$.

The Fig. 3 shows that quite naturally the HC droplets move faster to the nucleus’s center for an increased diffusion coefficient values. As can be seen, a decrease in the number of identical-types of HC droplets with increased values of diffusion coefficient, which reveals that the diffusion accelerates the kinetics of HC droplet coalescence. Despite the self-attractive interaction of fHC domains (weak mixing affinity of fHC and cHC), we find that the fHC-droplets are fully disconnected in all of the chromosome territories when one has a large diffusion coefficient of cHC relative to fHC (five times or higher). The cHC-droplets, on the other hand move rapidly to the nuclear center and coalesce.

Figure 3.

The role of HC mobility and fluctuations on the emergent subnuclear chromatin morphology and dynamics. (A) Shown are nuclear chromatin patterns for repulsive interaction between cHC and fHC HC types and weaker mixing affinity of both cHC and fHC. (B) Displacement fields of cHC and fHC corresponding to one-time units. Shown are simulation results from fast intermediate and slow diffusion coefficients corresponding to panels arranged from top to bottom, respectively. Colors correspond to cHC and fHC, respectively. (C) Calculated spatial correlation functions of displacement fields as a function of the displacement period. (D) Dynamic structure factors of HC domains. Shown are results from fast intermediate and slow diffusion coefficients corresponding to panels arranged from top to bottom, respectively. (E) Comparison of orientation displacement vectors from simulations (left panel) with the experimentally obtained spatial displacement autocorrelation functions from displacement correlation spectroscopy experiments (right panel) reported by Zidovska et al. (49). Arrows are colored by direction. (F) Reproduction of the figure of spatial displacement auto-correlation functions of in situ nuclear chromatin measured by displacement correlation spectroscopy as reported in Zidovska et al. (49). To see this figure in color, go online.

We calculate the spatial correlation functions for HC domains $C_{{\psi }_2}\left(r\right)=⟨{\mathbf{v}}_{{\psi }_2}^{\text{Δ}t}\left(r_i\right){\mathbf{v}}_{{\psi }_2}^{\text{Δ}t}\left(r_j\right)\delta \left(r_i-r_j-r\right)⟩$ by performing ensemble averaging over the velocity vectors ${\mathbf{v}}_{{\psi }_2}^{\text{Δ}t}$ defined over the physical domain $\text{Ω}$ for different displacement periods $\text{Δ}t$. The spatial autocorrelation functions for HC (either cHC or fHC) capture the experimentally observed growth and eventual decline of correlations over the nuclear scale (49). The growth of correlations has previously been shown in polymeric simulation with their origin clearly stemming from the microphase separation of chromatin types (11). However, the polymeric simulations have shown growth of correlation length with no decline. The observed nondecaying correlations ascribed to the finite size of single chromosome system and glassy dynamics of confined chromosomal chain. Herein, we are able to confirm that indeed the microphase separation of chromatin domains is responsible at least in part in generating correlation motions of domains and the emergence of mesoscopic dynamically associated domains (11), which do indeed decline over micron scales of nucleus. The impact of chromatin mobility seems to damp the smaller time-scale correlations with the overall long time-scale trends dictated by phase-separation kinetics. The most conspicuous role of the diffusion coefficient is revealed by analyzing dynamical structure factors $S\left(k,t\right)$ which show that chromatin domains with higher mobility favor the formation of connected mesoscopic channels over a fixed period of time (Fig. 3 D).

The differential mobility and lamina interactions of HC types contribute to conventional nuclear order

The crucial role of nuclear lamina and lamin-heterochomatin interactions in controlling nuclear chromatin architecuture is widely acknowledged (17,18). Here, we examine the impact of interactions of distinct forms of HC with lamina located at the nuclear envelope (NE) on the global nuclear chromatin patterning. To this end we have carried out simulations by varying the affinity coefficients ${\gamma }_1$ and ${\gamma }_2$ controlling the tethering of cHC and fHC domains to the NE, respectively. In the following simulations, we consider all the same interaction parameters of chromatin domains used in the previous sections. We note that the cHC is modeled at a higher effective temperature with respect to fHC set by the amplitude of flucutatons. Hence the model is capable and does break the symmetry of nuclear architectures over finite times for the swapping parameters of HC-lamina interactions.

