Active RNA synthesis patterns nuclear condensates

Abstract

Biomolecular condensates are membraneless compartments that organize biochemical processes in cells. In contrast to well-understood mechanisms describing how condensates form and dissolve, the principles underlying condensate patterning – including their size, number and spacing in the cell – remain largely unknown. We hypothesized that RNA, a key regulator of condensate formation and dissolution, influences condensate patterning. Using nucleolar fibrillar centers (FCs) as a model condensate, we found that inhibiting ribosomal RNA synthesis significantly alters the patterning of FCs. Physical theory and experimental observations support a model whereby active RNA synthesis generates a non-equilibrium state that arrests condensate coarsening and thus contributes to condensate patterning. Altering FC condensate patterning by expression of the FC component TCOF1 impairs ribosomal RNA processing, linking condensate patterning to biological function. These results reveal how non-equilibrium states driven by active chemical processes regulate condensate patterning, which is important for cellular biochemistry and function.

eTOC Blurb

Biomolecular condensates exhibit various patterns — that is, size, number, and spacing — within cells, but how condensate patterning contributes to cellular biochemistry remains unclear. Using nucleolar fibrillar centers as model condensates, we show that active synthesis of RNA maintains characteristic condensate patterning by arresting coarsening. Altered fibrillar center patterning compromises ribosomal RNA processing, which links condensate patterning to proper biochemical function.

Graphical Abstract

Introduction

Biomolecular condensates are membraneless compartments that concentrate proteins and nucleic acids with shared function to organize cellular biochemistry^1,2. Many vital cellular functions, including signaling, DNA repair, chromatin state, RNA biosynthesis, splicing, translation, and ribosomal biogenesis are organized into condensates^3–19. Condensates generate unique chemical environments that concentrate or exclude specific reactants to regulate biochemical reactions^20–22. Thus, condensate formation and dissolution are key processes important for cellular function and are subject to regulation.

Imaging of cellular condensates involved in various processes has revealed a remarkable diversity of sizes, numbers and spacing within cells^23–25. We collectively refer to these properties – size, number and spacing – as condensate patterning. Several mechanisms have been proposed by which condensate patterning may be influenced. Condensates are thought to form at nucleation sites such as specific chromatin loci^26,27 and their spatial localization can be tuned by features such as component concentration and relative binding affinity²⁸. The dissolution of condensates can be driven by changes in the concentration of components²⁹, protein modifications that alter the affinity of multivalent interactions^10,30, and changes in the solvent environment, such as those caused by changes in total RNA concentration³¹. Dysregulation or disruption of biological processes associated with a given condensate often leads to concomitant changes in the condensate’s organization and patterning^{24,29,32–34}. These observations highlight features contributing to condensate patterning, and suggest a link between condensate patterning and proper cellular function. However, a general understanding of how patterning contributes to a condensate’s function remains unclear.

Existing physical models have greatly improved our understanding of many fundamental condensate behaviors by invoking thermodynamic equilibrium based on reciprocal biomolecular interactions^1,35–38. However, many condensates concentrate enzymes and reactants involved in energy-consuming chemical reactions^{15,29,39–42}, which generate continuous fluxes of biomolecules and give rise to nonequilibrium states^43–53. For example, continuous production and turnover of condensate material can lead to condensate division and size control^54–57. A comprehensive understanding of condensate behaviors in cells must therefore incorporate non-equilibrium effects.

RNA has emerged as an important regulator of many condensates and their properties. RNA affects condensate formation and dissolution^29,31,58,59, composition^60,61, and material properties such as viscosity, elasticity, and surface tension^62,63. Features of RNA, such as sequence, structure and its highly negatively charged phosphate backbone, all contribute in select ways to condensate regulation⁶⁴. Cells expend a significant amount of energy and resources on biochemistry involving RNA, and many cellular condensates are involved in the synthesis, processing, decoding, and degradation of RNA. For instance, the nucleolus is a multiphasic condensate dedicated to the transcription, processing, and assembly of ribosomal RNAs (rRNAs) into mature ribosomes⁹. RNA serves as an important scaffold for the nucleolus, as inhibiting rRNA synthesis leads to disruption of normal nucleolar morphology^65–67. The processing of pre-rRNA and assembly of mature rRNA into ribosomes is associated with RNA production, processing, and an advective flux⁶⁸, each a hallmark of active systems. How these nonequilibrium features influence nucleolar organization, however, remains unknown. The non-equilibrium nature of condensates involved in RNA metabolism, combined with the prominent role that RNA plays in condensate regulation, led us to ask whether active RNA synthesis contributes to condensate patterning.

Here, we provide theoretical and experimental evidence that active RNA synthesis generates a non-equilibrium state that patterns nuclear condensates. Our model proposes that localized, active synthesis of RNA in a condensate gives rise to the emergent behavior of arrested coarsening, which maintains condensate patterning. The rate of RNA synthesis and degree of protein-protein and protein-RNA interactions serve as dials that tune condensate patterning. Predictions of the model are tested by perturbing fibrillar centers (FCs), intranucleolar condensates involved in the synthesis of pre-rRNAs⁹, which we here refer to as rRNA for simplicity. Simulations and experiments confirm that inhibition of rRNA synthesis disrupts condensate patterning, while resumption of rRNA synthesis reestablishes patterning. Altering FC patterning by overexpression of the FC constituent TCOF1 impairs rRNA processing, suggesting that FC patterning may be important for normal ribosome biogenesis. These results reveal the importance of condensate patterning for normal cellular function, and they show how active processes out of equilibrium can regulate the physical and functional properties of condensates in cells.

Results

Condensate patterning of nucleolar fibrillar centers is inconsistent with equilibrium models

To understand the role of active processes in condensate patterning, we chose FC condensates in nucleoli as an experimental model system. This choice was motivated by prior evidence in this system for directed, non-equilibrium currents of rRNA during ribosomal assembly⁶⁸, a hallmark of an active system. We first sought to characterize the patterning of FC condensates in nucleoli by generating a murine embryonic stem cell (mESC) line expressing the FC component UBTF tagged with GFP (STAR Methods). In mESCs, numerous FCs were distributed throughout a given nucleolus (Figure 1A). We quantified the patterning of FCs by measuring their size, spacing, and number using Airyscan super-resolution microscopy (Figure 1B, 1C, STAR Methods). Quantification showed that FCs exhibited a characteristic diameter of around 260 nm and a characteristic spacing of around 480 nm. On average, mESCs contained around 26 FCs per nucleolus and around 2–3 nucleoli per cell, consistent with what was reported in a prior study⁶⁹. Staining of FCs in a panel of human cell lines suggested that a distributed FC patterning is a general and conserved feature across species and cell types (Figure 1D, Figure S1A). Taken together, these results suggest that FCs are patterned inside the nucleolus with a characteristic size, spacing, and number.

Figure 1: Condensate patterning of fibrillar centers in the nucleolus.

A. Representative image of three nucleoli in a murine embryonic stem cell (mESC) containing multiple fibrillar centers (FCs). The nucleoli are demarcated through Hoechst staining and a UBTF-GFP fusion protein is used to mark the FCs.

B. A schematic representation of features related to patterning of FCs: size, spacing and number.

C. Size, spacing, and number of FCs in the nucleoli of mESCs. The FCs have a typical diameter of 0.26 μm and spacing of 0.48 μm. There are on average 26 FCs per nucleolus.

D. Representative images of nucleoli and FCs in different healthy and diseased human cell types. Each cell type has a characteristic patterning of FCs.

E. Comparing the FC size distribution obtained from imaging murine embryonic stem cells with the shapes of distributions expected from different equilibrium and non-equilibrium mechanisms. Approach to equilibrium by coarsening through Ostwald ripening or Brownian motion-induced coalescence will give rise to a Lifschitz-Slyozov-Wagner (LSW) distribution or a Log-Normal distribution respectively. If coarsening is arrested through non-equilibrium mechanisms leading to a typical size, we expect a normal distribution centered around this size. We account for cell-cell variability in factors that may affect FC size by normalizing the data obtained from imaging experiments on a per-nucleolus basis to obtain z-scores for the different distribution shapes, as described in STAR Methods. We used the Cramer-von Mises test to compare the distributions with data and calculate a likelihood. The likelihood values for the different distributions are: (i) LSW: p = 2.66×10⁻⁹ (ii) Log-Normal: p = 1.47×10⁻⁵ (iii) Gaussian: p = 0.34. A higher likelihood corresponds to a higher probability that the data conforms to that distribution shape.

We next considered whether purely thermodynamic forces could account for FC patterning in cells. If this were the case, then the FCs would grow larger over time and fit a characteristic size distribution determined by the kinetics of coarsening. Coarsening due to diffusion-limited Ostwald ripening, where large condensates grow by siphoning material from smaller condensates, leads to condensate sizes following a Lifschitz-Slyozov-Wagner (LSW) distribution^70,71. Alternatively, coarsening driven by coalescence, where individual condensates fuse via collisions⁷², would fit a Log-Normal condensate size distribution^73,74. In contrast, an analysis of the size distribution of FCs in mESCs revealed that a Gaussian distribution was a better fit for the data compared to an LSW or Log-Normal distribution (Figure 1E, STAR Methods). We confirmed this result using a different super-resolution microscopy method, lattice structured illumination microscopy, as well as comparing 2D to 3D segmentation and analysis of human cell lines (Figure S1B–D, Figure S2). Taken together, the data suggests that the FCs in each nucleolus have a characteristic size, which does not arise due to coarsening.

