RNA transcription modulates phase transition-driven nuclear body assembly

Significance

Living cells contain various membraneless organelles whose size and assembly appear to be governed by equilibrium thermodynamic phase separation. However, the dynamics of this process are poorly understood. Here, we quantify the assembly dynamics of liquid-phase nuclear bodies and find that they can be explained by classical models of phase separation and coarsening. In addition, active nonequilibrium processes, particularly rRNA transcription, can locally modulate thermodynamic parameters to stabilize nucleoli. Our findings demonstrate that the classical phase separation mechanisms long associated with nonliving condensed matter can mediate organelle assembly in living cells, whereas chemical activity may serve to regulate these processes in response to developmental or environmental conditions.

Abstract

Nuclear bodies are RNA and protein-rich, membraneless organelles that play important roles in gene regulation. The largest and most well-known nuclear body is the nucleolus, an organelle whose primary function in ribosome biogenesis makes it key for cell growth and size homeostasis. The nucleolus and other nuclear bodies behave like liquid-phase droplets and appear to condense from the nucleoplasm by concentration-dependent phase separation. However, nucleoli actively consume chemical energy, and it is unclear how such nonequilibrium activity might impact classical liquid–liquid phase separation. Here, we combine in vivo and in vitro experiments with theory and simulation to characterize the assembly and disassembly dynamics of nucleoli in early Caenorhabditis elegans embryos. In addition to classical nucleoli that assemble at the transcriptionally active nucleolar organizing regions, we observe dozens of “extranucleolar droplets” (ENDs) that condense in the nucleoplasm in a transcription-independent manner. We show that growth of nucleoli and ENDs is consistent with a first-order phase transition in which late-stage coarsening dynamics are mediated by Brownian coalescence and, to a lesser degree, Ostwald ripening. By manipulating C. elegans cell size, we change nucleolar component concentration and confirm several key model predictions. Our results show that rRNA transcription and other nonequilibrium biological activity can modulate the effective thermodynamic parameters governing nucleolar and END assembly, but do not appear to fundamentally alter the passive phase separation mechanism.

Living cells are composed of complex and spatially heterogeneous materials that partition into functional compartments called organelles. Many organelles are vesicle-like structures with an enclosing membrane. However, a large number of RNA and protein-rich bodies maintain a dynamic but coherent structure even in the absence of a membrane. These so-called RNA/protein granules are found in the cytoplasm and in the nucleus, where they are referred to as nuclear bodies. The mechanisms by which such structures form and stably persist are not well understood. However, recent evidence suggests that these membraneless organelles, such as P granules (1, 2), nucleoli (3), and stress granules (4), are liquid-phase droplets that may assemble by intracellular phase separation (5, 6). This concept is supported by work on synthetic systems, including repetitive protein domains that form droplets in vitro and in the cytoplasm (7) and intrinsically disordered protein domains that show signatures of liquid–liquid phase separation when expressed in the cell nucleus (8).

By examining the cell size-dependent assembly of nucleoli in early Caenorhabditis elegans embryos, we previously showed that nucleolar assembly is controlled by a concentration-dependent phase transition (9). Using a simple model based on the degree of supersaturation of nucleolar components, the predicted nucleolar sizes were in excellent quantitative agreement with experimental data. The picture that emerges is consistent with nucleoli behaving as liquid-phase droplets whose formation is governed by local equilibrium thermodynamic forces (10). Nevertheless, the cell and its organelles actively consume chemical energy (in the form of ATP and GTP) and are thus far from global thermodynamic equilibrium. Indeed, the nucleolus forms at ribosomal DNA sites, termed nucleolar organizing regions (NORs) (11), where ribosomal RNA (rRNA) is actively transcribed, processed, and assembled into ribosomal subunits (12). Moreover, the composition, morphology, and fusion dynamics of nucleoli are perturbed upon transcription inhibition and ATP depletion (3, 13).

In light of the proposed concentration-dependent phase transition controlling the assembly of nucleoli in early C. elegans embryos, it is interesting to assess to what degree classical theories of nucleation, growth, and coarsening associated with equilibrium first-order phase transitions (14) can quantitatively describe the assembly process. Such theories have a long history of application in materials phenomena such as solidification, spinodal decomposition, and the evolution of emulsions and liquid sols (15). Recently, these theories have also been applied to biological phenomena such as compositional domain formation in lipid membranes (16–18).

Here, we investigate the assembly dynamics of nucleoli and “extranucleolar droplets” (ENDs) via a combination of theory, simulation, and experiment. We find that classical models of phase separation and coarsening are sufficient for explaining key features of nucleolar and END assembly dynamics. However, active nonequilibrium processes, particularly rRNA transcription, modulate effective thermodynamic interaction parameters and lead to preferential condensation of nucleolar components at the NOR.

In Vivo Measurements of Nuclear Body Dynamics

To study nucleolar assembly dynamics, we use the C. elegans embryo system (19), which exhibits stereotyped cycles of nucleolar assembly and disassembly with each cell division (9). We acquired 3D time-lapse movies of embryos expressing a fluorescently tagged version of the conserved nucleolar protein fibrillarin (FIB-1::GFP). A typical nucleolar cycle for the AB lineage at the eight-cell stage is shown in Fig. 1. Soluble FIB-1::GFP is rapidly imported into the nucleus upon reassembly of the nuclear envelope (time 0 min). After a short delay, we observe condensation of dozens of small droplets throughout the nucleus (Fig. 1A). To quantify the size of these droplets, we integrated their fluorescence intensity, I, which is proportional to droplet volume [assuming fixed molecular density (9)], and found that the mean integrated intensity, $\langle I\rangle$, of all droplets increases over time (Fig. 1B). Two of these droplets increase in size more quickly than the rest and become nucleoli located at each of the two NORs in the diploid embryo. We refer to the other droplets as ENDs, which either coalesce with nucleoli (Fig. 1A, 7.5 min) or dissolve into the nucleoplasm (Fig. 1A, 10 min). Eventually, nucleoli also dissolve and their components return to the nucleoplasm before nuclear envelope breakdown (time 17–18 min). We observed a similar evolution process for the phosphoprotein DAO-5 (dauer and aging adult overexpression), another nucleolar component (Fig. S1A).

Fig. 1.

Assembly/disassembly of nucleoli and extranucleolar droplets. (A) Maximum-intensity projection of 3D image stack of an eight-cell-stage C. elegans embryo expressing FIB-1::GFP. A temporal montage of an AB-lineage nucleus is shown throughout the cell cycle. The nuclear envelope assembles at 0 min and breaks down at ∼17 min. (B) Mean integrated intensity of ENDs (red) and nucleoli (blue) over time. Data are pooled from n = 12 embryos (24 nuclei) and plotted on log–log axes. Error bars represent SD. At early times ($\lesssim$ 8 min), ENDs and nucleoli are indistinguishable. Lines indicate the theoretical predictions for DOR (dashed) and BMC (solid). The DLG prediction falls several orders of magnitude above the data. (C) Nuclear volume increases over time (solid red squares), resulting in a decreasing concentration of FIB-1 in the nucleoplasm (open green circles). The dotted line indicates the experimentally determined nucleolar saturation concentration, $c_s$ (9). (D) Nascent rRNA transcripts colocalize with nucleoli (gray arrows) but not ENDs (white arrowheads). (Scale bars, 5 μm.)

Fig. S1.

Assembly dynamics of nucleoli and extranucleolar droplets, labeled with DAO-5::GFP. (A) Maximum-intensity projections of 3D image stacks of control nucleus over time. (B) Maximum-intensity projections of 3D image stacks of C36E8.1(RNAi) nucleus over time. (C) Coarsening kinetics of DAO-5::GFP droplets.

As can be seen in Fig. 1A, the nucleus expands significantly throughout the cell cycle, which will strongly impact the total nuclear concentration. To quantify nuclear volume, we crossed the FIB-1::GFP line with a line expressing EMR-1::mCherry, an integral membrane protein that localizes to the inner nuclear membrane (Fig. 1C, Inset). Using this marker, we find that the volume of the nucleus increases throughout the cell cycle, expanding approximately threefold between the times of initial detection and dissolution of droplets (Fig. 1C). As the nucleus expands, the concentration of FIB-1::GFP in the nucleoplasm decreases correspondingly. This large change in nuclear concentration should have a significant effect on nucleolar assembly. Specifically, when the nucleoplasmic concentration falls below the saturation concentration, $c_n<c_s$, we expect that nucleoli will disassemble (9). Here, $c_s$ represents the saturation concentration of nucleolar droplets at the NORs. However, it remains unclear why ENDs initially dissolve whereas nucleoli increase in size, with nucleoli dissolving only at the latest stages of the cell cycle.

Fibrillarin has previously been observed in structures outside the nucleolus, such as nucleolar-derived foci (NDF) in the cytoplasm and prenucleolar bodies (PNBs) in the nucleus (20). However, several observations suggest that the ENDs we observe here are distinct from PNBs. First, ENDs form during interphase, rather than mitosis. Second, fibrillarin, a methyltransferase that participates in early rRNA processing steps, remains in ENDs for the majority of the cell cycle whereas it is one of the first proteins to leave PNBs in early G1 (21). Finally, unlike PNBs, ENDs appear to lack rRNA. We used oligonucleotide fluorescence in situ hybridization (oligoFISH) to visualize nascent rRNA transcripts in early embryos. In interphase nuclei, we found two bright spots of oligoFISH signal that colocalize with large FIB-1::GFP droplets, representing nucleoli, but we failed to detect rRNA in smaller ENDs (Fig. 1D). These results are consistent with a previous study that reported a punctate immunofluorescence pattern for FIB-1 and DAO-5 with no corresponding pre-rRNA localization (22). Nevertheless, NDFs and PNBs have primarily been observed in cycling mammalian cells, and it remains possible that ENDs are an embryonic variant of these structures. Regardless of classification, ENDs, NDFs, and PNBs are related structures that may reflect the intrinsic capacity of weakly “sticky” nucleolar components to phase separate.

In Vitro Phase Separation

Many nucleolar proteins, including FIB-1 and DAO-5, contain intrinsically disordered domains that are thought to promote assembly of RNA/protein droplets (2, 5, 8). To determine whether self-interactions between nucleolar components are sufficient for droplet assembly, we purified recombinant FIB-1 and examined its behavior in vitro. At near-physiological salt concentrations (150 mM NaCl), we find that FIB-1 condenses into protein-rich droplets. These droplets are highly spherical and upon contact with one another readily coalesce to form larger droplets, indicative of liquid-like behavior (Fig. 2A). We constructed a phase diagram by determining the threshold concentration of protein required to form droplets at various salt concentrations. The line connecting these points is called the phase boundary, which represents the boundary between the mixed and demixed regimes of the system (Fig. 2B). At low protein/high salt, FIB-1 remains dissolved in solution and no droplets are observed, whereas at high protein/low salt, FIB-1 condenses into droplets.

Fig. 2.