The nuclear morphology resulting from the phase separation, which is initiated by the chromatin type to type interactions, and the interactions with the nuclear lamina are shown in Fig. 4 A. The resulting morphology illustrates how the interaction strengths of different HC compartments with the NE affect the chromatin positioning relative to the nuclear periphery. We see that quite naturally the distinct types of HC, cHC, and fHC, move closer to the nuclear periphery and tether to it because of the attractive interactions $\left({\gamma }_i>0\right)$. The contact layer between HC regions and the NE tends to increase with the binding affinity coefficients. We quantify accumulation of HC at nuclear periphery via radial profiles (Fig. 4 B) which are computed as the azimuthally averaged histograms of HC types measured from the center of nucleus obtained from the final simulation mesh configuration: $P\left(r\right)=N_i\left(r\right)/M\left(r\right)$ where $N_i\left(r\right)$ is the pixel count of HC type I (cHC or fHC), which is at a distance $[r,r+\text{Δ}r]$ from the center of nucleus and $M\left(r\right)$ is the total pixel count of the mesh at a distance $[r,r+\text{Δ}r]$.

Figure 4.

The role of lamina-HC interactions on nuclear chromatin patterns. (A) Shown are nuclear morphologies generated for various strengths of nuclear lamina interactions with cHC domain ${\gamma }_1$ and fHC domain ${\gamma }_2$. (B) Radially averaged density profile of cHC measured from a center of nucleus. (C) Radially averaged density profile of fHC measured from a center of nucleus. To see this figure in color, go online.

Turning on the attractive interaction of cHC type with the lamina ${\gamma }_1>0$ in the absence of fHC lamina interaction ${\gamma }_2=0$, facilitates the formation of nuclear patterns in which cHC is localized at the periphery of NE with a thickness of layer depending on the strength of interaction parameter ${\gamma }_1$. Because ${\gamma }_2=0$, the fHC droplets move through their chromosomal territory in the direction of the nuclear center and coalesce with neighboring droplets forming a larger droplet. The cHC droplets are experiencing motion toward the NE because of the attractive interaction with the NE. However, the cHC droplets located initially in the nucleus center have less freedom to move to NE due to the strong interaction with nearest fCH droplets. Consequently, the fCH droplets far from the NE remain at the center of their chromosome territory. We obtain a similar patterning when turning on interaction of fHC types with lamina while turning off the same interaction for cHC ${\gamma }_1=0$ and ${\gamma }_2>0$. When an equal strength of NE attractive interaction is considered, i.e., ${\gamma }_1={\gamma }_2$, we find that the cHC and fHC droplets attach to the NE with alternating positions of the droplets. Furthermore, depending on the strength of NE attractive interaction with the cHC and fHC droplets, we can simulate a predominant attachment of a given HC type to the nuclear lamina. For instance, the fHC droplets are attached to almost the whole NE’s surface and arranged as a layer near it when ${\gamma }_2>{\gamma }_1$. However, only the cHC droplets very close to the NE attach to it in this case because they interact relatively weakly with the attractive NE compared with fHC droplets. From the simulations with varying lamina-HC interactions, we learn that the interplay between the chromatin type to type interactions located in different chromosomal territories and HC domain interactions with the nuclear lamina have a significant impact the global nuclear patterning (Fig. 4). Indeed, the competition between these interactions determines how droplets position relative to the NE as well as the thickness of HC layers attached to NE.

HC content controls the percolation threshold in the connectivity of chromatin droplets

The fractional content of chromatin types in the nucleus is expected to strongly impact the nuclear organization. Here, we focus specifically on the connectivity of droplets and the ability to form mesoscale chromatin channels (15,23), which are natural candidates for gene regulation through chromatin architecture. To investigate the role of HC fraction in the nuclear architecture, we performed simulations by varying the total HC fraction, which includes the constitutive and facultative components for two limiting strengths of the cHC-fHC type interactions. The lamina interactions are set to ${\gamma }_2=2\times {\gamma }_1=1.1$ which allow tethering of HC components to the NE. The self-interactions of cHC and fHC domains are set to the same values used in previous section. The simulated nuclear architectures are presented in Fig. 5 A. The corresponding profiles of the phase-field variables ${\psi }_1$ and ${\psi }_2$ along the major-axis of the nucleus are shown in Fig. 5 B.

Figure 5.

Variation of the average density of cHC and fHC HC types. (A) Shown are simulated nuclear chromatin patterns for different values of HC content fraction in the nucleus as a function of interactions strength between cHC-fHC domains. (B) Profiles of the phase-field variables ${\psi }_1$ and ${\psi }_2$ along the major-axis of the nucleus as a function of HC density. The profile plots are associated with the simulated morphology presented in (A) in the same order as the strength of the cHC-fHC interactions. To see this figure in color, go online.