A non-equilibrium physical model of nucleolar fibrillar centers

Fibrillar centers are sites of active rRNA transcription. The RNA Polymerase I enzyme transcribes rRNA by using nucleotides in a nonequilibrium chemical reaction⁹. Prior studies show that the active transcription of rRNA is essential to the formation and maintenance of nucleolar compartments^9,75. Such non-equilibrium chemical reactions are known to modulate the behavior of phase separating systems^{43,45,48,49,51,52,56,57,76–78}. Motivated by these observations, we wondered if the non-equilibrium process of rRNA transcription also underlies and controls FC patterning.

We adopted a field-theory approach^{43,48,49,52,72,76–78} building on prior models of transcriptional condensates^29,44,79 and developed a coarse-grained model informed by the current understanding of nucleolar organization⁹. We treat the nucleolus as a ternary mixture containing three categories of biomolecules: FC molecules, non-FC molecules and nascent rRNA (Figure 2A and 2B). The category of FC molecules represents proteins involved in the synthesis of rRNA such as RNA polymerase I and transcription factors^69,80,81. To describe the molecular interactions and the entropy of the mixture, we invoke the Flory-Huggins theory of solutions. Given the volume fractions ${\varphi }_P$ of FC proteins, $1-{\varphi }_P$ of non-FC molecules, and ${\varphi }_R$ of RNA, the mixture is characterized by the following simplified free energy (STAR Methods, Figure 2B):

$$\frac{ℱ}{k_{B}T}=\int \frac{d^3\mathit{x}}{V_m}\underset{f_P\left({\varphi }_P\right)V_m/k_{B}T}{\underbrace{\left[{\varphi }_P\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_P+\frac{1-{\varphi }_P}{l_P}\mathrm{l}\mathrm{o}\mathrm{g}\left(1-{\varphi }_P\right)-\beta {\varphi }_P^2\right.}}+w^2\beta {\left|\nabla {\varphi }_P\right|}^2+\frac{{\varphi }_R}{l_R}\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_R+\mathrm{\Delta }\chi {\varphi }_P{\varphi }_R].$$ (1)

Figure 2: A non-equilibrium model for fibrillar center patterning mediated by rRNA transcription.

A. A schematic representation of FCs in the nucleolus. Zooming in, we build a simplified model considering three classes of biomolecules present in the nucleolus – the nascent ribosomal RNA (rRNA), the FC proteins which are constituents of the fibrillar center, and the non-FC molecules which are constituents of the rest of nucleolus including the dense fibrillar component and the granular component. Their corresponding composition variables are ${\varphi }_R$, ${\varphi }_P$ and $1-{\varphi }_P$ respectively.

B. A non-equilibrium phase-field model that couples the dynamics of phase separation of FCs with active rRNA transcription.

C. The dynamics of FC protein concentration profiles obtained from numerical simulations of the phase-field model. This model can exhibit both coarsening dynamics or arrested coarsening leading to a typical FC size depending on the choice of model parameters.

D. Impact of rRNA-protein interactions on coarsening: Concentration profiles of FC proteins and rRNA obtained from numerical simulations of the phase-field model after a long time for different possibilities of rRNA-protein interactions. If rRNAs preferentially attract FC proteins ($\mathrm{\Delta }\chi <0$), our model predicts coarsening. If rRNAs preferentially attract non-FC proteins ($\mathrm{\Delta }\chi >0$), our model predicts arrested coarsening leading to a typical FC size.

E. Impact of rRNA transcription on coarsening: Concentration profiles of FC proteins and rRNA obtained from numerical simulations of the phase-field model after a long time for three different cases. If there is no rRNA in the system or if rRNA is just present but is not being actively transcribed, the model predicts coarsening. When there is active transcription of rRNA in a FC protein-dependent manner, our model predicts that coarsening is arrested leading to a typical FC size.

F. Radial distribution (line plot) of rRNA (magenta) around an FC (green) in our simulations.

G. Scheme depicting transcription-dependent condensate patterning without (left) or with (right) active RNA synthesis at FCs.

H. An effective free-energy-based understanding of arrested coarsening with transcription at FCs. The inward RNA concentration gradient introduces a nontrivial protein-RNA interaction energy and reshapes the total free energy landscape, leading to a local minimum at a finite condensate radius. The effective free energy per FC is given by $F_{FC}:={ℱ}_0+𝒰/2$, where ${ℱ}_0\left[{\varphi }_P\right]:=ℱ\left[{\varphi }_P,{\varphi }_R=0\right]$ indicates the free energy in absence of RNA and $𝒰$ is the protein-RNA interaction energy (Methods). The number of FCs is proportional to the total volume of FC material, $V$, divided by the volume of each FC, $V_{FC}$.

We estimate the ratio of typical molecular volumes between rRNA and FC proteins $l_R\sim 20$, and between non-FC molecules and FC proteins $l_P\sim 0.016$. Note that we use the thermal energy $k_{B}T$ and the molecular volume $V_m$ of FC proteins for normalization. The bulk free energy density $f_P\left({\varphi }_P\right)$ drives the de-mixing of FC proteins from the non-FC molecules in the nucleolus, with ${\varphi }_c$ the critical point (STAR Methods). The interactions among FC proteins are characterized by $\beta \sim \left({\chi }_{PS}-{\chi }_{SS}\right)/l_P$ and the range $w$, whereas $\mathrm{\Delta }\chi \sim \left(1+{\chi }_{SS}-{\chi }_{RS}\right)/l_P-\beta$ denotes the solubility of rRNA among non-FC compared to FC molecules. Expecting that the pairwise interactions ${\chi }_{ij}$ between different molecules are weak, we use $l_P\beta \sim 0.648$ and $l_P\mathrm{\Delta }\chi \sim 0.64$. We choose the interface width $w~6.3\mathrm{n}\mathrm{m}$, which is roughly the length scale of a protein, and recapitulates the typical FC size seen in experiment.

Next, we described the dynamics of FCs based on diffusion due to thermodynamic forces and non-equilibrium chemical reactions required for nucleolar function. The thermodynamic forces associated with the free energy defined in Equation 1, ${\mathit{f}}_P=-\mathbf{\nabla }\left(V_m\delta ℱ/\delta {\varphi }_P\right)$ and ${\mathit{f}}_R=-\mathbf{\nabla }\left(V_m{l}_R\delta ℱ/\delta {\varphi }_R\right)$, drive molecular fluxes. The FC proteins are conserved on the timescales relevant for our simulations, so they follow gradient flow dynamics⁸² (Equation 2; STAR Methods). In contrast, rRNA is produced by transcription and undergoes processing. Because rRNA transcription requires FC proteins, we assumed its rate proportional to the FC volume fraction ${\varphi }_P$ and the rate constant $k_t\sim 0.05{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$. Similarly, we modeled rRNA processing with a rate proportional to the rRNA volume fraction ${\varphi }_R$ and the rate constant $k_p\sim 0.05{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$. These parameter estimates are summarized in Table 1 and discussed in the STAR Methods, where we also corroborate our results with an alternative model in which rRNA is transcribed at the FC interface and processed in the non-FC region⁸³. In summary, the dynamics of the nascent rRNA in the limit ${\varphi }_R\ll 1$ (STAR Methods) are given by Equation 3.

Table 1.

Model parameters

Model parameter	Description	Value	Justification
	Free energy functional
$l_P\beta$	Effective strength of attraction between FC proteins that promotes phase separation (cf. Eq. S14).	0.648 (kBT)	Estimated based on measured FC protein Partition coefficient ~ 5 – 10 (King et al.81) and based on small volume fractions of FC proteins
$l_P$	Measure of the ratio between the typical molecular volumes of non-FC molecules and FC proteins.	0.016	See above
$l_p\mathrm{\Delta }\chi$	Affinity between FC proteins and nascent rRNA. If $\mathrm{\Delta }\chi <0$, FC proteins preferentially attract rRNA. If $\mathrm{\Delta }\chi >0$, the solvent non-FC molecules preferentially attract rRNA (cf. Eq. S16).	0.64 (kBT)	Fit to approximately match FC size in experiments
$w$	Lengthscale for the interaction range of FC molecules, which sets the interface width	6.3 nm	Fit to match FC size in experiments
	Concentration dynamics
$D_P=\overline{D_P}$	Diffusivity of FC proteins (UBTF) in nucleolus	4 × 10−3 μm2/s	Ide et al.88, see discussion in STAR Methods
$D_R=\overline{D_R}{l}_R$	Diffusivity of a nascent rRNA monomer in nucleolus	2 × 10−5μm2/s × 20 = 4 × 10−4μm2/s	Riback et al.68
$k_t$	Transcription rate of nascent rRNA per FC protein (UBTF) molecule	0.05 min−1	Lafontaine et al.9, see discussion in STAR Methods
$k_p$	Processing rate of nascent rRNA in the nucleolus	0.5 min−1	See above