In vitro phase separation of a nucleolar protein. (A) Time-lapse images of FIB-1 droplets coalescing on a surface. (B) Phase diagram of FIB-1 in the presence and absence of RNA. The star indicates the protein and salt concentrations of the systems shown in A and C. Data points are averages of three independent trials with the SDs as error bars. (C) Maximum-intensity projections of 3D image stacks of a solution of FIB-1 droplets coarsening over time. (D) Droplet evolution kinetics from experimental data (points) and theoretical predictions (dotted line and hatched area). DLG, black dotted line; BMC, green (−RNA) and red (+RNA); DOR, gray; power law data fits, dashed lines. See Table 1 for numerical values of K and n.

Upon crossing the phase boundary, in vitro FIB-1 droplets are initially small and numerous, but over time their average size increases and their number decreases. To quantify these changes, we acquired 3D images of the droplets over time (Fig. 2C) and used custom image analysis routines to calculate the mean droplet radius $\langle R\rangle$ as a function of time. The mean droplet size increases monotonically with power law scaling, indicated by a straight line on the log–log plot in Fig. 2D.

Nucleoli and many other membraneless organelles contain RNA, but it is not clear whether RNA contributes to phase separation or is merely “along for the ride” (5, 23). To distinguish between these two alternatives, we added total yeast RNA to our purified protein solution and measured its effect on droplet stability and coarsening. At a given salt concentration, we observed droplet formation at a lower protein concentration when RNA was present (Fig. 2B), indicating that RNA promotes condensation, possibly by stabilizing interactions between FIB-1 molecules. RNA specificity did not increase the magnitude of this effect. We found quantitatively similar results with an in vitro-transcribed segment of C. elegans rRNA that is a direct substrate of FIB-1 (24) (Fig. S2). Moreover, in addition to shifting the phase boundary, RNA accelerated droplet coarsening. Whereas the power law scaling remained unchanged, the droplet volume fraction and coarsening rate both increased in the presence of RNA (Fig. 2D). Thus, RNA plays a direct role in phase separation in vitro, quantitatively altering both thermodynamics and droplet assembly kinetics.

Fig. S2.

Effect of RNA on in vitro phase separation is not sequence specific. Shown is threshold concentration of FIB-1 required to phase separate at 250 mM NaCl in the presence or absence of RNA.

Dynamic Scaling in Vitro.

A general feature of classical phase-separating systems is that domain morphology exhibits asymptotic power law scaling behavior. For example, the increase in average droplet size during phase separation can often be characterized by power law scaling of the form (14, 15)

$$\langle R\rangle ={\left(Kt\right)}^n,$$ [1]

where K is the dynamic prefactor and n is the growth or coarsening exponent. The formal definitions of growth and coarsening (15) are used throughout this article. Growth refers to an increase in the volume fraction of droplet phase mediated by transport of molecules from a supersaturated soluble phase into droplets. Coarsening refers to droplet evolution that occurs after the supersaturation within the soluble phase has been depleted, such that the volume fraction of droplet phase remains fixed at its equilibrium value while the average droplet size increases.

By quantifying the magnitude of n and K, as well as the functional dependences of K on various physical parameters, one can identify and distinguish between various potential transport mechanisms. In liquid-like systems that form droplets, as opposed to connected bicontinuous domains, the three most relevant mechanisms expected to give rise to power law scaling are (i) diffusion-limited growth (DLG) (15), (ii) diffusion-limited Ostwald ripening (DOR) (25, 26), and (iii) coarsening via Brownian motion-induced coalescence (BMC) (27) (Fig. 3).

Fig. 3.

Schematic illustrations of passive transport mechanisms. DLG: diffusion-limited growth. Molecules move from the supersaturated bulk fluid into droplets. DOR: Diffusion-limited Ostwald ripening. Molecules move from small droplets to large droplets. BMC: Brownian motion-induced coalescence. Small droplets fuse to form larger droplets. For all mechanisms, the average droplet size $\langle R\rangle$ in steady state increases as a power law in time with exponent n.

We extracted values for K and n from the in vitro experimental data shown in Fig. 2D (Table 1) and found a scaling exponent of $n\simeq 1/3$, which is consistent with droplet coarsening occurring through BMC and/or DOR. Based on measured (Fig. S3) and estimated parameter values, we predict that DLG would deplete the initial supersaturation within $\sim$ 5 s of nucleation, inducing a crossover from growth to coarsening before measurements begin and before droplets are detectable. SI Text provides parameter values and further details concerning determination of these values. Consistent with these expectations, we do not observe an early-stage DLG ($n=1/2$) regime.

Table 1.

Comparison of theoretical and measured growth/coarsening exponents n and prefactors K (given in units of $\left[{\mathrm{\mu }\text{m}}^{1/n}/\text{s}\right]$)

Model/experiment type	n	$K_{\text{theor}}$	$K_{\text{in}\hspace{0.17em}\text{vivo}}$	$K_{\text{in}\hspace{0.17em}\text{vitro}}^{-\text{RNA}}$	$K_{\text{in}\hspace{0.17em}\text{vitro}}^{+\text{RNA}}$
DLG	1/2	$2D{S}_0$*	0.04	0.045	0.16
DOR	1/3	$8\sigma D{c}_{\infty }{V}_m^2/\left(9N_{\text{A}}{k}_{\text{B}}T\right)$†	0.00002	$6.0\times {10}^{-6}$	$4.5\times {10}^{-6}$
BMC	1/3	$2k_{\text{B}}T\theta /\left(\pi \eta \right)$‡	0.0005	0.00063	0.0023
In vivo	$0.33\pm 0.14$	N/A	0.0001 ($n=1/3$), 0.0003 ($n=1/2$)	N/A	N/A
In vitro	$0.32\pm 0.11$ (−RNA)	N/A	N/A	0.00018	N/A
	$0.34\pm 0.08$ (+RNA)	N/A	N/A	N/A	0.00054

N/A, not applicable.

^*$D\equiv$ diffusivity of droplet-forming proteins, $S_0=\left(c_n-c_s\right)/\left(c_d-c_s\right)\equiv$ supersaturation, where $c_n\equiv$ average solute concentration, $c_s\equiv$ equilibrium solute concentration in the nucleoplasm or soluble phase, and $c_d\equiv$ equilibrium solute concentration in the droplet phase.

^†$\sigma \equiv$ droplet surface tension, $c_{\infty }\equiv$ equilibrium molar solute concentration in the soluble phase with a flat interface, $V_m\equiv$ molar volume of droplet forming molecules, $N_{\text{A}}\equiv$ Avogadro constant, $k_{\text{B}}\equiv$ Boltzmann constant, and $T\equiv$ temperature.

^‡$\theta \equiv$ droplet volume fraction and $\eta \equiv$ solvent or nucleoplasmic viscosity.

Fig. S3.

Fluorescence correlation spectroscopy (FCS) results.

The measured dynamic prefactors K are quantitatively consistent with those predicted for BMC, whereas within estimated uncertainty, the largest predicted values of $K_{\text{DOR}}$ are ∼10 times smaller than those measured (Fig. 2D). The estimated uncertainty in the predicted prefactors, plus or minus approximately one order of magnitude, arises primarily from uncertainty in the value of droplet surface tension (DOR) and inherent approximations of the derived expression for $K_{\text{BMC}}$ (27) (see SI Text for further discussion). A more detailed dynamic scaling analysis, also outlined in SI Text, demonstrates that all measured scaling exponents are consistent with those of BMC and DOR (which have nearly identical scaling behaviors), whereas the lognormal shape of the droplet size distributions and linear dependence of K on the droplet volume fraction are consistent with those of BMC only (Figs. S4 and S5 and Table S1). We therefore conclude that in vitro FIB-1 droplets evolve in a manner that is most consistent with steady-state Brownian coalescence (BMC); however, Ostwald ripening (DOR) could also play a nonnegligible role in the observed evolution.

Fig. S4.

In vitro droplet distributions and scaling. (A) Number density of droplets vs. time. Fits are to the solid symbols for $t>12$ min (+RNA) and $t>25$ min (−RNA), respectively. (B) Distribution height ($A_{\text{fit}}$) vs. time based on Gaussian and lognormal fits to the distributions (as shown in D). (C) Bare +RNA droplet number density distributions $N\left(R,t\right)/V$, where V is the volume imaged at each time point. (D) Scaled +RNA droplet number density distributions, displayed as $N\left(R/\langle R\rangle ,t\right)/\left(V{A}_{\text{fit}}\right)$, where $A_{\text{fit}}$ is the amplitude of the best lognormal fit to each distribution.

Fig. S5.

Comparison of kinetic prefactors obtained from BMC simulations and predictive expressions used in the analysis of experimental data.

Table S1.

Comparison of theoretical and measured dynamic scaling behaviors for in vitro droplets (K values given in units of $\left[{\mathrm{\mu }\text{m}}^3/\text{s}\right]$)

Parameter	DOR	BMC	In vitro −RNA	In vitro +RNA
n	$\frac{1}{3}$	$\frac{1}{3}$	0.32	0.34
K value −RNA	0.0000045	0.00063	0.00018	N/A
K value +RNA	0.000006	0.00229	N/A	0.00054
K	$\frac{8\sigma D{c}_{\infty }{V}_m^2}{9N_{\text{A}}{k}_{\text{B}}T}$	$\frac{2k_{\text{B}}T\theta }{\pi \eta }$	$\sim \theta$
γ	$-1$	$-1$	$-0.98$	$-1.10$
δ	$-\frac{4}{3}$	$-\frac{4}{3}$	$-1.29$	$-1.25$
$N\left(R,t\right)$	$\frac{3^{4}e}{2^{5/3}}\frac{{\left(R/\langle R\rangle \right)}^2\text{exp}\left[-\left(1/\left(1-2R/3\langle R\rangle \right)\right)\right]}{{\left(R/\langle R\rangle +3\right)}^{7/3}{\left(3/2-R/\langle R\rangle \right)}^{11/3}}$	$\sim \frac{1}{x}{e}^{-\left({\left(\ln x-b\right)}^2/2{\sigma }^2\right)}$	$\sim$ Lognormal	$\sim$ Lognormal

N/A, not applicable.

SI Text

In Vitro Experiments with RNA.

To determine whether the role of RNA in phase separation in vitro depends on the sequence of RNA being used, we added two different varieties of RNA to purified FIB-1 protein—either commercial yeast RNA (Roche) consisting of total RNA from Saccharomyces cerevisiae or an in vitro-transcribed 50-nt fragment of the 26S rRNA gene from C. elegans, which is a direct substrate of FIB-1 (24). Gel electrophoresis indicated that the yeast RNA was ∼30–50 nt in length. In both cases, we observed quantitatively similar effects: RNA lowered the protein concentration required to phase separate at a given salt concentration.

Determination of Physical Parameter Values.

In vitro.

The in vitro diffusion coefficient of GST::FIB-1 (mass $\sim 64\hspace{0.17em}\text{kDa}$/molecule) is assumed to be similar to that of GFP (mass $\sim 27\hspace{0.17em}\text{kDa}$/molecule), which was determined by FCS to be $D\simeq 9\times {10}^{-11}\hspace{0.17em}{\text{m}}^2/\text{s}$ (Fig. S3). This value is used as an approximate upper bound for the larger GST::FIB-1 molecules.