The resulting morphology shows a nonlinear dependence on the relative volumes occupied by cHC and fHC domains and the cHC-fHC interaction parameter ${\beta }_{{\psi }_1,{\psi }_2}$ (Fig. 5 B). For weak interaction between the cHC and fHC, we find that the cHC droplets are stretched along the NE forming a thin connected layer. The connectedness and thickness of the layer is a function of fHC content. We find that for large fractional contents there is a well percolation threshold for the dominant form of HC. This percolation threshold is modulated by the cHC-fHC. Specifically, the strong interaction between the cHC and fHC prevents the fusion of cHC droplets by limiting their freedom to move along the NE. From simulation results, we can conclude that both the lamina interactions and the HC density contained in the nucleus can have a significant consequence in how intranuclear droplets are arranged and how much they will be dynamically constrained during the interphase cycle of the cell.

Discussion

Chromatin organization, dynamics, and genetic activity require the coordinated action of hundreds of transcription factors, RNA molecules, and genomic loci spanning multiple time and length scales in the nucleus. How dynamic gene regulatory programs are enabled and to what extent are they caused by chromatin organization and dynamics remains a central question that fuels intensive studies by genomics and computational methods. An emerging explanation for chromatin compartmentalization is seen in the interplay of two processes: active DNA loop extrusion by cohesion motors and passive segregation of epigenetically distinct chromatin loci (4,68, 69, 70). The phase separation of chromatin polymer, however, more complex compared with phase separation of synthetic polymers and is likely driven and regulated by several different factors including epigenetic marks, transcription factors, motorized machinery, and RNA molecules (40,71). Over the last decade, molecular models of varying resolution have been employed to shed light on the hierarchical nature of chromatin compartmentalization and its emergent dynamical behaviors. Detailed computer simulations of nucleosomal arrays have shown that interplay of histone linker fluctuations, transient multivalent internucleosome interactions, and epigenetic tags can induce phase separation of chromatin fiber into fluid-like nanoscale amorphous nucleosomal conformations (72, 73, 74). Computer simulations with kilo base scale resolution single chromosome models likewise have shown that topologically associated domains result from protein mediated chromatin phase separation (12, 13, 14,70,75, 76, 77, 78, 79, 80), which alongside chromosomal looping is sufficient for accurately reproducing experimental Hi-C contact maps. Many recent observations suggest that on the micrometer scales state of chromatin can likewise be rationalized as multiphase liquid condensate.

A natural question therefore is given the success of molecular models of chromatin phase separation whether one can extend such nanoscale descriptions to obtain micron scale models of eukaryotic nuclei to describe long-timescale dynamical processes in the nucleus? In the present contribution, we have developed a phenomenological field-theoretic model of nucleus resolved in terms of multiphase liquid chromatin states. With a multiphase liquid model of nuclear chromatin, we are able to explicitly track motions and interactions of chromatin types both within and outside chromosome territories as well as with the nuclear lamina. Employing a parameterized model of Drosophila nucleus, we show that an interplay of dynamic phase separation and the intermingling of chromosomal territories gives rise to a wide range of architectural patterns which have been observed in interphase eukaryotic nuclei. Specifically, we are able to recapitulate patterns with segregating HC droplets as well as mesoscopic channels recently observed in the superresolution nuclear imaging experiments (23). Besides generating distinct nuclear architectures, we also show that the model readily captures many dynamical observables which emerge from the interplay of phase separation and interactions of HC types with the lamina. Remarkably, the model is able to capture the full spectrum of time scales for the experimentally observed coherent motion of HC domains which displays an initial growth phase followed by an eventual decay of spatial autocorrelation functions (49).

Through simulations, we also make new and concrete predictions on the contribution of heterogeneous HC-lamina interactions in interphase nuclear architecture. Specifically, we show that HC-lamina interactions, besides the more obvious consequence of generating spatial gradients of HC, also have a less obvious effect of raising the effective mobility of EC which facilitates microphase separation and correlated motions of unbound HC domains.

In summary, our multiphase liquid model of nucleus has shown promise at capturing some of the more salient features of emerging nuclear order and dynamics. This motivates to pursue a line of quantitative refinements where one could, for instance, begin to account for the observed nonequilibrium motorized forces due to nuclear actin, interconversion between HC types as well as spatially dependent droplet viscosity to establish a more direct link with chromatin phase density and hydrodynamics.

Data availability

The model has been coded in the C++ language using the finite element numerical library MOOSE (62,63). The data visualizations have been generated via Python-based Paraview software (81). The source code is available upon a reasonable request from the corresponding authors.

Editor: Tom Misteli.