$$\begin{gathered} \frac{\partial {\varphi }_P(\mathit{x},t)}{\partial t}=-\mathbf{\nabla }\cdot \underset{{\mathit{J}}_P}{\underbrace{\left[-\frac{M_P\left({\varphi }_P\right)}{k_{B}T}\mathbf{\nabla }\left(V_m\frac{\delta ℱ}{\delta {\varphi }_P}\right)\right]}} \\ \approx D_P\mathbf{\nabla }\cdot \left[\left(1+\frac{1-l_P}{l_P}{\varphi }_P-2\beta {\varphi }_P\left(1-{\varphi }_P\right)-2\beta w^2{\varphi }_c\left(1-{\varphi }_c\right){\mathbf{\nabla }}^2\right)\mathbf{\nabla }{\varphi }_P+\mathrm{\Delta }\chi {\varphi }_P\left(1-{\varphi }_P\right)\mathbf{\nabla }{\varphi }_R\right] \end{gathered}$$ (2)

$$\frac{\partial {\varphi }_R(\mathit{x},t)}{\partial t}=-\mathbf{\nabla }\cdot \underset{{\mathit{j}}_R}{\underbrace{\left[-\frac{M_R\left({\varphi }_R\right)}{k_{B}T}\mathbf{\nabla }\left(V_m{l}_R\frac{\delta ℱ}{\delta {\varphi }_R}\right)\right]}}+k_t{\varphi }_P-k_p{\varphi }_R\approx D_R{\mathbf{\nabla }}^2{\varphi }_R+k_t{\varphi }_P-k_p{\varphi }_R$$ (3)

For polymer solutions such as mixtures of FC and non-FC molecules, and across a wide range of densities, the mobility function $M_P\left({\varphi }_P\right)=D_P{\varphi }_P\left(1-{\varphi }_P\right)$ with effective diffusion coefficient $D_P$ is a good approximation^84–87. This specific form comes from friction between the FC protein fluxes and the corresponding non-FC counter-fluxes required by incompressibility (STAR Methods). We estimate $D_P\sim 4\times {10}^{-3}\mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ based on single-molecule tracking of chromatin-bound UBTF⁸⁸, but neglecting the subdiffusion of chromatin. Considering that the volume fraction of rRNA is small so that counter-fluxes are negligible, the mobility of rRNA is approximately $M_R\left({\varphi }_R\right)\approx D_R{\varphi }_R$ with effective diffusion coefficient $D_R$. Note that, because the FC size and the rRNA size are similar, this diffusion coefficient corresponds to the motion of rRNA monomers in our model (in other words, their hopping rate within a lattice gas description). Starting from the center-of-mass diffusion coefficient reported by Riback et al.⁶⁸, considering rRNA as a Rouse chain, and assuming a lattice gas model where the rRNA is coarse-grained into monomers with the size of an FC protein, we estimate $D_R\sim 4\times {10}^{-4}\mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$. Together, the underlying free energy functional (Equation 1) and dynamic equations (Equations 2 and 3) constitute a non-equilibrium model for the nucleolar FCs.

To study the predictions of the non-equilibrium model under various conditions, we numerically simulated the partial differential equations (2) and (3) (STAR Methods). In the absence of rRNA transcription ($k_t=0$), the simulations show that FCs exhibit coarsening dynamics and approach thermodynamic equilibrium as expected (Figure 2C, Supplementary Movie 1). In contrast, active rRNA transcription ($k_t>k_t^{\star }$ where $k_t^{\star }>0$ is a critical level of transcription) leads to an arrest in the coarsening of FCs (Figure 2C, Supplementary Movie 2). The simulations reach a non-equilibrium steady state with multiple FCs of characteristic size (Figure 2C), reminiscent of the typical FC size observed in imaging experiments (Figure 1A and E). Arrested coarsening was observed only in regimes where non-FC proteins have a higher affinity than FC proteins for rRNA ($\mathrm{\Delta }\chi >0$, Figure 2D, Supplementary Movie 2; cf. $\mathrm{\Delta }\chi <0$ Supplementary Movie 3). This result is consistent with the observation that in the nucleolus the dense fibrillar component (DFC) and the granular component (GC), rather than the FC, contain the bulk of the proteins that are involved in rRNA processing and assembly⁸⁹. Treatment of cells with ammonium acetate, which screens RNA-protein interactions and would shift $\mathrm{\Delta }\chi$, led to FC coarsening, as expected from the model (Figure S3A, S3B). Importantly, patterning of FCs requires active synthesis of rRNA, rather than its mere presence (Figure 2E, Supplementary Movies 2 & 4). This is because analysis of our model reveals that FC patterning has a characteristic spacing (wavelength) $4R^{\star }$, with

$$R^{\star }=\frac{\pi }{2}{\left(\frac{D_R}{k_p}\right)}^{1/2}{\left[{\left(\frac{\mathrm{\Delta }\chi k_t{D}_R}{2\beta w^2{k}_p^2}\right)}^{1/2}-1\right]}^{-1/2},$$ (4)

only if the transcription rate constant exceeds a critical value $k_t>2\beta w^2{k}_p^2/\left(D_R\mathrm{\Delta }\chi \right)$ (STAR Methods). Simulations in the presence of rRNA without transcription or processing (${\varphi }_R(\mathit{x},t=0)=\overline{{\varphi }_R}>0,k_t=0$ and $k_p=0$) confirm coarsening as expected. These results show active rRNA transcription can arrest FC coarsening and control condensate patterning.

The underlying mechanism for arrested coarsening can be interpreted as follows. In the absence of active rRNA synthesis, classical nucleation theory posits that FC condensates grow to minimize the interfacial energy between FC and non-FC molecules. As a consequence of this demixing, the total free energy decreases monotonically as a function of the radius of FC condensates. The presence of active rRNA synthesis generates a concentration gradient of rRNA (Figure 2F), with rRNA concentrations highest at the FC (Figure 2G). In this setting, non-FC proteins that preferentially interact with rRNA will be attracted towards the FCs, thus favoring mixing and monotonically increasing the total free energy with FC radius. The sum of these two opposing contributions leads to a non-monotonic profile of the total free energy, with a minimum at a characteristic radius consistent with arrested coarsening (Figure 2H).

Under steady-state conditions, the model predicts that a state of arrested coarsening would generate a lattice pattern of FCs, smeared out by noise if one takes into account fluctuations, which we observed when imaging native FCs (Figure S3C). The model also predicts that the length scale of FC spacing should be about twice the length scale of FC size, which we also observe (Figure 1C). These observations are consistent with steady-state arrested coarsening, which prompted us to test perturbations of the model and qualitatively compare simulation to experiment.

Active RNA synthesis maintains condensates in a state of arrested coarsening in cells

If active RNA synthesis arrests coarsening and patterns condensates in accordance with the model, we would expect the cessation and resumption of rRNA synthesis to disrupt FC patterning as predicted (Equation 4). To test this notion, we investigated the cessation and subsequent resumption of RNA synthesis in model simulations. Simulations showed that upon reduction of rRNA synthesis, FC size and spacing increased (Equation 4) while the number of FCs per cell decreased (Figure 3A, Supplementary Movie 5). To experimentally test this prediction, we treated cells with the rRNA transcription inhibitor CX-5461 and conducted time-course imaging⁹⁰ (Supplementary Movie 6). The results showed that FCs became larger over the course of 1 hour (Figure 3B), and upon sustained treatment, generated nucleolar caps (Figure S4A). The size and spacing of FCs increased while the number of FCs per nucleolus decreased over the 1 hour treatment (Figure 3C). Thus, the behavior of FCs upon inhibition of rRNA transcription is consistent with the predictions of the model (Supplementary Movies 5 & 6).

Figure 3: rRNA synthesis maintains fibrillar centers in a state of arrested coarsening.

A. Dynamics of FC protein concentration profiles obtained from numerical simulations of the phase-field model upon first turning off, and later turning on rRNA transcription. Turning off transcription leads to coarsening which is accompanied by an increase in FC size and spacing and a decrease in their number. Turning on rRNA transcription breaks apart these coarsened FCs and arrests coarsening. The FCs increase in number and go back to their smaller size and spacing similar to the starting point.

B. Representative images of FCs, marked by a UBTF-GFP fusion protein, after inhibiting rRNA transcription in murine embryonic stem cells (mESCs) using the drug CX-5461.

C. Transcriptional inhibition by CX-5461 alters FC patterning by increasing FC size and spacing while decreasing their number. p < 0.0001 (t-test) when comparing control vs inhibited for size, spacing, and number.

. Representative images of FCs after restoring rRNA transcription by re-feeding starved mESCs.

E. Restoring rRNA transcription alters FC patterning by decreasing FC size and spacing while increasing their number. p < 0.0001 (t-test) when comparing starved vs fed for size, spacing, and number.

The non-equilibrium model also predicts that resuming rRNA transcription should decrease FC size and spacing while increasing the number of FCs, leading to the re-establishment of normal FC patterning (Figure 3A). Since treatment with CX-5461 is irreversible⁹¹, we used an alternative experimental approach to reduce and then resume rRNA synthesis. We first nutrient-starved mESCs, a treatment known to reduce rRNA transcription⁹², and then resumed rRNA synthesis by refeeding cells. Refeeding cells decreased FC size and spacing and increased FC number compared to starved cells (Figure 3D, 3E). Thus, the behavior of FCs upon resumption of rRNA synthesis is consistent with the predictions of the model (Supplementary Movies 5 & 7). The FC volume fraction also remained the same between distributed and coarsened states upon inhibition or resumption of rRNA synthesis (Figure S4B–D). Taken together, results from simulations and experiment agreed and were consistent with a non-equilibrium model of condensate patterning driven by RNA synthesis.