The initial GST::FIB-1 supersaturation is determined from the measured droplet volume fractions as $S_0\simeq \theta \simeq 0.0009$ +RNA and 0.00025 −RNA.

The equilibrium molar solute concentration in the soluble phase with a flat interface is taken as equivalent to the measured saturation concentrations in the soluble phase, $c_{\infty }\simeq c_s\simeq 2.45\hspace{0.17em}\mathrm{\mu }\text{M}$ +RNA and $3.25\hspace{0.17em}\mathrm{\mu }\text{M}$ −RNA.

The droplet surface tension is estimated from the droplet fusion events observed in image sequences such as that shown in Fig. 2A of the main text. Following the analysis of Brangwynne et al. (3), the surface tension of two fusing liquid droplets can be written in terms of droplet viscosity ${\eta }_d$, diameter $\ell$, and characteristic fusion time τ as $\sigma \simeq {\eta }_d\ell /\tau$. Typical values of τ for $\ell \simeq 6-7\hspace{0.17em}\text{μm}$ are $\sim 25-30\hspace{0.17em}\text{s}$. ${\eta }_d$ is unknown but we estimate its value based on those of other nuclear and cytoplasmic bodies: cytoplasmic P granules $\gtrsim 1\hspace{0.17em}\text{Pa}\cdot \text{s}$ (1), Laf-1 droplets (nucleolar protein, in vitro) $\sim 10\hspace{0.17em}\text{Pa}\cdot \text{s}$ (2), and Xenopus nucleoli $\sim \mathrm{1,000}\hspace{0.17em}\text{Pa}\cdot \text{s}$ (3). Using an intermediate value ${\eta }_d\simeq 100\hspace{0.17em}\text{Pa}\cdot \text{s}$, we estimate $\sigma \simeq 3\times {10}^{-5}\hspace{0.17em}\text{J}/{\text{m}}^2$ $\pm$ approximately one order of magnitude. Surface tension can also be estimated on general physical grounds as $\sigma \simeq k_{\text{B}}T/{\xi }^2$, where ξ is a typical molecular length scale. For $\xi \simeq 10\hspace{0.17em}\text{nm}$, we obtain $\sigma \simeq 4\times {10}^{-5}\hspace{0.17em}\text{J}/{\text{m}}^2$. Although reducing the estimated uncertainty in σ would strengthen the quantitative scaling analysis outlined below in SI Text, Dynamic Scaling Analysis of in Vitro Droplet Evolution, the magnitude of $K_{\text{DOR}}\sim \sigma$ is only one aspect of this analysis. Our conclusion that behavior consistent with BMC is observed, while contributions from DOR may also play a role, should therefore not be significantly changed by reducing the magnitude of uncertainty in σ. Only the degree of the potential DOR contribution would be better quantified.

Fibrillarin is classified as a globular protein, and the average partial specific volume of globular proteins falls within a narrow range near $0.73\hspace{0.17em}{\text{cm}}^3/\text{g}$. Globular GST::FIB-1 ($\sim 64\hspace{0.17em}\text{kDa}$/molecule) would therefore have a molar volume of $V_m\simeq 0.046\hspace{0.17em}{\text{m}}^3/\text{mol}$.

The viscosity η of the soluble phase, which is composed primarily of water, is taken to be that of water, $\eta \simeq 0.001\hspace{0.17em}\text{Pa}\cdot \text{s}$.

In vivo.

Parameter values for the in vivo system are determined analogously to those of the in vitro system. The only qualitative difference is that just one type of molecule, GST::FIB-1, concentrates within in vitro droplets, whereas many types of protein in addition to FIB-1 appear to concentrate within in vivo droplets. DAO-5, for example, condenses as FIB-1, as shown in Fig. S1. It is assumed in the following that all participating molecules have physical properties sufficiently similar to those of FIB-1 to be regarded as approximately thermodynamically equivalent to FIB-1. The true concentrations of droplet-forming molecules are therefore significantly larger than the measured FIB-1 concentrations in this scenario, but the interpretation is otherwise unchanged.

The diffusion coefficient of FIB-1::GFP was measured in vivo as $D\simeq 2\times {10}^{-12}\hspace{0.17em}{\text{m}}^2/\text{s}$ by FCS (Fig. S3). This value is used as an approximate lower bound for the smaller (unlabeled) FIB-1 molecules.

The initial FIB-1 supersaturation estimated from the directly observed droplet volume fractions is $S_0\simeq \theta \simeq 0.01$. Similarly, using the measured FIB-1 concentrations within each phase, we estimate $S_0=\left(c_n-c_s\right)/\left(c_d-c_s\right)\simeq \left(0.12\hspace{0.17em}\text{μM}-0.08\hspace{0.17em}\text{μM}\right)/\left(8\hspace{0.17em}\text{μM}-0.08\hspace{0.17em}\text{μM}\right)\simeq 0.01$.

The fundamental physical arguments applied to the in vitro case can again be used to estimate droplet surface tension as $\sigma \simeq k_{\text{B}}T/{\xi }^2\simeq {10}^{-5}\hspace{0.17em}\text{J}/{\text{m}}^2$.

The molar volume of FIB-1 is estimated as $V_m\simeq 0.023\hspace{0.17em}{\text{m}}^3/\text{mol}$, based again on an average partial specific volume of $0.73\hspace{0.17em}{\text{cm}}^3/\text{g}$ for globular proteins and a mass of $\sim 32\hspace{0.17em}\text{kDa}$/molecule.

We estimate the viscosity η of the soluble nucleoplasmic phase from the GFP::FIB-1 FCS diffusivity measurements. For a spherical particle of radius R, the Stokes–Einstein equation relates diffusivity to viscosity as $D=kT/\left(6\pi R\eta \right)$. Approximating GFP::FIB-1 as a sphere with $R\simeq 2.5\hspace{0.17em}\text{nm}$ and using the measured value $D\simeq 2\times {10}^{-12}\hspace{0.17em}{\text{m}}^2/\text{s}$, we obtain $\eta \simeq 0.05\hspace{0.17em}\text{Pa}\cdot \text{s}$.

Dynamic Scaling Analysis of in Vitro Droplet Evolution.

The theoretical dynamic scaling behaviors for DOR and BMC are summarized in Table S1. The results of our in vitro GST::FIB-1 droplet measurements are also summarized for comparison. As noted in the main text, the measured values of $n\simeq 0.34$ with RNA and $n\simeq 0.32$ without RNA are consistent with steady-state DOR, BMC, or a combination of both. The measured K values are consistent with those predicted for BMC within expected error, whereas the largest predicted values for DOR fall slightly below the expected lower measurement bound.

The observed correlation between K and droplet volume fraction θ is also nearly linear, as is expected for BMC. θ increases by $360\text{\%}$, from $\sim 0.0025$ without RNA to $\sim 0.009$ with RNA, whereas K similarly increases by $300\text{\%}$. The corresponding effect of θ on DOR rates is predicted to be much weaker. For example, an empirical expression for DOR (42), $K\sim {\left(1-{\theta }^{1/3}\right)}^{-1}$, predicts only a $4\text{\%}$ increase in K upon increasing θ by $360\text{\%}$, from 0.00025 to 0.0009. Accounting for the measured $25\text{\%}$ decrease in $c_{\infty }$ upon addition of RNA, the predicted increase in K is still far below the observed $300\text{\%}$ increase.

During both BMC and DOR, the droplet number density $N\left(t\right)$ is predicted to decay as $t^{\gamma }$ with $\gamma =-1$. As shown in Fig. S4A, we find that ${\gamma }^{+\text{RNA}}\simeq -1.10$ and ${\gamma }^{-\text{RNA}}\simeq -0.98$, consistent with both mechanisms. The droplet size distribution $N\left(R,t\right)$ is also predicted to scale dynamically as $t^{\delta }N\left(R/\langle R\rangle ,t\right)$ with $\delta =-4/3$. We also find that ${\delta }^{+\text{RNA}}\simeq -1.25$ and ${\delta }^{-\text{RNA}}\simeq -1.29$, as shown in Fig. S4B.

The droplet size distributions, as shown in Fig. S4 C and D, exhibit approximately lognormal shapes with moderately extended tails at large R. This is consistent with established BMC distributions, whereas DOR would generate a distinctly different functional form with a tail at small R and no droplets larger than $R=3\langle R\rangle /2$. Based on this compilation of evidence, we conclude that in vitro FIB-1 droplets evolve in a manner that is consistent with steady-state Brownian coalescence (BMC). Ostwald ripening (DOR), although apparently not dominant, could also play a nonnegligible role in the observed evolution.

The expression for $K_{\text{BMC}}$ that we use in the main text, from ref. 27, assumes that all droplets have the same radius, i.e., does not consider an evolving distribution of sizes. It also does not consider potential effects of long-range van der Waals attractions, which would act in general to promote coalescence, as well as hydrodynamic effects, which tend to resist coalescence when two droplets approach each other. Nonetheless, more detailed treatments appear to generate similar results. That of Wang and Davis (43) and Friedlander and Wang (44), for example, leads to the scaling relation

$${\langle R\rangle }^3-{\langle R\rangle }_0^3\simeq \frac{3.1935k_{\text{B}}T\theta \left(\widehat{\eta }+1\right)Et}{\pi \eta \left(3\widehat{\eta }+2\right)},$$ [S9]

where ${\langle R\rangle }_0$ is the number-averaged droplet radius at time $t=0$, η is the solvent viscosity, $\widehat{\eta }=\eta \prime /\eta$ is the viscosity ratio, $\eta \prime$ is the viscosity of the droplet phase, and E is a droplet collision efficiency that may vary from 0 to 1. Estimates put typical values of E in the range 0.5–1, and in the present in vitro system $\eta \prime \gg \eta$, such that $\left(\widehat{\eta }+1\right)/\left(3\widehat{\eta }+2\right)\simeq 1/3$. For $E=1$, which is equivalent to neglecting both hydrodynamic and van der Waals effects, this result deviates from that of ref. 27 by only a multiplicative constant of ∼1/2. Thus, we use the simpler result, as uncertainties in collision efficiencies, etc., are expected to be on the order of the difference between the two expressions.

We have also performed independent BMC simulations, following the approach of Meakin (45), which verify that these expressions predict prefactor values $K_{\text{BMC}}$ quite accurately. Good agreement, within a multiplicative factor of ∼2, is obtained for droplet volume fractions $\theta \lesssim 0.1$, to the lowest value examined at $\theta =0.0009$ (Fig. S5). Taking these considerations together, we conclude that the estimated values of $K_{\text{BMC}}$ quoted in Table 1 should, conservatively, be accurate to within plus or minus one order of magnitude.

Finally, we note that several other mechanisms, in addition to DLG, DOR, and BMC, are capable of producing power law scaling. These include surface attachment-limited growth ($n=1$), surface attachment-limited Ostwald ripening ($n=1/2$), viscous hydrodynamic coarsening ($n=1$), inertial hydrodynamic coarsening ($n=2/3$), and coalescence via laminar shear flow (exponential growth of $\langle R\rangle$) (e.g., refs. 14, 15, 32). The surface attachment-limited mechanisms are most likely to be relevant in ordered/solid phases, the noted hydrodynamic mechanisms are only operable in connected, bicontinuous structures (not systems of dispersed droplets), and laminar flow effects are not expected to be large in the low $\text{Re}$ systems studied here.

Effect of Interaction Parameter χ on Dynamic Prefactor K.

As discussed in the main text, one generally anticipates that a larger thermodynamic driving force increases the rate of phase change. For the specific growth and coarsening mechanisms considered in the present work, the effect of χ on the dynamic prefactor K can be established as follows. First, we note that for DLG, $K\sim S_0$, and the supersaturation $S_0$ increases with χ due to the widening of the miscibility gap. Second, for DOR, $K\sim \sigma$, and the second term in Eq. 2 in the main text implies that the surface tension σ increases with χ. Finally, for BMC, $K\sim \theta$, and $\theta \sim S_0$ again increases with χ.

Effects of Chemical Activity: Ternary Reaction–Diffusion Model.

The physical description of the nucleoplasm presented in the main text incorporates the effects of localized activity at NORs (rRNA transcription and/or the accumulating presence of rRNA) through altered local thermodynamic parameters. This interpretation captures the central features of END and nucleolar evolution, but it remains unclear whether an explicit, kinetic treatment of chemical activity at NORs may lead to measurably different assembly or disassembly dynamics. In an attempt to address this issue, we have conducted simulations of a ternary extension of our Flory–Huggins regular solution model with chemical reaction kinetics built explicitly into the equations of motion. In this description, ${\varphi }_A$, ${\varphi }_B$, and ${\varphi }_C$ represent the volume fractions of rRNA, all soluble molecular species, and all droplet-forming molecular species, respectively. The free energy is

$$F\left[\varphi \left(\overrightarrow{r}\right)\right]={\displaystyle \int d\overrightarrow{r}\left[{\varphi }_A\ln {\varphi }_A+{\varphi }_B\ln {\varphi }_B+{\varphi }_C\ln {\varphi }_C+{\chi }_{AB}{\varphi }_A{\varphi }_B+{\chi }_{AC}{\varphi }_A{\varphi }_C+{\chi }_{BC}{\varphi }_B{\varphi }_C+{\displaystyle \sum _i{\left({\lambda }_i\nabla {\varphi }_i\right)}^2}\right]},$$ [S10]

where ${\varphi }_i\left(\overrightarrow{r}\right)$ is the order parameter field representing the volume fraction of molecular population $i=A,B,$ or C; ${\chi }_{ij}\left(\overrightarrow{r}\right)$ controls the strength of interaction between i and j molecules; and ${\lambda }_i$ is the surface energy coefficient for population i. The fluid is taken to be incompressible such that ${\varphi }_A+{\varphi }_B+{\varphi }_C=1$.

The equations of motion are

$$\frac{\partial {\varphi }_i\left(\overrightarrow{r}\right)}{\partial t}=M_i{\nabla }^2\frac{\delta F}{\delta {\varphi }_i}-{\mathrm{\Gamma }}_j{\varphi }_i+{\mathrm{\Gamma }}_i{\varphi }_j+\sqrt{D}{\eta }_i,$$ [S11]

where here $i,j=A$ or B only due to incompressibility, and $i\ne j$. $M_i$ is the mobility of population i, t is dimensionless time, and ${\mathrm{\Gamma }}_i$ is the rate at which i molecules are produced in the first-order chemical reaction $A{\rightleftharpoons }_{{\mathrm{\Gamma }}_A}^{{\mathrm{\Gamma }}_B}B$. In steady state, the ratio ${\mathrm{\Gamma }}_A/{\mathrm{\Gamma }}_B$ sets the ratio of A to B concentrations, such that ${\overline{\varphi }}_A^{\text{ss}}/{\overline{\varphi }}_B^{\text{ss}}={\mathrm{\Gamma }}_A/{\mathrm{\Gamma }}_B$. The last term in Eq. S11 is a Gaussian stochastic noise term with $\langle \eta \rangle =0$, $\langle {\eta }_i\left({\overrightarrow{r}}_1,t_1\right){\eta }_i\left({\overrightarrow{r}}_2,t_2\right)\rangle =-{\nabla }^2\delta \left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)\delta \left(t_1-t_2\right)$, and magnitude $D\sim k_{\text{B}}T$.

Simulations are conducted analogously to those of the main text. ${\overline{\varphi }}_C$ is increased (${\overline{\varphi }}_B$ decreased) as shown in Fig. S6A to simulate nuclear envelope assembly and initiate droplet formation. Chemical reactions are then initiated locally within the NOR-like domain such that an excess of A molecules (rRNA) is created at the expense of local B molecules. Beginning from the point at ${\overline{\varphi }}_A=0.08$, ${\overline{\varphi }}_B=0.74$ shown in Fig. S6A, a ratio ${\mathrm{\Gamma }}_A/{\mathrm{\Gamma }}_B=0.30/0.52$ drives the steady-state concentrations of A and B phases to 0.30 and 0.52, respectively. Thus, “rRNA” accumulates at the NOR-like domains through this chemical activity, altering local thermodynamics and potentially kinetics as well. ${\overline{\varphi }}_C$ is then decreased (${\overline{\varphi }}_B$ increased) at a rate of $0.0004/t$ to simulate the effect of nuclear envelope expansion and drive droplet dissolution.

Fig. S6.

(A) Ternary model phase diagram. Circles represent the two-phase binodal, connected by tie lines (dotted lines). Squares represent the states cycled between in simulations. The vertical path is traversed by homogeneous concentration changes, whereas the sloped path along ${\overline{\varphi }}_A+{\overline{\varphi }}_B=0.82$ is traversed by reaction exchange. (B) Collective droplet evolution from representative simulations with and without reactions. All droplets, solid lines; ENDs, dashed lines. The qualitative behavior is unchanged from that of the model examined in the main text.

Chemical reaction kinetics of this type are known to suppress domain coarsening at a length scale that decreases as reaction rates increase (29). Here we use values of $\mathrm{\Gamma }={\mathrm{\Gamma }}_A+{\mathrm{\Gamma }}_B$ that are sufficiently small to ensure that the chemically limited maximum droplet size is larger than the NOR-like domain. Larger reaction rates would prevent the nucleoli-like droplets from reaching their terminal size.

As can be seen in Fig. S6, the results of such simulations display similar trends to those presented in the main text; chemical reaction kinetics of this type and strength do not alter the collective assembly–disassembly kinetics in any readily apparent way. We therefore conclude that any explicit effects of this type of chemical activity on droplet kinetics are currently likely to be experimentally indistinguishable from those due to local modulation of equilibrium thermodynamic parameters, as outlined in the main text.

Estimate of Nucleolar Collision Timescale.

Within our description, multiple nucleoli can be stabilized against DOR, but given sufficient time they may still collide and coalesce via BMC. If we consider a spherical nucleus of radius $R_n$ with two equal-sized droplets that undergo simple Brownian motion, the characteristic timescale for their collision is ${\tau }_{\text{BMC}}\simeq \pi \eta R_n^3/\left(2k_{\text{B}}T{E}_0\right)$, where $E_0$ is a characteristic collision efficiency (assumed to be 1, making this estimate of ${\tau }_{\text{BMC}}$ a lower bound) (43). Substituting $R_n\simeq 3-4\hspace{0.17em}\text{μm}$ and $\eta \simeq 0.05\hspace{0.17em}\text{Pa}\cdot \text{s}$, we obtain ${\tau }_{\text{BMC}}\simeq 500-1300\hspace{0.17em}\text{s}$. Immediately following END dissolution, two nucleoli undergoing simple Brownian motion must therefore survive $\gtrsim 8-21$ min to have a large probability of coalescence. Given a typical lifetime of $\lesssim 5$ min after END dissolution, only a minority of typical WT nuclei would exhibit nucleolar collision and coalescence, which is consistent with observations. This probability should increase notably in smaller nuclei, which is also consistent with observations.

Concentration Dependence of Maximum END Size and Number Density.

For BMC, with $K\sim \theta \sim c_n-c_s$, the maximum average droplet size ${\langle R\rangle }_{\text{max}}^3$ obtained over a fixed time interval $\delta t$ increases as $K\sim c_n-c_s$. More precisely, ${\langle R\rangle }_{\text{max}}^3={\langle R\rangle }_0^3+K_{\text{BMC}}\delta t$, where $K_{\text{BMC}}=2k_{\text{B}}T\left(c_n-c_s\right)/\left[\pi \eta \left(c_d-c_s\right)\right]$ when $\theta =\left(c_n-c_s\right)/\left(c_d-c_s\right)$. Substituting $\eta \simeq 0.05\hspace{0.17em}\text{Pa}\cdot \text{s}$, $c_d-c_s\simeq 10\hspace{0.17em}\text{μM}$, ${\langle R\rangle }_0^3\simeq 0$, and $\delta t\simeq 600\hspace{0.17em}\text{s}$, we obtain the prediction ${\langle R\rangle }_{\text{max}}^3\simeq 3\times {10}^{-18}\left(c_n-c_s\right){\text{m}}^3/\text{μM}$. This prediction, as well as that using the measured value $K_{\text{exp}}\simeq {10}^{-22}\hspace{0.17em}{\text{m}}^3/\text{s}$, is plotted in terms of integrated intensity I in Fig. 5D of the main text.

Fig. 5.

Inhibition of rRNA transcription suppresses nucleolar coarsening; END assembly depends on nuclear concentration. (A) Maximum-intensity projections of 3D image stacks of an eight-cell-stage AB-lineage nucleus over time following RNAi knockdown of C36E8.1. (B) Mean integrated intensity of ENDs pooled from n = 15 C36E8.1(RNAi) embryos (n = 30 nuclei) and plotted on log–log axes. Error bars represent SD. Lines indicate the theoretical predictions for DOR (dashed) and BMC (solid). The DLG prediction falls several orders of magnitude above the data. (C) AB-lineage nuclei from four-cell-stage embryos following RNAi to manipulate $c_n$, the nuclear concentration of FIB-1, and other nucleolar components. Note that nucleoli but not ENDs assemble at intermediate $c_n$ in ani-2(RNAi) embryos. (D) Maximum integrated intensity of ENDs increases with increasing $c_n$ in eight-cell-stage embryos. Solid line is a linear fit to the data. The x intercept represents the nucleolar saturation concentration, $c_s$. Dashed and dotted lines indicate the predictions given experimentally observed ($K_{\text{exp}}$) and theoretical ($K_{\mathrm{BMC}}$) parameter values, respectively.

The number density of demixed droplets is also expected to increase with $c_n$ due in part to the accelerated approach to the resolution threshold, i.e., reaching visibility before as many ENDs have coarsened out or dissolved. For BMC, the time to detection is $t*\sim K^{-1}\sim {\theta }^{-1}\sim {\left(c_n-c_s\right)}^{-1}$. The maximum number density of detected ENDs should occur near $t*$ ($N\sim 1/t$) and therefore increase as $N_{\text{max}}\sim 1/t*\sim c_n-c_s$. Measurements of droplet number density in AB-lineage nuclei at the eight-cell stage are also consistent with this prediction, as shown in Fig. S7.