Altered condensate patterning of fibrillar centers impairs RNA processing

We next tested whether altered patterning influences condensate functionality. To test whether changes in FC patterning were associated with changes in ribosomal biogenesis, we developed a cell line capable of inducible expression of GFP-tagged TCOF1, a nucleolar protein that interacts with RNA polymerase I and regulates transcription⁹³. Overexpression of TCOF1 is known to increase FC sizes⁹⁴, and we confirmed that overexpression of TCOF1 led to larger FCs (Figure 4A). Quantification of these data showed an increase in FC size and spacing concomitant with a decrease in the number of FCs per nucleolus (Figure 4B). Expression of TCOF1 retained the spatial nesting of FC, DFC and GC phases (Figure S5), and as a control, expression of DFC (FBRL) or GC (NPM1) components did not alter FC patterning (Figure S6). We next evaluated the rates of rRNA transcription and processing upon TCOF1 overexpression. Whereas TCOF1 overexpression reduced the rate of rRNA transcription by 15% relative to cells without overexpression (Figure 4C), rRNA processing was reduced by 50% compared to the control (Figure 4D). These results suggest that changes in FC patterning are associated with modest reductions in rRNA transcription and more substantial reductions in rRNA processing.

Figure 4: Altered condensate patterning of fibrillar centers impairs rRNA processing.

A. Representative images of FCs in murine embryonic stem cells (mESCs) marked by a TCOF-GFP fusion protein upon overexpression of TCOF, a constituent protein of FCs.

B. TCOF overexpression alters condensate patterning by increasing the size and spacing of FCs while reducing their number. p < 0.0001 (t-test) when comparing −TCOF vs +TCOF for size, spacing, and number.

C. TCOF overexpression does not significantly alter the rRNA transcriptional efficiency in mESCs. p = 0.04 (t-test); Cohen’s d = 0.88.

D. TCOF overexpression alters the rRNA processing efficiency in mESCs. This indicates that a larger fraction of rRNA passes through the DFC without being processed. p = 0.0002 (t-test); Cohen’s d = 1.73.

E. Representative image with TCOF-GFP fusion protein (green) marking the FCs and EU stain (magenta) marking the nascent rRNA for the control mESCs. Cells are fixed ten minutes post EU removal.

F. Dynamics of nascent rRNA diffusion from the FCs for the control mESCs and those with TCOF overexpression. The TCOF-GFP and EU intensity profiles for each time point are obtained by averaging across several FCs in several cells.

G. The average EU intensity inside FCs vs. time for the control and TCOF overexpression case. TCOF overexpression, which leads to larger FCs, results in a more gradual decline in the EU intensity.

We sought to understand the relationship between FC patterning and rRNA processing. Processing of rRNA is intimately linked with its transport out of the FC into the surrounding DFC region of the nucleolus^9,95. We hypothesized that altered FC patterning in TCOF-overexpressed cells may therefore also impact the transport of rRNA from the FC to the DFC. To measure the rate of rRNA efflux in the setting of disrupted FC patterning, we metabolically labeled nascent rRNA within normal or TCOF1-overexpressed FCs (Figure 4E). Tracking the radial distribution of nascent rRNA signal outward from the FC as a function of time (Figure 4F) provided a measure of the rRNA efflux rate. Consistent with our hypothesis, tracking rRNA signal within FCs over time revealed that nascent rRNA was retained longer within FCs in the setting where patterning was disrupted by overexpression of TCOF1 as compared to FCs with normal patterning (Figure 4G). Moreover, we also used a mathematical first-passage time analysis to compare the expected residence time of rRNA in the DFC in TCOF1-overexpressed and wild-type cells (STAR Methods). Contrasting a scenario with many distributed FCs inside of a DFC versus a setup with a single consolidated FC inside of a spherical DFC revealed that with TCOF1-overexpression, the residence time of rRNA in the DFC should decrease to a value between 29% and 64% of the wild-type reference. Assuming a constant rRNA processing rate in the DFC, this is consistent with a dramatic decrease in the processing efficiency and recent reports⁹⁶. Together, these results provide a link between condensate patterning and rRNA processing.

Discussion

In this study, we propose a physical model supported by experimental evidence that active RNA synthesis can regulate condensate patterning in cells. In this model, non-equilibrium processes are essential for describing condensate dynamics, and predictions from the model are consistent with behaviors observed for fibrillar centers in cells. Such non-equilibrium processes arrest the growth and coalescence of condensates, which arranges condensates in a distributed rather than centralized pattern (Figure 5A). Perturbation of active transcription disrupted condensate patterning, and altered condensate patterning disrupted rRNA biosynthesis. Together, these results provide a mechanistic and regulatory link between active processes, condensate patterning and biochemical function.

Figure 5: Non-equilibrium control of condensate patterning and function.

A. Active RNA synthesis maintains condensates in a non-equilibrium state of distributed patterning, with many condensates having a typical size and spacing. In the absence of RNA synthesis, condensates approach equilibrium through coarsening leading to a centralized patterning with a few large droplets.

B. Centralized patterning leads to slower transport of molecules to condensate interface, smaller surface area per unit volume, and limited availability of products of reactions happening within the condensate. Distributed patterning on the other hand leads to faster transport to condensate interface, larger surface per unit volume, and enhanced availability of products.

Our results indicate that the physical basis of FC condensate patterning involves arrested coarsening arising due to active RNA biosynthesis. This arrested coarsening arises because the diffusively moving RNA mediates effective long-ranged interactions among FC and non-FC proteins, reminiscent of a screened Coulomb potential (STAR Methods). In other systems, Coulombic interactions can also arise from active production and degradation of the condensate material itself^{48,78,97–99} as first studied by Glotzer et al.^51,52, or from stochiometric constraints in passive block copolymer melts¹⁰⁰. Several additional active and passive mechanisms may also impede condensate coarsening in cells. The intracellular environment contains physical barriers, such as the cytoskeleton or chromatin, that limit condensate mobility and coalescence^101,102. Condensate coalescence may also be limited if the condensate forms on or associates with an immobile surface, such as an intracellular membrane¹⁰³. Proteins binding to the condensate surface may act as emulsifying agents that stabilize surface tension and impair coalescence¹⁰⁴. Finally, cell cycle-regulated formation and dissolution³⁰ may impair condensate coarsening by dissolving them before growth by coarsening can take place. Each of these diverse mechanisms can affect the patterning of a condensate and may be regulating proper biological function.

The regulation of condensate patterning may serve useful biological functions in the cell. Previous efforts have demonstrated that disruption of biological processes often leads to immediate changes in condensate patterning. For instance, inhibition of RNA splicing causes nuclear speckle condensates to grow, become more spherical and coalesce in the interchromatin space³⁴. Our findings suggest that condensate patterning can in turn regulate biochemical processes, and the arrangement of such patterning could influence the efficiency of consecutive biochemical reactions, especially when those reactions happen near interfaces (Figure 5B). Patterning can affect the transport of reactants and products, the effective surface area of interfaces, and the local availability of biomolecules throughout the cell (Figure 5B). A recent study on nuclear speckle condensates highlighted the importance of these patterning features, as speckle proximity was found to enhance RNA splicing efficiency due to localized gradients of splicing factors¹⁰⁵. Thus, condensate patterning may be an independent and emergent means by which cells regulate biochemical processes.

The results of this study should be relevant to other condensates involved in RNA biogenesis. For instance, histone locus bodies are condensates that concentrate proteins involved in the transcription and processing of mRNAs that code for histones^106,107. There are several histone locus bodies in the nurse cells of Drosophila, and their number and size remain the same as these cells grow¹⁰⁷. Transcriptional condensates are involved in the regulation of genes transcribed by RNA polymerase II^7,8,29 and are located in close spatial proximity to splicing condensates which concentrate splicing factors¹⁰. We speculate that RNA transcription might be involved in controlling the size and patterning of these condensates through similar mechanisms. Indeed, inhibition of transcription or splicing alters the patterning and dynamics of transcriptional and splicing condensates^10,29.

In the case of FCs, our work suggests that active maintenance of condensate patterning is required for proper rRNA biogenesis. In our model, RNA is an important driver that mediates arrested coarsening. Given that FC patterning is controlled by rRNA production and patterning itself contributes to proper rRNA processing, this patterning mechanism may have evolved to serve as a positive feed forward regulatory step. Beyond the FCs, our results provide a foundation to study the function and regulation of condensate patterning through active processes, which is likely to be involved in many cellular condensates and disrupted in human disease²⁴.

Limitations of the Study

In the present study, we have considered the category of FC proteins as biomolecules that undergo regular diffusion. If the difference in the diffusion coefficients of FC proteins and rRNA is large enough, then our model in principle admits oscillatory dynamics¹⁰⁸ instead of a stationary state. However, FCs also contain chromatin, a large polymer consisting of DNA in complex with proteins, which undergoes subdiffusion. Future research is required to investigate how subdiffusion of chromatin affects FC dynamics.