Fig. S7.

Maximum number density of ENDs increases with increasing $c_n$ in eight-cell-stage embryos. The solid line is a linear fit to the data. The x intercept represents the saturation concentration, $c_s$.

Effect of a Lower Size Measurement Cutoff.

Altered droplet statistics.

Measurements of several quantities will be influenced by how much of the true droplet size distribution falls above and below the resolution cutoff. These quantities include the number of droplets N, average droplet size $\langle R\rangle$ or $\langle R^3\rangle$, average droplet volume fraction θ, and average nucleoplasmic concentration $c_s$. To understand how these measurements will be affected, we have derived analytical and numerical results for evolving droplet size distributions subjected to a resolution cutoff. The distribution analyzed is the asymptotic Lifshitz–Slyozov–Wagner or DOR distribution, given by

$$n\left(R,t\right)=\frac{3^{4}e}{2^{5/3}}{\langle R\rangle }^{-4}\frac{{\left(R/\langle R\rangle \right)}^2\text{exp}\left[-\left(1/\left(1-2R/3\langle R\rangle \right)\right)\right]}{{\left(R/\langle R\rangle +3\right)}^{7/3}{\left(3/2-R/\langle R\rangle \right)}^{11/3}},$$ [S12]

where we use the exact solution $\langle R\rangle ={\left(4t/9\right)}^{1/3}$. Examples of $n\left(R,t\right)$ at various t are shown in Fig. S8A.

Fig. S8.

Effect of measurement resolution on droplet statistics during DOR. (A) Evolution of the DOR distribution. Inset shows the same curves on a log–log scale. (B) Number of droplets detected $N_{\text{cut}}$ vs. $t-t_0$ for four values of $R_{\text{cut}}$. The dotted line is the exact result corresponding to $R_{\text{cut}}=0$. Inset shows the same curves on a log–log scale. (C and D) Measured average droplet radius ${\langle R\rangle }_{\text{cut}}$ and measured average droplet volume ${\langle R^3\rangle }_{\text{cut}}$ vs. $t-t_0$, respectively. The dotted lines are the exact results corresponding to $R_{\text{cut}}=0$. Insets show the same curves on a log–log scale. (E) Measured droplet volume fractions ${\theta }_{\text{cut}}$ vs. $t-t_0$, for $\theta =1/10$. (F) Measured nucleoplasmic concentration $c_s^{\text{cut}}$ vs. $t-t_0$, for $\theta =1/10$, $c_n=1/4$, and $c_s=1/20$.

Droplet number and size.

For this distribution, the number of droplets is given by $N={\displaystyle {\int }_0^{\infty }n\left(R,t\right)dR=4/\left(9t\right)}$, the average droplet radius is given by $\langle R\rangle ={\displaystyle {\int }_0^{\infty }n\left(R,t\right)RdR/N={\left(4t/9\right)}^{1/3}}$, and the average droplet volume is given by $\langle R^3\rangle ={\displaystyle {\int }_0^{\infty }n\left(R,t\right)R^{3}dR/N\sim t}$. To quantify the effect of a lower R cutoff, $R_{\text{cut}}$, we can modify the bounds of these integrals as $N_{\text{cut}}=N\left(R_{\text{cut}}\right)={\displaystyle {\int }_{R_{\text{cut}}}^{\infty }n\left(R,t\right)dR}$, ${\langle R\rangle }_{\text{cut}}={\displaystyle {\int }_{R_{\text{cut}}}^{\infty }n\left(R,t\right)RdR/N\left(R_{\text{cut}}\right)}$, and ${\langle R^3\rangle }_{\text{cut}}={\displaystyle {\int }_{R_{\text{cut}}}^{\infty }n\left(R,t\right)R^{3}dR/N\left(R_{\text{cut}}\right)}$. Example results are shown in Fig. S8 B–D. Simple analytic forms for $N_{\text{cut}}$, ${\langle R\rangle }_{\text{cut}}$, and ${\langle R^3\rangle }_{\text{cut}}$ do not emerge, but the general effect is that the measured value $N_{\text{cut}}$ is significantly smaller than the actual N, and the measured ${\langle R\rangle }_{\text{cut}}$ and ${\langle R^3\rangle }_{\text{cut}}$ values are larger and grow more slowly than the $R_{\text{cut}}=0$ values at early t. All quantities eventually converge toward the exact results at a time that scales as $t*\sim R_{\text{cut}}^3$.

One can also argue that $R_{\text{cut}}$ will have a significant effect over a time roughly equal to that required to double $\langle R\rangle$ from ${\langle R\rangle }_{\text{cut}}$ to $2{\langle R\rangle }_{\text{cut}}$. Approximately $90\text{\%}$ of the DOR distribution will have shifted above the resolution limit during this time period of $\sim 8t_0$, where $t_0$ is the time of initial droplet detection. If $t_0\simeq 2$ min, the effect of $R_{\text{cut}}$ should then become secondary for $t\gtrsim 16$ min. This analysis suggests that measurements of N, $\langle R\rangle$, and $\langle R^3\rangle$ could be biased by such effects over much of the measurement period, which spans $\sim 15$ min. We find in fact that in vivo measurements of these quantities exhibit qualitative early-time trends that are consistent with the predicted $R_{\text{cut}}$ effects, but that such effects appear to abate within roughly 4–8 min.

Droplet volume fraction and nucleoplasmic concentration.

$R_{\text{cut}}$ can also bias measurements of the droplet volume fraction ${\theta }_{\text{cut}}$ and the average nucleoplasmic concentration $c_s^{\text{cut}}$. At early times when $\langle R\rangle$ is very small, voxels interpreted as bulk nucleoplasm may contain some unresolvable fraction of droplet phase. This will make the measured nucleoplasmic intensity larger than the actual intensity. The effects of $R_{\text{cut}}$ on measurements of $c_s$ and θ have been quantified in Fig. S8 E and F. The actual (equilibrium) θ and $c_s$ values are not approached until a time that scales as $t*\sim R_{\text{cut}}^3$ or again roughly $t*\simeq 8t_0$. This indicates that a portion of the measured decrease in $c_s$ with time may be due to $R_{\text{cut}}$. If θ is low, then the magnitude of this bias may be minimized, but in general the measured values $c_s^{\text{cut}}$ will decrease as $\langle R\rangle$ increases and will tend to be greater than the true value.

Summary.

The time dependence of measurement error is summarized in Fig. S9. If the fractional error in quantity X is given by ${\langle X\rangle }_{R_{\text{cut}}=Y}/{\langle X\rangle }_{R_{\text{cut}}=0}-1$, then the errors decrease as a power law with time, although the rates of decrease generally slow at very late times. Different effective power law exponents imply that the relative errors in different quantities may change with time. For example, in the case of Fig. S9B at $t-t_0=3$, $c_s$ has the largest fractional measurement error, followed by $\langle R^3\rangle$, θ, N, and $\langle R\rangle$, respectively. At $t-t_0=10$, $c_s$ still has the largest error but is followed by N, $\langle R^3\rangle$, $\langle R\rangle$, and θ, respectively. At $t-t_0=20$, N has the largest error and is followed by $\langle R^3\rangle$, $c_s$, $\langle R\rangle$, and θ, respectively. At $t-t_0\gtrsim 100$, $\langle R^3\rangle$ has the largest error and is followed by N, $\langle R\rangle$, $c_s$, and θ, respectively. Thus, $c_s$ and θ may be in significant error at very early times but will converge toward the actual values relatively quickly. The error in $\langle R^3\rangle$ may initially be smaller than that in $c_s$ and θ, but will persist at later times. These trends are consistent with the observed plateau in $\langle R^3\rangle$ at early times, the decrease in $c_s$ at intermediate times, and the initial lag in reaching a plateau θ value.

Fig. S9.

Error in DOR droplet statistics due to measurement resolution. (A) Fractional error in measurement of $\langle R\rangle$ for various $R_{\text{cut}}$. The quantity plotted is ${\langle R\rangle }_{R_{\text{cut}}=1.5}/{\langle R\rangle }_{R_{\text{cut}}=0}-1$. The lines obey a power law with exponent $-1$. (B) Fractional error in measurement of various quantities for $R_{\text{cut}}=1.5$. The dashed lines are power law fits of the form indicated. Inset shows the time required to reach $10\text{\%}$ fractional error, $t\ast$, vs. $R_{\text{cut}}$ for the same quantities. All scale as $R_{\text{cut}}^3$.

The time required to reach a given fractional error in all quantities also scales as $R_{\text{cut}}^3$, indicating that a halving of $R_{\text{cut}}$ will reduce the window of significant measurement error by a factor of 8.

Effect of a time-dependent concentration.

These calculations do not consider the effect of a time-dependent total concentration, but the observed dissolution at late times eventually shifts $n\left(R,t\right)$ back toward and below $R_{\text{cut}}$, reamplifying the error due to finite resolution. $R_{\text{cut}}$ produces an artificial acceleration of the measured decreases in N, $\langle R\rangle$, $\langle R^3\rangle$, and θ at late times.

Potential uncertainty in END phase behavior.

Our quantification of in vivo droplet miscibility boundaries is based on the apparent absence of ENDs and/or nucleoli at sufficiently low nucleoplasmic concentrations $c_n$, as well as the measured soluble phase concentrations in the presence of droplets. It is possible that droplets smaller than the experimental resolution may exist in some systems with $c_n$ less than the apparent $c_s$ value. Even if this is the case, the trend of decreasing maximum droplet size is consistent with that expected for a BMC-dominated system with lowering droplet volume fraction. If detected, proportionally smaller droplets would therefore be consistent with our description, with the phase boundary simply shifting to a lower concentration than its apparent value.

Analysis and Physical Description of in Vivo Nuclear Bodies

Our in vitro experiments show that nucleolar protein and RNA components can drive liquid–liquid phase separation, supporting the hypothesis that a similar process occurs in vivo. To test this, we used a similar scaling analysis, as well as numerical simulations, to determine whether these passive mechanisms could also account for the assembly of nucleoli and ENDs.

Dynamic Scaling in Vivo.

We first extracted K and n parameters from Fig. 1B and compared these values to the predictions of Eq. 1 (Table 1). Based on measured and estimated parameter values (see SI Text), the predicted rate of DLG would rapidly deplete the initial supersaturation, within ∼10 s of droplet nucleation, making coarsening the dominant mode of evolution by the time droplets are detectable. For times $\lesssim$ 8 min, ENDs and nucleoli are indistinguishable and appear to increase in average size according to a power law. The observed exponent, $n\simeq 0.33\pm 0.14$, and prefactor K are both consistent with those predicted for BMC and DOR (Table 1 and Fig. 1B). We estimate that $K_{\text{BMC}}>K_{\text{DOR}}$, and direct observations of droplet coalescence (Fig. 1A) confirm that BMC indeed occurs at measurable rates. Differences between experimental and theoretical coarsening rates may be due to the complex, viscoelastic nature of the nucleoplasm, which can kinetically arrest RNA/protein emulsions (28). Nevertheless, the assembly kinetics and initial coarsening rates of ENDs and nucleoli are quantitatively consistent with those predicted for passive thermodynamic processes.