RESOURCE AVAILABILITY

Lead Contact:

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Arup K. Chakraborty (arupc@mit.edu).

Materials Availability:

Plasmids and cell lines generated in this study will be made available upon reasonable request.

Data and Code Availability:

• Source data statement. Data generated in this study will be made available upon reasonable request.

• Code statement. All original code has been deposited at Zenodo and is publicly available as of the date of publication. DOIs are listed in the key resources table.

• Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

KEY RESOURCES TABLE.

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Antibodies
UBTF	Santa Cruz Biotechnology	sc-13125; RRID:AB_671403
TCOF1	Sigma	WH0006949M2; RRID:AB_1843902
Fibrillarin	Abcam	ab5821; RRID:AB_2105785
Goat anti-Mouse IgG Alexa Fluor 488	Invitrogen	A11029; RRID:AB_2534088
Goat anti-Rabbit IgG Alexa Fluor 647	Invitrogen	A21244; RRID:AB_2535812
Chemicals, peptides, and recombinant proteins
DMSO	Sigma	D2438
CX-5461	MedChemExpress	HY-13323
Doxycycline	Sigma	D9891-5G
Poly-L-ornithine	Sigma	P4957-50ML
Laminin (Mouse)	Thermo Scientific	CB40232-1MG
Hoechst 33342	Thermo Scientific	62249
Critical commercial assays
PowerTrack SYBR Green Master mix	Applied Biosystems	A46109
NEBuilder HiFi DNA Assembly Master Mix	NEB	E2621S
RNeasy Plus Kit	QIAGEN	74134
Phusion polymerase	NEB	M0531S
Lipofectamine 3000 Transfection Agent	ThermoFisher	L3000001
Qiaquick Gel Extraction Kit	QIAGEN	28706
Click-iT™ RNA Alexa Fluor™ 594 Imaging Kit	Invitrogen	C10330
Click-iT™ Nascent RNA Capture Kit	Invitrogen	C10365
SuperScript® VILO™ cDNA synthesis kit	Invitrogen	11754-050
Experimental models: Cell lines
V6.5 murine embryonic stem cells (mESCs)	Jaenisch laboratory, Whitehead Insitute/MIT	N/A
UBTF-mEGFP mESCs	This paper	N/A
doxycycline inducible TCOF1-GFP mESCs	This paper	N/A
Human embryonic stem cell	ATCC	SCRC-2002
A549	ATCC	CRM-CCL-185
HCT116	ATCC	CCL-247
U2OS	ATCC	HTB-96
HEK293	ATCC	CRL-1573
Mel JuSo	Golub laboratory, Broad Insitute/MIT	N/A
Oligonucleotides
Primers for qPCR	This paper	See methods section
sgRNA for CRISPR knock-in of mEGFP to UBTF C-terminus	This paper	See methods section
Recombinant DNA
px330_mcherry_sgRNA-UBTF	This paper	N/A
pUC19_RT_UBTF-mEGFP_C.term	This paper	N/A
pbfh_TCOF1-EGP	Banani et al., 2022	Modified from pbfh_GFP
Software and algorithms
Fiji image processing package	Schindelin et al., 2012	https://fiji.sc/
Prism Version 9.4.0 Software and algorithms	GraphPad
Jupyter Notebook	Project Jupyter	https://jupyter.org/
MATLAB v2021b	MathWorks
Cellpose v0.7.2	Stringer et al., 2021	https://www.cellpose.org/
Python code generated in the study for simulations and data analysis	This paper	https://doi.org/10.5281/zenodo.18892655
Other
35 mm glass-bottom imaging dishes	Mattek Corporation	P35G-1.5-20-C
Glass-bottom 24-well plates	Mattek Corporation	P24G-1.5-13-F

STAR Methods

EXPERIMENTAL MODEL AND STUDY PARTICIPANT DETAILS

Mouse stem cell culture

Source.

V6.5 murine embryonic stem cells (mESCs) were a gift from the Jaenisch lab at Whitehead Institute.

Culturing Media.

DMEM-F12 (Life Technologies, 11320082), 0.5X B27 supplement (Life Technologies, 17504044), 0.5X N2 supplement (Life Technologies, 17502048), an extra 0.5mM L-glutamine (Gibco, 25030-081), 0.1mM b-mercaptoethanol (Sigma, M7522), 1% Penicillin Streptomycin (Life Technologies, 15140163), 0.5X nonessential amino acids (Gibco, 11140-050), 1000 U/ml LIF (Chemico, ESG1107), 1μM PD0325901 (Stemgent, 04-0006-10), 3μM CHIR99021 (Stemgent, 04-0004-10). This recipe is so-called “2i media”.

Growing condition.

37°C with 5% CO2 in a humidified incubator.

Surface coating for daily culture.

Tissue culture plates were pretreated with 0.2% gelatinized (Sigma, G1890) for at least 15 minutes at 37°C. Plates were rinsed once with PBS before seeding cells.

Surface coating for imaging experiments.

Glass dishes (Mattek Corporation P35G-1.5-20-C) or glass-bottom 24-well plates (Mattek Corporation P24G-1.5-13-F) were coated with 5 μg/ml of poly-L-ornithine (Sigma-Aldrich, P4957) for 30 min at 37°C and with 5μg/ml of Laminin (Corning, 354232) for 2hrs-16hrs at 37°C.

Detaching.

Cells were washed in PBS (Life Technologies, AM9625), followed by treating with TrypLE (Life Technologies, 12604021) to detach from plates. TrypLE was quenched with FBS/LIF-media, DMEM K/O (Gibco, 10829-018), 1X nonessential amino acids, 1% Penicillin Streptomycin, 2mM L-Glutamine, 0.1mM β-mercaptoethanol, and 15% Fetal Bovine Serum (FBS) (Sigma Aldrich, F4135). Cells were spun at 1000rpm for 3 min at room temperature (RT), resuspended in 2i media and replated.

Tips to maintain stemness.

Cells were passaged with a 1:10 to 1:20 dilution immediately when reaching >80% confluency. Gentle pipetting was applied to minimize mechanical stress. Culturing media were refreshed every 1–2 days. Culturing media were used within 2 weeks after adding LIF and two inhibitors (PD0325901 and CHIR99021). 50 mL aliquots were made for the 2i media and the cycles of prewarming to 37°C were minimized for each aliquot.

METHOD DETAILS

Endogenously-tagged UBTF-GFP generation

CRISPR/Cas9 was used to generate endogenously-mEGFP-tagged UBTF. Oligos coding for guide RNAs targeting the N-terminus of mouse UBTF were cloned into a px330 vector expressing Cas9 and mCherry (gift from R. Jaenisch). The guide RNA sequence is:

5’ CTGGAGAATGAACGGAGAAG 3’

Repair templates were cloned into a pUC19 vector (NEB) containing mEGFP, a GS linker and 800 bp homology arms flanking the N-terminus of mouse Ubtf. mESCs were plated in a 6-well plate and transfected with 1.25 μg px330 vector and 1.25 μg repair templates using lipofectamine 3000. Cells were sorted 2 days after transfection for mCherry and 1 week after first sort for mEGFP. Twice sorted cells were then sparsely plated in a 15 cm plate and colonies were picked after 4 days and seeded into a 96 well plate. 2–4 days after colony picking, cells were passaged into 3 plates. 1 plate was used for genotyping, and 2 were frozen down at −80°C in 10% DMSO, 10% FBS and 80% 1x 2i media. Genotyping was done by amplifying the region surrounding the N-terminus, looking for increase in amplicon size as a result of tagging. One of the colonies with homozygous GFP insertion was picked and expanded for experiments.

Doxycycline inducible expression of TCOF1-GFP generation

PiggyBac transposon for doxycycline inducible expression of TCOF1-GFP, described in our previous publication²⁴, was co-transfected with transposase following the instruction of manufacturer (PB210PA-1, System Biosciences). Three days following co-transfection, cells were hygromycin selected and expanded. Cells were then doxycycline treated and sorted for single colonies. Six single colonies were expanded and verified, and one with the best titration range (titratable in-between 1 ng/mL–1 μg/mL of doxycycline) was picked for final experiments.

Treatments

RNA Pol-I inhibition.

RNA Polymerase I transcription inhibitor CX-5461 (MedChemExpress HY-13323) was reconstituted in DMSO with 10 mM stock concentration and stored in −80°C. The inhibitor was 1:1,000 diluted into culturing media (final concentration as 10 μM) for over the course of 1 hour. For vehicle control, 0.2% of Glacial Acetic Acid was added to culturing media.

Nutrient starvation and refeeding.

Amino acid-starved media was made with the same composition as the mESC culturing media, except with the substitution of amino acid-depleted DMEM F-12 (USBiological D9807-10). Amino acid-starved media and control culturing media were fresh prepared in parallel. For nutrient starvation, 2i media was either replaced with nutrient-starved media or control media, followed by culturing for 6 hours and proceeding to the downstream experiments. For refeeding, nutrient-starved media was replaced with control media, followed by culturing for 1.5 hours and proceeding to the downstream experiments.

TCOF1-GFP overexpression.

0.1 μg/mL of doxycycline was added to culturing media for 20 hours, leading to TCOF1-GFP overexpression (also referred to as +TCOF1). For negative control, 1 ng/mL of doxycycline was added to culturing media for 20 hours, leading to TCOF1-GFP basal expression (also referred to as −TCOF1).

Live cell imaging

UBTF-mEGFP proteins were endogenously expressed in mESCs. The live-cell imaging experiments were performed with ZEISS 980 with a 63x 1.47NA objective and with the Airyscan super-resolution mode, and processed by ZEN Blue. For each frame, a z-stack was collected sectioning a monolayer of cells with the optimized interval suggested by ZEN Blue and 3D processed. The maximum intensity projection of the processed z-stack was generated for illustration and quantification.

For time-lapse movies, frames were taken at a 5-minute interval, with each frame being a z-stack.

For images containing nuclear staining, Hoechst 33342 (Thermo Scientific, 62249) was 1:2,000 diluted in media, and stained the sample for five minutes and removed prior to imaging.

Quantifying FC size, spacing, and number

Following Airyscan processing, 2D images or maximum intensity projection of 3D images were Gaussian smoothed (sigma=1), and applied with Laplace of Gaussian transformation (sigma=3) using the scikit-image package in Python. Puncta were identified above a threshold intensity 3 standard deviations above the mean of the image as previously described²⁹. Following puncta identification, patterning was quantified as three values: size, spacing (within each nucleolus), and number (per nucleus).

Each image had a field of view that was 80 μm × 80 μm with around 10–30 nucleoli with several FCs which clustered together within each nucleolus. We extracted the area of the regions of high intensity and the x-y position coordinates of the centroids of these regions. Using these coordinates, we clustered the FCs using density-based clustering (DBSCAN) using a threshold of finding at least 3 members in 2 μm neighborhood to cluster FCs. Since each cluster corresponded to a nucleolus, we reported the number of FCs in that cluster as the number of FCs present in that nucleolus. The diameters of the FCs were calculated as $d=\sqrt{A/\pi }$ where $A$ is the area of the FC in μm². The spacing between FCs was calculated as the distance between the centroid of each FC and its nearest neighbor. The size increments of small FCs appear discrete near the resolution limit of the microscope.

Controlling for variability between nucleoli

In Figure 1C, we have shown the aggregated distribution of measured FC sizes across several nucleoli. Depending on the underlying mechanism, the kinetics of phase separation leads to a characteristic shape of the condensate size distribution (Table 2).

Table 2.

Physical mechanisms leading to different FC size distributions

Mechanism	Distribution	Probability density function	Parameters
Ostwald ripening	Lifshitz-Slyozov-Wagner (LSW)	$p(r;\mu )=\frac{4}{9\mu }\frac{{\left(\frac{r}{\mu }\right)}^2\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{2r/\mu }{2r/\mu -3}\right)}{{\left(1+\frac{r}{3\mu }\right)}^{7/3}{\left(1-\frac{2r}{3\mu }\right)}^{11/3}}$	$\mu$ (Mean of FC size)
Brownian-motion coalescence	Log-Normal	$p\left(r;{\mu }_L,{\sigma }_L\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{L}r}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\left(\mathrm{l}\mathrm{n}r-{\mu }_L\right)}^2}{2{\sigma }_L^2}\right)$	${\mu }_L$ (Mean of log FC size) ${\sigma }_L$ (Standard deviation of log FC size)
Size control	Gaussian	$p(r;\mu ,\sigma )=\frac{1}{\sqrt{2\pi }\sigma }\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{(r-\mu {)}^2}{2{\sigma }^2}\right)$	$\mu$ (Mean of FC size) $\sigma$ (Standard deviation of FC size)

In addition, nucleolus-to-nucleolus variability in the transcription rates, FC protein amounts, or even the time of onset of phase separation, can broaden the distribution of FC sizes measured across all nucleoli. This can lead to confounding effects if we directly compare the aggregated size distribution of all FCs across several nucleoli (Figure 1C) with the shapes of the distributions described in the above table. Before we can perform such a comparison, we therefore need to deconvolve the FC size distribution from the inter-nucleolar variability by adequately normalizing the FC sizes. To that end, we computed z-scores of the FC sizes within each nucleolus. For a cluster containing $N$ FCs having sizes $r_i(i=1\dots N)$, the z-scores are computed for each of the above distributions as follows:

1. LSW: $\mu =\frac{1}{N}{\sum }_{i=1}^N{r}_i$ and $z_{LSW,i}=\frac{r_i}{\mu }$;

2. Log-Normal: ${\mu }_L=\frac{1}{N}{\sum }_{i=1}^N\mathrm{l}\mathrm{n}{r}_i$, ${\sigma }_L=\sqrt{\frac{1}{N-1}{\sum }_{i=1}^N{\left(\ln r_i-{\mu }_L\right)}^2}$, and $z_{L,i}=\frac{\ln r_i-{\mu }_L}{{\sigma }_L}$;

3. Gaussian: $\mu =\frac{1}{N}{\sum }_{i=1}^N{r}_i$, $\sigma =\sqrt{\frac{1}{N-1}{\sum }_{i=1}^N{\left(\ln r_i-\mu \right)}^2}$, and $z_i=\frac{r_i-\mu }{\sigma }$.

Even if the inter-nucleolar variability precludes us from directly comparing FC sizes across nucleoli, we can compare the z-scores which account for this variability. Therefore, to get an indication for the mechanisms which give rise to the measured FC sizes, we constructed a distribution of the aggregated z-scores and compared it with the proposed distributions in the above table. To compute a likelihood that the computed z-scores conform to a specific distribution, and to compare the likelihoods of different distributions, we used the Cramer-von Mises test.

The estimated volume fraction of nascent rRNA is small

Our model system of mouse embryonic stem cells contains $N=3$ nucleoli per cell with an average diameter of around $d=3.5\mathrm{\mu }\mathrm{m}$ (Figure 1A). Imaging studies that visualize nascent rRNA transcription from rDNA show that there are about $N_R=100$ nascent rRNA transcripts on each copy of rDNA¹⁰⁹. Moreover, there are about $N_D=200$ copies of rDNA in mouse cells^110,111.

Estimate for the lower bound of rRNA volume fraction.

Using these numbers, and assuming that nascent rRNA is distributed around the entire nucleolus, we can estimate the concentration of nascent rRNA in the nucleolus as:

$$C_R=\frac{N_R{N}_D}{N\frac{\pi }{6}{d}^3}\approx 300\text{nascent}\text{rRNA}/{\mathrm{m}\mathrm{m}}^3.$$ (S1)

In humans, each rDNA gene is a 45 kb long unit with a 31 kb long intergenic spacer^112,113. Thus, a nascent 47S rRNA molecule has a length of roughly 14 kb. With a typical molecular mass of 340 Da per nucleotide (Thermo Fisher Scientific), this leads to a total molecular mass of 4.76 MDa. With 1.89 g/cm³ the mass density of RNA¹¹⁴, this leads to a molecular volume of $v_R\approx 4200{\mathrm{n}\mathrm{m}}^3$. Taken together, this leads to a total volume fraction of rRNA of

$${\varphi }_R^{\mathit{low}}=v_R{C}_R\approx 0.001\ll 1.$$ (S2)

Estimate for the upper bound of rRNA volume fraction.

If we consider the same number of rRNA transcripts distributed over the volume of the $N_{FC}=26$ FCs in each nucleolus, with an average FC diameter of $d_{FC}=0.26\mathrm{\mu }\mathrm{m}$, then we have a nascent rRNA concentration in the vicinity of FCs of

$$C_R=\frac{N_R{N}_D}{N{N}_{FC}\frac{\pi }{6}{d}_{FC}^3}\approx 3\times {10}^4\text{nascent}\mathrm{r}\mathrm{R}\mathrm{N}\mathrm{A}/{\mathrm{\mu }\mathrm{m}}^3,$$ (S3)

and a total volume fraction of rRNA of

$${\varphi }_R^{high}=v_R{C}_R\approx 0.1\ll 1.$$ (S4)

With the estimated volume fraction of rRNA lying in the range between 0.001 and 0.1, we take an intermediate value of ${\varphi }_R\approx 0.01$, which corresponds to a concentration of $C_R=3000$ nascent rRNA/μm³ consistent with Riback et al.⁶⁸.

In comparison, the typical volume fraction occupied by proteins is roughly an order of magnitude higher. The average protein concentration in mouse cells¹¹⁵ is around $C_P=1.5\times {10}^6$ molecules/μm³. The typical diameter of a protein¹¹⁶ is roughly 5 nm. Taken together, this leads to a typical volume fraction of proteins of

$${\varphi }_P=C_P\times \pi /6\times 5^3{\mathrm{n}\mathrm{m}}^3=0.1.$$ (S5)

Mouse UBTF, for example, has a molecular weight of roughly 89.5 kDa and forms dimers (UniProt P25976-1). With 1.37 g/cm³ the mass density of proteins¹¹⁷, this gives a molecular volume of $v_P/2=108{\mathrm{n}\mathrm{m}}^3$ per monomer and a volume fraction of ${\varphi }_P=v_P{C}_P/2\approx 0.16$ where we assume that $C_P$ corresponds to the concentration of monomers. These estimates are consistent with Ref.¹¹⁸, which measured 0.215 g/cm³ for the protein density in dense fibrillar centers in Xenopus oocyte nucleoli. Taking into account the mass density of proteins¹¹⁷, 1.37 g/cm³, this would correspond to a protein volume fraction of ${\varphi }_P\approx 0.16$. This is also consistent with the observation that most nuclear condensates including the nucleolus contain a low density of proteins¹¹⁹. Taken together, our estimates suggest that the nascent rRNAs occupy only a small fraction of the nucleolar volume compared to proteins. Below, we will use the assumption that ${\varphi }_R\ll 1$ to derive a simplified expression for the free energy functional.