After this initial period, nucleoli diverge from the $n=1/3$ scaling regime and rapidly increase in size. Concomitantly, ENDs appear to cease coarsening and temporarily plateau at a roughly fixed size. Eventually all droplets dissolve, with ENDs disappearing first, followed by nucleoli. Because these observations cannot be explained by a simple dynamic scaling analysis, we now turn to numerical simulations to investigate the physical basis underlying the distinct coarsening rates and dissolution times of ENDs and nucleoli.

Model of Nuclear Bodies as Passive Liquid-Phase Droplets.

In our model, the nucleus is described as a continuum binary fluid composed of droplet-forming molecules, termed component A (i.e., FIB-1 and other nucleolar proteins), and all other molecules composing the nucleoplasm, termed component B. The dimensionless protein concentration field is defined as $\varphi \left(\overrightarrow{r}\right)={\varphi }_A/\left({\varphi }_A+{\varphi }_B\right)$, where ${\varphi }_i$ is the volume fraction of molecular population i, and the fluid is taken to be incompressible such that ${\varphi }_A+{\varphi }_B=1$. A Flory–Huggins regular solution free energy functional is then written

$$F={\displaystyle \int d\overrightarrow{r}\left(f_{\text{RS}}+\chi {\lambda }^2{\left|\nabla \varphi \right|}^2+H\chi \varphi \right)},$$ [2]

where $f_{\text{RS}}=\varphi \ln \varphi +\left(1-\varphi \right)\text{ln}\left(1-\varphi \right)+\chi \varphi \left(1-\varphi \right)$. The first two terms in $f_{\text{RS}}$ account for entropy of mixing between A and B molecules and therefore favor a homogeneous $\varphi \left(\overrightarrow{r}\right)$, whereas the third term accounts for molecular self-affinity and can favor either demixing (for $\chi >0$, where A and B repel each other) or mixing (for $\chi <0$, where A and B attract). The second term in Eq. 2 imposes an energetic cost for interfaces, such that χ and λ control the strength and range of intermolecular interactions, respectively (29).

The last term in Eq. 2 is included to localize the droplet phase within NOR-like domains when a spatially varying χ is used, as discussed further in the following; the prefactor H is a constant that sets the strength of this term. Models related to Eq. 2 have been used, e.g., in studies of spatial organization in the cytoplasm (30) and centrosome formation (31).

We use a model for the dynamics of ϕ that incorporates diffusion, hydrodynamic coupling between droplets, and thermal fluctuations; these effects can be expressed in the form of a type of generalized advection–diffusion equation. Because the Reynolds number, Re, of our in vivo system is estimated to be $\ll 1$, an appropriate description is the Stokes or creeping flow limit (low Re) (16, 32) of so-called model H dynamics (33),

$$\frac{\partial \varphi \left(\overrightarrow{r},t\right)}{\partial t}+\overrightarrow{v}\cdot \nabla \varphi \left(\overrightarrow{r},t\right)=M{\nabla }^2\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}+{\xi }_{\varphi }\left(\overrightarrow{r},t\right),$$ [3]

where $\overrightarrow{v}\left(\overrightarrow{r},t\right)$ is the fluid velocity field, M is treated as a constant mobility, and ${\xi }_{\varphi }\left(\overrightarrow{r},t\right)$ is a Gaussian stochastic noise variable, defined in Materials and Methods. Eq. 3 is a stochastic nonlinear advection–diffusion equation, wherein hydrodynamic effects are incorporated via the $\overrightarrow{v}\cdot \nabla \varphi$ term. The velocity field $\overrightarrow{v}\left(\overrightarrow{r},t\right)$ is obtained, in the overdamped Stokes limit, from $v_i\left(\overrightarrow{r},t\right)={\displaystyle \int d^{3}r\prime T_{ij}\left(\overrightarrow{r}-\overrightarrow{r}\prime \right)\left\{\left[\delta F/\delta \varphi (\overrightarrow{r}\prime )\right]{\nabla }_j\prime \varphi \left(\overrightarrow{r}\prime ,t\right)+{\xi }_j\left(\overrightarrow{r}\prime ,t\right)\right\}}$. Here, $T_{ij}\left(r\right)$ is the Green’s function or Oseen tensor, which can be expressed in Fourier space as $T_{ij}\left(\overrightarrow{k}\right)=\left({\delta }_{ij}-k_i{k}_j/k^2\right)/\left(\eta k^2\right)$, where η is the fluid viscosity, k is wavenumber, and ${\delta }_{ij}$ is the Kronecker delta (the Einstein summation convention is used in these equations). ${\xi }_j$ is an additional Gaussian stochastic noise variable, defined in Materials and Methods.

The advective component of Eq. 3 thus incorporates, through $T_{ij}$, the long-ranged ($\sim 1/k^2$) hydrodynamic correlations induced by the local motion and flow field of any one droplet on the local flow field and motion of all other droplets.

At sufficiently low volume fraction of condensed phase ($\theta \lesssim 0.3$), this class of models generally describes the sequence of droplet nucleation, early-stage DLG, and late-stage DOR. Numerical instabilities at large $k_{\text{B}}T/\left(\eta \lambda \right)$ prevent direct simulation of the BMC-dominated regime. To bypass this limitation, we simultaneously solve Eq. 3 and couple the trajectory of each droplet to a Brownian motion algorithm, as described in SI Materials and Methods.

This binary fluid model results in an equilibrium phase diagram as depicted in Fig. 4A. At a given value of the interaction parameter χ, the saturation concentration ${\varphi }_s$ (which is related to $c_s$ in Fig. 1C) defines the concentration above which the system phase separates. This point falls on the binodal curve (solid line in Fig. 4A) and represents the miscibility threshold. Motivated by our experimental results, which suggest that the in vivo system cycles into and out of the two-phase region, simulations of END and nucleolar evolution were conducted as follows.

Fig. 4.

Results from representative simulations in which BMC, BMC+DOR, and DOR, respectively, dominate droplet evolution. (A) Phase diagram showing bulk (END) and NOR miscibility boundaries (solid line), spinodal boundaries (dashed line), and tie lines (dotted lines). Simulations cycle between the points indicated. (B) Mean concentration of A molecules within soluble and droplet phases vs. t for the representative DOR simulation. (C) Average droplet volume ${\langle R\rangle }^3$ vs. $t-t_0$, where $t_0$ is the time of initial droplet nucleation. All droplets, upper solid lines; ENDs, lower solid lines; θ, dashed lines; NOR-less DOR simulation, gold solid line. (D) Time sequence of droplet configurations during a DOR-dominated simulation with two NOR-like domains (highlighted at time 1,150).

A uniformly mixed “nucleoplasmic” fluid is first equilibrated outside of the two-phase miscibility gap shown in Fig. 4A. To simulate the effect of nuclear assembly and FIB-1 import, the average concentration of A molecules, $\overline{\varphi }$, is uniformly increased above the lower miscibility boundary at a constant rate ${\dot{\overline{\varphi }}}^{+}$, inducing droplet nucleation and growth. $\overline{\varphi }$ is then uniformly decreased back toward and below the lower miscibility threshold ${\varphi }_s$ at a constant rate ${\dot{\overline{\varphi }}}^{-}$ to simulate the observed effect of nuclear envelope expansion (Fig. 1 A and C).

The postulated effect of rRNA on nucleolus formation is examined by designating a small spherical region (∼3$\text{\%}$ by volume) as a NOR-like domain and raising the value of the protein–protein interaction parameter to ${\chi }_{\text{NOR}}>{\chi }_{\text{bulk}}$ within this sphere, where ${\chi }_{\text{bulk}}$ is the value outside the sphere. ${\varphi }_s$ within this region is shifted to a lower concentration than that of the bulk (Fig. 4A), analogous to the shift observed in vitro upon addition of RNA (Fig. 2B). A range of $k_{\text{B}}T$ values, from 0 to 2, was used to independently examine systems in which evolution is dominated by DOR, BMC, or a mixture of both. Further simulation details can be found in SI Materials and Methods.

Locally Enhanced Protein Interactions Drive Differential Coarsening and Dissolution.

Four representative sets of simulation results are displayed in Fig. 4. As expected, an initial $n\simeq 1/2$ DLG regime is followed by the $n\simeq 1/3$ scaling of DOR and/or BMC. Biased coarsening then sets in as nucleoli-like droplets outcompete END-like droplets, which eventually shrink and dissolve. As was observed in vivo, a rapid increase in ${\langle R\rangle }^3$ occurs at this stage as the droplet size distribution loses its low R contributions and only nucleolus-like droplets survive. Thus, we see qualitatively identical behavior to experiments in terms of initially homogeneous coarsening followed by spatially biased coarsening and temporally separated dissolution, no matter what is the dominant evolution mechanism.

These dynamics can be understood as follows. In systems undergoing phase separation, one expects a larger driving force (represented by χ) in general to correspond to larger growth rates. For the three transport mechanisms considered in this study, the dynamic prefactor K can indeed be shown to increase with the magnitude of the driving force χ (SI Text). Thus, the heterogeneity in $\chi \left(\overrightarrow{r}\right)$, which introduces a local thermodynamic preference for protein accumulation at the NOR-like domain, leads directly to larger local growth and coarsening rates. This differential coarsening is further accelerated by the continuous decrease in $\overline{\varphi }$, as described in the following subsection.

Furthermore, in the coarsening regime, the NOR-like domain experiences an effective supersaturation $S_0^{\text{NOR}}=\left({\varphi }_s^{\text{bulk}}-{\varphi }_s^{\text{NOR}}\right)/\left({\varphi }_d^{\text{NOR}}-{\varphi }_s^{\text{NOR}}\right)>0$, which leads to DLG in its vicinity. (All variables here are analogous to $S_0$, $c_s$, and $c_d$ defined in Table 1, and superscripts distinguish between NOR-like domains and the bulk fluid.) The survival of the nucleolus-like droplet is thus ensured as the system’s path toward thermodynamic equilibrium splits into a growing nucleoli-like branch and an ultimately diminishing END-like branch. As the preferred droplet becomes larger, the rate of divergence between its size and that of the END-like droplets increases such that the final stages of END dissolution and nucleolar enhancement (temporally separated dissolution) can become quite rapid. Our simulation results indicate that a rather modest local enhancement of χ, on the order of ${\chi }_{\text{NOR}}-{\chi }_{\text{bulk}}\simeq 0.1k_{\text{B}}T$, is sufficient to produce the observed spatially heterogeneous coarsening and temporally separated dissolution dynamics.

Decreasing Protein Concentrations Enhance Differential Coarsening and Dissolution.