These estimates also suggest that the ratio of molecular volumes between rRNA and FC proteins is roughly

$$l_R=\frac{v_R}{v_P}\approx 20.$$ (S6)

Free energy of a ternary mixture containing FC proteins, non-FC molecules, and rRNA

In the following, we derive the free energy functional of FCs reported in the main text. We consider the nucleolus as a ternary mixture containing three categories of biomolecules: (1) FC molecules with molecular volume $v_P$, concentration $c_P$, and occupying the volume fraction ${\varphi }_P=v_P{c}_P$; (2) nascent rRNA with molecular volume $v_R$, concentration $c_R$, and occupying the volume fraction ${\varphi }_R=v_R{c}_R$; and (3) non-FC molecules with molecular volume $v_S$, concentration $c_S$, and occupying the volume fraction ${\varphi }_S=v_S{c}_S=1-{\varphi }_P-{\varphi }_R$. The free energy of this ternary mixture is given by^86,120

$$\frac{ℱ}{k_{B}T}=\int d^3\mathit{x}\left[\frac{f_s}{k_{B}T}+\frac{f_e}{k_{B}T}+\frac{f_{\kappa }}{k_{B}T}\right],$$ (S7)

with $f_s$ the free energy density of mixing due to entropy, $f_e$ the energy density of local molecular interactions, and $f_{\kappa }$ penalizing concentration gradients at the interfaces between different phases due to the finite range of molecular interactions.

According to the Flory-Huggins theory^121,122 of polymer solutions, the free energy density of mixing is given by:

$$\frac{f_s}{k_{B}T}=\sum _{i\in \{P,R,S\}}{c}_i\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_i.$$ (S8)

Again following the Flory-Huggins theory^121,122, we take pairwise interactions between molecules into account by the local interaction energy density

$$\frac{f_e}{k_{B}T}=\sum _{i,j\in \{P,R,S\}}\frac{1}{2}{\chi }_{ij}\left(c_i{\varphi }_j+c_j{\varphi }_i\right),$$ (S9)

which is given by the concentration $c_i$ of one species, weighted with the probability $\sim {\varphi }_j$ of encountering a molecule of another species, multiplied with the (mean-field) interaction energies ${\chi }_{ij}{k}_{B}T$ between pairs of molecules of these species where ${\chi }_{ij}$ is the Flory-Huggins parameter. The sign convention is such that negative Flory-Huggins parameters indicate favorable pairwise attraction.

Finally, because the pairwise interactions among molecules have a finite interaction range $w$, there is an interfacial energy

$$\frac{f_{\kappa }}{k_{B}T}=-w^2\sum _{i,j\in \{P,R,S\}}\frac{1}{2}{\chi }_{ij}\left(\mathbf{\nabla }{c}_i\mathbf{\nabla }{\varphi }_j+\mathbf{\nabla }{c}_j\mathbf{\nabla }{\varphi }_i\right),$$ (S10)

which penalizes concentration gradients. Note that the interaction range $w$ controls the width of the interface between the dense and light phases of the FC proteins when they phase separate. We will next simplify the above Flory-Huggins free energy, which has ten parameters in total ($w$, six independent ${\chi }_{ij}={\chi }_{ji}$, and three $v_i$), by using that the rRNA volume fraction is small, ${\varphi }_R\ll 1$, and rRNAs are much larger than nucleolar proteins (Eq. S6) as we discussed in the previous section.

Simplified free-energy functional with a reduced set of parameters

Since ${\varphi }_R\ll 1$, one has ${\varphi }_S=1-{\varphi }_P-{\varphi }_R\approx 1-{\varphi }_P$ and the entropy of mixing can be simplified via a Taylor expansion,

$$\frac{f_s}{k_{B}T}\approx \frac{1}{v_P}\left[{\varphi }_P\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_P+\frac{{\varphi }_R}{l_R}\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_R+\frac{1-{\varphi }_P}{l_P}\mathrm{l}\mathrm{o}\mathrm{g}\left(1-{\varphi }_P\right)+\frac{{\varphi }_R^2}{2l_P}+\frac{{\varphi }_P{\varphi }_R}{l_P}\right],$$ (S11)

where we have defined $l_R:=v_R/v_P$ and $l_P:=v_S/v_P$ and only kept the lowest order relevant terms (quadratic terms, since constant and linear terms do not affect the thermodynamic stability). Because ${\varphi }_R\ll 1$, one can also neglect the term $\propto {\varphi }_R^2$. As discussed in the previous section, the length of the nascent rRNA molecule is around $l_R\approx 20$ times the length of a typical intrinsically disordered region of a protein. The ratio $l_P$ between the typical molecular volumes of non-FC molecules, which include solvent, and FC proteins is expected to be small. To nondimensionalize the in-silico dynamics of the FCs, we choose $V_m=v_P$ as a reference measure of volume. This choice is arbitrary, and has no effect on the results. Moreover, as in the main text, we use the thermal energy $k_{B}T$ as a reference value for the energy.

Analogously, the energy due to pairwise molecular interactions expands to:

$$\begin{array}{ll}\frac{f_e}{k_{B}T}=& \frac{1}{v_P}\left[{\chi }_{PP}{\varphi }_P^2+\frac{{\chi }_{RR}}{l_R}{\varphi }_R^2+\frac{{\chi }_{SS}}{l_P}{\left(1-{\varphi }_P-{\varphi }_R\right)}^2+{\chi }_{RS}\left(\frac{1}{l_R}+\frac{1}{l_P}\right){\varphi }_R\left(1-{\varphi }_P-{\varphi }_R\right)\right.\\ & \left.+{\chi }_{RP}\left(1+\frac{1}{l_R}\right){\varphi }_R{\varphi }_P+{\chi }_{PS}\left(1+\frac{1}{l_P}\right){\varphi }_P\left(1-{\varphi }_P-{\varphi }_R\right)\right].\end{array}$$ (S12)

Since terms in the energy that are either constant or linear in ${\varphi }_i$ do not affect the thermodynamic stability of the system or phase separation, we retain only the quadratic terms of the form ${\varphi }_i^2$ and ${\varphi }_i{\varphi }_j$ in the above expression. Rearranging and grouping coefficients of the quadratic terms, the above expression reduces to:

$$\begin{array}{ll}\frac{f_e}{k_{B}T}\approx & \frac{1}{v_P}\left\{-\left[{\chi }_{PS}\left(1+\frac{1}{l_P}\right)-{\chi }_{PP}-\frac{{\chi }_{SS}}{l_P}\right]{\varphi }_P^2\right.\\ & \left.+\left[2\frac{{\chi }_{SS}}{l_P}+{\chi }_{RP}\left(1+\frac{1}{l_R}\right)-{\chi }_{RS}\left(\frac{1}{l_R}+\frac{1}{l_P}\right)-{\chi }_{PS}\left(1+\frac{1}{l_P}\right)\right]{\varphi }_R{\varphi }_P\right\},\end{array}$$ (S13)

where we have also used ${\varphi }_R\ll 1$ to neglect terms $\propto {\varphi }_R^2$. By the above grouping, we obtained the following reduced set of independent parameters:

1. The interaction energy of FC proteins with non-FC molecules relative to their respective self-interaction energies is characterized by

   $$\beta :=\left[{\chi }_{PS}\left(1+\frac{1}{l_P}\right)-{\chi }_{PP}-\frac{{\chi }_{SS}}{l_P}\right]\sim \frac{1}{l_P}\left[{\chi }_{PS}-{\chi }_{SS}\right],$$ (S14)

   where we have used that the non-FC molecules (mainly solvent) are much smaller than FC molecules ($l_P\ll 1$). This parameter indicates if FC proteins can spontaneously phase separate. For $\beta <0$, the FC proteins have favorable interactions with the non-FC molecules and the solution remains well-mixed. For $\beta >{\beta }_c>0$, the FC proteins have unfavorable interactions with the non-FC molecules and can therefore phase separate.