The steady decrease in $\overline{\varphi }$ further accelerates these differential processes, enhancing both heterogeneous coarsening and temporally separated dissolution, regardless of the dominant evolution mechanism. As $\overline{\varphi }$ decreases, the steady-state volume fraction of END-like droplets decreases at a faster rate than that of nucleoli-like droplets. The END-like dissolution threshold is reached first, due to the expanded miscibility gap within the NOR-like domain. END-like droplet dissolution then deposits excess concentration into the soluble phase, which the nucleoli-like droplets rapidly incorporate, leading to continued size increase despite the decreasing $\overline{\varphi }$. These simulation results explain both the plateau in mean END size and the rapid increase in nucleolar size observed in vivo (at $\simeq 10$ min in Fig. 1B). Based on previous characterization of the nucleolar phase boundary (9), expansion of the WT nucleus to roughly twice its average volume would lead to dissolution of both ENDs and nucleoli. This is comparable to the expansion quantified in Fig. 1D. Our in vivo observations of differential coarsening and temporally separated dissolution of nucleoli and ENDs are therefore consistent with the effects of locally enhanced interactions and time-dependent concentrations seen in our simulations. Potential effects of chemical activity on this behavior are explored in SI Text and Fig. S6.

Multiple-Droplet Stability.

After END dissolution, two nucleoli of comparable size and stability typically remain. Within our description, these nucleoli will both be stable as long as the additional surface energy needed to maintain two droplets, a factor of $\sim 2^{1/3}$, is less than the energy difference between a fully depleted NOR domain and one with a droplet. Multiple nucleoli-like droplets are thus favored for sufficiently low surface energies and/or sufficiently large droplets. The nucleoli may also collide via Brownian motion, in which case one nucleolus may become energetically preferable to two if the NORs are structurally able to merge. Our calculations indicate that the average collision rate for two nucleoli undergoing simple Brownian motion would produce nucleolar collision and potentially coalescence in only a minority of typical nuclei. This is consistent with in vivo observations, as is the expectation that this probability should increase in smaller nuclei. Multiple droplets can thus be stabilized indefinitely against DOR by local energetic differences such as that associated with ${\chi }_{\text{NOR}}$ and stabilized in effect against BMC by low collision rates.

SI Materials and Methods

Numerical Simulation of the Hydrodynamic Flory–Huggins Regular Solution Theory.

The governing equations for our overdamped hydrodynamic model of the nucleoplasmic fluid (Eq. 3 of the main text and in-text equations immediately after) are written

$$\frac{\partial \varphi \left(\overrightarrow{r},t\right)}{\partial t}+\overrightarrow{v}\cdot \nabla \varphi \left(\overrightarrow{r},t\right)=M{\nabla }^2\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}+{\xi }_{\varphi }\left(\overrightarrow{r},t\right)$$ [S1]

$$v_i\left(\overrightarrow{r},t\right)={\displaystyle \int d^{3}r\prime T_{ij}\left(\overrightarrow{r}-\overrightarrow{r}\prime \right)\left[\frac{\delta F}{\delta \varphi (\overrightarrow{r}\prime )}{\nabla }_j\prime \varphi \left(\overrightarrow{r}\prime ,t\right)+{\xi }_j\left(\overrightarrow{r}\prime ,t\right)\right]},$$ [S2]

where $T_{ij}\left(\overrightarrow{k}\right)=\left({\delta }_{ij}-k_i{k}_j/k^2\right)/\left(\eta k^2\right)$, $\langle {\xi }_{\varphi }\rangle =\langle {\xi }_j\rangle =0$, $\langle {\xi }_{\varphi }\left({\overrightarrow{r}}_1,t_1\right){\xi }_{\varphi }\left({\overrightarrow{r}}_2,t_2\right)\rangle =-2M{k}_{\text{B}}T{\nabla }^2\delta \left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)\delta \left(t_1-t_2\right)$, and $\langle {\xi }_i\left({\overrightarrow{r}}_1,t_1\right){\xi }_j\left({\overrightarrow{r}}_2,t_2\right)\rangle =-2k_{\text{B}}T\eta {\nabla }^2{\delta }_{ij}\delta \left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)\delta \left(t_1-t_2\right)$. Our numerical discretization of these equations follows the fast Fourier transform (FFT) method of Koga and Kawasaki (16, 41). The system is evolved numerically on a uniform grid, using an explicit pseudospectral finite difference algorithm and periodic boundary conditions. This algorithm was parallelized using message passing interface (MPI), and simulations were conducted using from 2 to 16 CPU cores. For an $L\times L\times L$ system, the dynamics of the Fourier modes ${\varphi }_k={\displaystyle {\int }_0^L{\displaystyle {\int }_0^L{\displaystyle {\int }_0^L{d}^{3}r\varphi \left(\overrightarrow{r}\right)e^{-i\overrightarrow{k}\cdot \overrightarrow{r}}}}}$ can be written

$$\frac{\partial {\varphi }_k\left(t\right)}{\partial t}+{\left\{\overrightarrow{v}\cdot \nabla \varphi \left(\overrightarrow{r},t\right)\right\}}_k=-M{\mathrm{ℒ}}^2{\left\{\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}\right\}}_k+{\xi }_{{\varphi }_k}\left(t\right)$$ [S3]

$$v_{k,i}\left(t\right)=T_{ij}\left(\overrightarrow{k}\right){\left\{\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}{\nabla }_j\varphi \left(\overrightarrow{r},t\right)+{\xi }_j\left(\overrightarrow{r},t\right)\right\}}_k$$ [S4]

$$\langle {\xi }_{{\varphi }_k}\left(t_1\right){\xi }_{{\varphi }_{k\prime }}^{\text{*}}\left(t_2\right)\rangle =2M{k}_{\text{B}}T{L}^3{\mathrm{ℒ}}^2{\delta }_{k,k\prime }\delta \left(t_1-t_2\right)$$ [S5]

$$\langle {\xi }_{k,i}\left(t_1\right){\xi }_{k\prime ,j}^{\text{*}}\left(t_2\right)\rangle =2k_{\text{B}}T{L}^3\eta {\mathrm{ℒ}}^2{\delta }_{ij}{\delta }_{k,k\prime }\delta \left(t_1-t_2\right),$$ [S6]

where ${\left\{f\left(\overrightarrow{r}\right)\right\}}_k$ is the Fourier transform of $f\left(\overrightarrow{r}\right)$, ${\mathrm{ℒ}}^2={\left\{{\nabla }^2\right\}}_k=-2\left(1-\cos k\right)/{\left(\mathrm{\Delta }x\right)}^2$, $\mathrm{\Delta }x$ is the real-space numerical grid spacing, and * denotes complex conjugation. Local real-space operations are computed in $\overrightarrow{r}$ space, and real-space derivatives and convolutions are computed in $\overrightarrow{k}$ space. FFTs are used to convert terms between the two representations as necessary.

For example, the chemical potential

$$\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}=\text{ln}\left(\frac{\varphi }{1-\varphi }\right)+\chi \left(H+1-2\varphi \right)-2{\lambda }^2\left[\chi {\nabla }^2\varphi +\left(\nabla \chi \right)\left(\nabla \varphi \right)\right]$$ [S7]

is computed by first evaluating the Laplacian and gradient terms in Fourier space as ${\left\{{\nabla }^2\right\}}_k{\varphi }_k={\mathrm{ℒ}}^2{\varphi }_k$, ${\{\nabla \}}_k{\chi }_k=ik{\chi }_k/\text{Δ}x$, and ${\{\nabla \}}_k{\varphi }_k=ik{\varphi }_k/\text{Δ}x$ and then inverse Fourier transforming to obtain ${\nabla }^2\varphi$, $\nabla \chi$, and $\nabla \varphi$, respectively. The remaining terms are computed and summed in real space. Once all terms in Eq. 3 are evaluated in this way, ${\varphi }_k$ is evolved with a simple explicit Euler scheme,

$${\varphi }_k\left(t+\mathrm{\Delta }t\right)={\varphi }_k\left(t\right)+\mathrm{\Delta }t\left[-{\left\{\overrightarrow{v}\cdot \nabla \varphi \left(\overrightarrow{r},t\right)\right\}}_k-M{\mathrm{ℒ}}^2{\left\{\frac{\delta F}{\delta \varphi \left(\overrightarrow{r},t\right)}\right\}}_k+{\xi }_{{\varphi }_k}\left(t\right)\right],$$ [S8]

where $\mathrm{\Delta }t$ is the numerical time step.

Model parameters used in the main text are ${\chi }_{\text{bulk}}=3$, ${\chi }_{\text{NOR}}=3.25$ (if active), $\lambda =1/2$, $H=-1$, $M=1$, $\eta =1/20$, and $R_{\text{NOR}}=25$. Numerical discretization parameters are $\mathrm{\Delta }x=1$, $\mathrm{\Delta }\tau =0.005$, and $L=128$. Concentrations were evolved from $\overline{\varphi }=0.03$ to $\overline{\varphi }=0.25$ at a rate ${\dot{\overline{\varphi }}}^{+}=0.00055/t$ and then back to $\overline{\varphi }=0.03$ at rates varying from ${\dot{\overline{\varphi }}}^{-}=0.00044/t$ to $0.000044/t$, such that $K/{\dot{\overline{\varphi }}}^{-}$ remains nearly constant.

Coupled Brownian Motion Algorithm.

In principle, the regime in which Brownian coalescence dominates droplet evolution dynamics can be accessed by selecting parameter values that favor BMC over DOR, namely by maximizing the ratio $k_{\text{B}}T/\left(\eta \lambda \right)$. In practice, numerical instabilities begin to appear at values of $k_{\text{B}}T/\left(\eta \lambda \right)$ ∼ 10–100 times smaller than that at which $K_{\text{BMC}}\simeq K_{\text{DOR}}$. These instabilities quickly drive the required numerical grid spacing and time step to impractical values. We therefore take an alternative approach to accessing the BMC regime, by simultaneously solving the hydrodynamic equations of motion and coupling the trajectory of each droplet to a Brownian motion algorithm.

Once nucleation occurs, each droplet i is coupled to a local effective field that increases its internal interaction parameter to ${\chi }_{\text{drop}}={\chi }_{\text{bulk}}+\mathit{ϵ}\left(k_{\text{B}}T\right)$ over a spherical core of radius $R_i^{\text{B}}\left(t\right)\simeq 3R_i\left(t\right)/4$. $R_i\left(t\right)$ is the radius of droplet i at time t, and $\mathit{ϵ}\left(k_{\text{B}}T\right)\sim 0.1$ is a small number chosen to ensure that the position of droplet i is adequately coupled to that of the spherical core within $R_i^{\text{B}}$. After m iterations of Eq. 3 of the main text by time step $\mathrm{\Delta }t$, the radius of each droplet $R_i$ is recomputed, and the position of its core $R_i^{\text{B}}$ is displaced in a random direction by a distance $\delta r_i=\sqrt{6D_{i}m\mathrm{\Delta }t}$, where $D_i=k_{\text{B}}T/\left(6\pi \eta R_i\right)$. When droplets overlap, their $R_i^{\text{B}}$ cores are merged with conservation of volume and center of mass. This algorithm induces the droplets to follow Brownian trajectories, but does not significantly alter the growth and coarsening mechanisms produced by Eqs. 2 and 3 of the main text. Long-range hydrodynamic effects are suppressed by the imposed overdamped Brownian trajectories, but short-range hydrodynamic effects remain operable. With increasing $k_{\text{B}}T$, the late-stage behavior crosses over from DOR-dominated, to mixed DOR and BMC, to BMC-dominated evolution, respectively.

Fluorescence Correlation Spectroscopy.

Fluorescence correlation spectroscopy (FCS) was performed using a Zeiss LSM 780 microscope, equipped with a 40× water immersion objective. The FCS curves shown in Fig. S3 reflect the average of 5–10 autocorrelation curves from multiple embryos. Data were collected at the two- to four-cell stage where nucleoli and ENDs are largely absent, to obtain data on the diffusivity of soluble protein. Averaged autocorrelation curves were fitted to a single-component model, and the decay time was converted into a diffusion coefficient, using the known point-spread function parameters of the objective.

In Vivo Validation of Model Predictions

Effect of rRNA Transcription.

Our in vitro results (Fig. 2D) led us to hypothesize that the presence of rRNA at NORs is responsible for the rapid coarsening kinetics of nucleoli compared with those of ENDs (Fig. 1B). Indeed, the distinct behavior of nucleoli-like droplets in our simulations is dependent on a localized enhancement of interaction energy (${\chi }_{\text{NOR}}$) (Fig. 4).

To test this prediction in vivo, we inhibited transcription of rRNA by knocking down C36E8.1, an RNA polymerase I transcription initiation factor that is essential in yeast and mice (34, 35). In C36E8.1(RNAi) embryos, we observed spontaneous formation of FIB-1::GFP and DAO-5::GFP droplets, but no spatially heterogeneous coarsening (Fig. 5A and Fig. S1B). In other words, ENDs assembled as in control embryos but none of these droplets coarsened into nucleoli. Nevertheless, the kinetics of END nucleation, coarsening, and dissolution in the absence of nucleoli were similar to those observed in the presence of transcription and nucleolar coarsening (Fig. 5B; analogous simulation results are in Fig. 4C). Thus, the self-assembly of FIB-1::GFP into droplets in vivo does not require transcriptional activity. However, either the act of transcription or the presence of rRNA appears necessary for the accelerated coarsening of nucleoli. This is consistent with recent evidence showing that an rDNA transcription unit is required for the formation of synthetic nucleoli in human cells (36).

Nucleoli and ENDs Have Different Phase Boundaries.

Our model predicts that for a certain range of concentrations, nucleoli will assemble in the absence of ENDs, due to the enhanced interactions at NORs and consequent broader nucleolar miscibility gap. To test this prediction, we tuned the concentration of nucleolar components by manipulating embryo size with RNAi. We previously reported that upon knockdown of the anillin ANI-2, four-cell-stage nuclei fall near the nucleolar phase boundary (9). In these embryos, we find that ABa and ABp nuclei assemble small nucleoli, but we did not observe accompanying ENDs (Fig. 5C). In contrast, ima-3(RNAi) embryos, which have a higher concentration of nucleolar components, assemble both large nucleoli and many ENDs at the four-cell stage. These results indicate that the END-phase boundary falls between 0.18 μM and 0.25 μM, the nuclear concentrations of FIB-1 in ani-2(RNAi) and ima-3(RNAi) embryos, respectively. Thus, as seen in our simulations, enhanced interactions at NORs result in a wider miscibility gap for nucleoli compared with ENDs, with ani-2(RNAi) embryos falling in between these distinct phase boundaries. Furthermore, these distinct phase boundaries indicate that the saturation concentration for ENDs is greater than for NORs ($c_s^{\text{END}}>c_s^{\text{NOR}}$), accounting for the dissolution of ENDs before nucleoli as the nucleoplasmic concentration decreases during the cell cycle (Fig. 1).

Concentration Dependence of END Dynamics.

In addition to droplet phase stability, the maximum average size of ENDs also depends on concentration. For BMC, with $K_{\text{BMC}}\sim \theta \sim c_n-c_s$, the maximum average droplet size ${\langle R\rangle }_{\text{max}}^3$ obtained over a fixed time interval increases as $K\sim c_n-c_s$. As predicted, we find an approximately linear increase in the maximum size of ENDs, ${\langle I\rangle }_{\text{max}}$, with increasing $c_n-c_s$ in AB-lineage nuclei at the eight-cell stage, where $c_s\simeq$ 0.076 μM (Fig. 5D). This increase can be predicted quantitatively by using the analytical expression for $K_{\text{BMC}}$ given in Table 1 (see SI Text for details of the calculation). The prediction, plotted in terms of integrated intensity I in Fig. 5D, somewhat overestimates the measured increase in ${\langle R\rangle }_{\text{max}}^3$, but is in very reasonable agreement given the assumptions involved. If we use the measured value $K_{\text{exp}}\simeq {10}^{-22}\hspace{0.17em}{\text{m}}^3/\text{s}$ rather than the predicted value of $K_{\text{BMC}}\simeq 5K_{\text{exp}}$, even closer agreement with the measured increase in ${\langle R\rangle }_{\text{max}}^3$ is obtained, as seen in Fig. 5D. Similar arguments lead us to expect a linear increase in the number density of demixed droplets with $c_n-c_s$. Measurements of droplet number density in AB-lineage nuclei at the eight-cell stage are also consistent with this prediction (Fig. S7). An analysis of the effects of imaging resolution on these and other measured quantities relevant to evolving droplet distributions is also given in SI Text and Figs. S8 and S9.

Discussion

The nucleolus, the most prominent nuclear body, was first described in 1835 (37) and has since been the subject of thousands of studies. Although much has been learned about its biochemical activity (38), the biophysics of its assembly—and the link to nucleolar function—remain poorly understood. To address this gap in knowledge, we combined in vivo and in vitro experiments with theory and simulation to characterize the dynamics of nucleolar assembly. We find that the nucleoplasm of early C. elegans embryos behaves as a fluid that is periodically driven between homogeneous and inhomogeneous states during the cell cycle. Our results indicate that the formation of nucleoli and ENDs can be understood in terms of classical nucleation, growth, coarsening, and dissolution processes associated with first-order phase transitions. Transcription and other active nonequilibrium processes that occur at the NOR appear to locally modulate thermodynamic parameters, rather than qualitatively change the mechanism of organelle assembly.

A growing body of evidence points to phase separation as a general mechanism for the assembly of nuclear bodies and other membraneless organelles (5, 6, 39). Living cells appear to use several different strategies to control these physicochemical processes. In principle, a cell could move around in phase space through changes in gene expression, posttranslational modifications, system size, and/or chemical reaction rates. The particular regulatory mechanism likely depends on a cell’s proximity to the phase boundary. For example, we showed previously that the concentration of nucleolar components in wild-type embryos is close to the saturation concentration (9), such that relatively small changes in system size (through nuclear expansion) are sufficient to control nucleolar assembly. A similar argument has been proposed for biological membranes, which may be positioned near a critical point such that small changes in temperature or composition lead to large-scale spatial heterogeneity within the 2D lipid bilayer (40). In contrast, an autocatalytic chemical reaction was recently shown to be required for the nucleation and sigmoidal growth of centrosomes, membraneless organelles important for organizing the mitotic spindle (31). Such an active regulatory mechanism may enable the cell to tightly control centrosome assembly and thereby prevent their spontaneous nucleation, which could result in multipolar spindles and missegregation of DNA.

In addition to controlling organelle assembly, these regulatory mechanisms may impact function. By preferentially coarsening at the NOR, nucleoli appear to accelerate rRNA splicing and other modifications, directly at the transcription site. These findings are likely to be relevant not only for other nuclear bodies, but also for cytoplasmic RNP bodies such as P granules, stress granules, and processing bodies, which share many similarities with nucleoli. Many of these structures also exhibit dynamic liquid-like behaviors and function to control the flow of genetic information by regulating RNA stability, modifications, and/or translational control. It is likely that RNA can also serve to nucleate and stabilize these structures, through a similar impact on the intermolecular affinity (χ) and growth/coarsening dynamics. The molecular affinity is expected to be sensitive to the degree to which RNA has been modified, providing potentially important feedback on the assembly and coarsening dynamics. Elucidating this link between phase separation and the active biological processes that are consequently impacted will be important for understanding the physicochemical regulation of biological function.

Materials and Methods

Worm Maintenance and Imaging.

C. elegans lines were grown at 20 °C on NGM plates seeded with OP50 bacteria. Lines CPB001 and CPB064 were described previously (9); line CPB089 [ptnsIs050 (DAO-5::GFP)] was constructed using CRISPR/Cas9. Embryos were dissected from gravid hermaphrodites and imaged on M9-agarose pads. Four-dimensional confocal datasets were acquired on an inverted Zeiss Axio Observer Z1 microscope equipped with a Yokogawa CSU-X1 confocal spinning disk (Intelligent Imaging Innovations), a QuantEM 512SC camera (Photometrics), and a 100×/NA1.4 oil immersion objective.

Image Analysis.

Images were analyzed with custom software in Matlab. Droplets were detected using a 3D bandpass filter and the total fluorescence intensity within each segmented object was calculated at each time point. To compare in vivo data with classical theories of nucleation, growth, and coarsening, we assumed that the density of FIB-1::GFP molecules is independent of droplet size such that the integrated intensity is proportional to droplet volume.

Oligonucleotide Fluorescent in Situ Hybridization.

OligoFISH was performed as described in ref. 22, using Stellaris probes (Biosearch Technologies) homologous to the ITS1 and ITS2 sequences of C. elegans pre-rRNA.

RNAi.

RNAi experiments were performed by picking L4 larvae onto NGM plates containing 1 mM IPTG and 100 μg/mL ampicillin, seeded with feeding clones from the Ahringer library. Worms were allowed to feed and grow at 20 °C for 45–50 h before embryos were harvested for imaging.

In Vitro Phase Separation.

GST::FIB-1 was expressed in Escherichia coli and purified using standard protocols. Purified protein was stored in high salt to prevent phase separation. A trace amount of fluorophore-labeled polyU50 RNA (1 nM) was used for fluorescence imaging of droplets. Total yeast RNA or a 50-mer of in vitro-transcribed C. elegans rRNA, known to interact directly with FIB-1 (24), was added to a final concentration of ∼5 μM to conduct experiments with RNA.

Stochastic Variables.

The Gaussian stochastic noise variables associated with Eq. 3, ${\xi }_{\varphi }$ and ${\xi }_j$, are defined such that $\langle {\xi }_{\varphi }\rangle =\langle {\xi }_j\rangle =0$, $\langle {\xi }_{\varphi }\left({\overrightarrow{r}}_1,t_1\right){\xi }_{\varphi }\left({\overrightarrow{r}}_2,t_2\right)\rangle =-2M{k}_{\text{B}}T{\nabla }^2\delta \left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)\delta \left(t_1-t_2\right)$, and $\langle {\xi }_i\left({\overrightarrow{r}}_1,t_1\right){\xi }_j\left({\overrightarrow{r}}_2,t_2\right)\rangle =-2k_{\text{B}}T\eta {\nabla }^2{\delta }_{ij}\delta \left({\overrightarrow{r}}_1-{\overrightarrow{r}}_2\right)\delta \left(t_1-t_2\right)$.