2. The effective interaction energy between FC proteins and RNA is characterized by

   $$\mathrm{\Delta }\chi :=\left[\frac{1}{l_P}+{\chi }_{RP}\left(1+\frac{1}{l_R}\right)-{\chi }_{PS}\left(1+\frac{1}{l_P}\right)+2\frac{{\chi }_{SS}}{l_P}-{\chi }_{RS}\left(\frac{1}{l_R}+\frac{1}{l_P}\right)\right].$$ (S15)

The first term comes from the expansion of the entropy of mixing around ${\varphi }_R\approx 0$. The next two terms indicate how energetically favorable it is to replace non-FC molecules with RNA as interaction partner for FC proteins. The last two terms indicate how energetically favorable it is to replace RNA with non-FC molecules as interaction partner for non-FC molecules. In summary, $\mathrm{\Delta }\chi k_{B}T$ is the interaction energy of RNA with FC proteins relative to the interaction energy of RNA with non-FC molecules. For $\mathrm{\Delta }\chi <0$, it is more energetically favorable to localize RNA to FC protein-rich regions as opposed to non-FC molecule-rich regions. In contrast, for $\mathrm{\Delta }\chi >0$, it is energetically favorable to localize RNA to non-FC molecule-rich regions as opposed to FC protein-rich regions. If all Flory-Huggins parameters have similar magnitudes, then we can use $l_P\ll 1$ and $l_R\gg 1$ to approximate:

$$\mathrm{\Delta }\chi \sim \frac{1}{l_P}\left[1+2{\chi }_{SS}-{\chi }_{RS}-{\chi }_{PS}\right]=\frac{1}{l_P}\left[1+{\chi }_{SS}-{\chi }_{RS}\right]-\beta ,$$ (S16)

This indicates that it can be generally expected that $\mathrm{\Delta }\chi >0$ as long as the interactions between RNA and non-FC molecules are sufficiently attractive (solvation), that is, if ${\chi }_{RS}\ll {\chi }_{SS}$ and $\beta$ is sufficiently small.

Finally, we will also simplify the interfacial energy term. The protein gradients are steepest, as measured by $\left|\mathbf{\nabla }{\varphi }_P\right|$, at the interfaces between the dense and light phases of the phase-separated FC proteins. The width of these interfaces is, as noted earlier, controlled by the interaction range 𝑤 which is related to the typical size of the FC proteins. In contrast, the rRNAs are much longer than the proteins, $l_R\approx 20$, and, moreover, occupy only a small volume fraction. Due to connectivity constraints that arise from the polymeric nature of rRNA, the rRNA gradients will scale as $\left|\mathbf{\nabla }{\varphi }_R\right|\propto {\varphi }_R/l_R$, so that $\left|\mathbf{\nabla }{\varphi }_R\right|\ll \left|\mathbf{\nabla }{\varphi }_P\right|$. In this limit, the interfacial energy term can be approximated as:

$$\frac{f_{\kappa }}{k_{B}T}\approx \frac{1}{v_P}{w}^2\beta {\left|\mathbf{\nabla }{\varphi }_P\right|}^2.$$ (S17)

Putting Eqs. (S7), (S11), (S13), and (S17) together and grouping the different terms by category of biomolecule leads to the simplified free energy functional with just three parameters ($\beta$, $\mathrm{\Delta }\chi$, and $w$) reported in the main text:

$$\frac{ℱ}{k_{B}T}\approx \int \frac{d^3\mathit{x}}{V_m}\underset{:=f_P\left({\varphi }_P\right)V_m/k_{B}T}{\underbrace{[{\varphi }_P\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_P+\frac{1-{\varphi }_P}{l_P}\mathrm{l}\mathrm{o}\mathrm{g}\left(1-{\varphi }_P\right)-\beta {\varphi }_P^2}}+w^2\beta {\left|\mathbf{\nabla }{\varphi }_P\right|}^2+\frac{{\varphi }_R}{l_R}\mathrm{l}\mathrm{o}\mathrm{g}{\varphi }_R+\mathrm{\Delta }\chi {\varphi }_P{\varphi }_R],$$ (S18)

where we have chosen $V_m=v_P$ as a reference measure of volume and defined the effective bulk free energy density of FC proteins, $f_P\left({\varphi }_P\right)$.

Figure 2B describes the physical intuition captured by the different terms in the above free energy functional and its three key parameters: $\beta$, $w$, and $\mathrm{\Delta }\chi$. The phase separation of FC proteins is controlled by their effective interaction energy $\beta$ with the non-FC molecules. Positive values of $\beta$ signify unfavorable interactions between the FC and non-FC molecules. When this interaction energy exceeds a critical value, $\beta >{\beta }_c$ with

$${\beta }_c=\frac{1}{2l_P}{\left(1+\sqrt{l_P}\right)}^2>\frac{1}{2l_P},$$ (S19)

the unfavorable interactions will cause FC proteins to spontaneously demix and phase separate from the non-FC molecules^65,80. This critical value scales as ${\beta }_c\sim 1/l_P$ analogously to the interaction parameters $\beta$ and $\mathrm{\Delta }\chi$. Since FC proteins like UBTF, RNA Pol I, and TCOF1 were shown to demix both in vitro and in vivo from non-FC molecules like Nucleolin, Nucleophosmin, and Fibrillarin^80,81, we choose $\beta >{\beta }_c$ for the present study. The critical point corresponds to the following volume fraction of FC proteins:

$${\varphi }_c=\frac{l_P-1+2\beta l_P}{4\beta l_P}.$$ (S20)

Note that, for $\beta \approx {\beta }_c$, the critical volume fraction is ${\varphi }_c=\sqrt{l_P}/\left(1+\sqrt{l_P}\right)$, which gives an upper bound for the critical point. Thus, the critical point can have a small value when the non-FC molecules are smaller than the FC proteins, $l_P<1$. The category of non-FC molecules includes nucleophosmin, which has a molecular weight of 32.6 kDa (UniProt Q61937) and is thus ~0.18 the size of a UBTF dimer (UniProt P25976-1) as well as water. Thus, we expect $l_P\ll 1$.

The formation of the FC interface is penalized by an interface stiffness $\beta w^2$ where $w$ is proportional to the interface width. Finally, $\mathrm{\Delta }\chi$ is a key parameter that measures the differential affinity between the nascent rRNA and FC proteins relative to non-FC proteins. More specifically, when $\mathrm{\Delta }\chi <0$, the FC proteins have a very high affinity for rRNA. This is physically implausible as demonstrated by Eq. (S16), because rRNA molecules are very large, and $\mathrm{\Delta }\chi$ is instead controlled mostly by the interactions of rRNA with non-FC material. Specifically, when $\mathrm{\Delta }\chi >0$, the non-FC proteins have a higher affinity for rRNA than for other non-FC proteins, indicating solvation.

Dynamics of FC proteins

Compared to the dynamics of rRNAs which have a residence time of around 30 minutes in the nucleolus⁹, the turnover of proteins takes several hours and is therefore slow^123,124. Hence, the dynamics of FC proteins must conserve their total mass and follow the model B formalism as described by Hohenberg and Halperin⁸². In this formalism of non-equilibrium thermodynamics^125,126, gradients in the chemical potential of FC proteins,

$$\frac{{\mu }_P}{k_{B}T}=\frac{v_P}{k_{B}T}\frac{\delta ℱ}{\delta {\varphi }_P}=1-\frac{1}{l_P}+\log {\varphi }_P-\frac{1}{l_P}\log \left(1-{\varphi }_P\right)-2\beta {\varphi }_P-2\beta w^2{\mathbf{\nabla }}^2{\varphi }_P+\mathrm{\Delta }\chi {\varphi }_R,$$ (S21)

where $v_P=V_m$, give rise to thermodynamic forces ${\mathit{f}}_P=-\mathbf{\nabla }{\mu }_P$. These thermodynamic forces result in diffusive fluxes ${\mathit{j}}_P=-M_P\left({\varphi }_P\right)\mathbf{\nabla }{\mu }_P/k_{B}T$ where the mobility (or Onsager) coefficient $M_P\left({\varphi }_P\right)$ generally depends on the volume fraction ${\varphi }_P$. It typically has the form $M_P\left({\varphi }_P\right)=D_P{\varphi }_P\left(1-{\varphi }_P\right)$ for polymer solutions^84–87, as was shown in studies on dense two-component lattice gas models^84,85, where $D_P$ refers to the diffusion coefficient of a lattice unit of protein. Since the density of FC proteins is low, we actually only need the part of the mobility function which is proportional to the volume fraction. This mobility function can be justified by considering the interdiffusion of two different components, which are labeled with $i$ and $n$ in the following. The first component has molecular volume $v_i$ and volume fraction ${\varphi }_i$. The second component has molecular volume $v_n$ and volume fraction ${\varphi }_n={\varphi }_0-{\varphi }_i$, where ${\varphi }_0$ is the total volume fraction available to both components. Mass exchange is driven by equal and opposite forces ${\mathit{f}}_i+{\mathit{f}}_n=0$ that act on the two components. Interdiffusion of the two components corresponds to a frictional force proportional to the velocity difference, leading to the force balance equation ${\mathit{f}}_i-\zeta \left({\mathit{v}}_i-{\mathit{v}}_n\right)=0$, with friction coefficient $\zeta$. The first component has the effective velocity ${\mathit{v}}_i={\mathit{j}}_i/{\varphi }_i$ where ${\mathit{j}}_i$ is a volumetric flux, and the second component has the effective velocity ${\mathit{v}}_n={\mathit{j}}_n/\left({\varphi }_0-{\varphi }_i\right)$. Importantly, mass conservation implies ${\mathit{j}}_i+{\mathit{j}}_n=0$, that is, a flux of the first component will lead to a counter-flux of the second component. With these conditions, one can solve the force balance equation for the current, ${\mathit{j}}_i={\zeta }^{-1}{\varphi }_i\left(1-{\varphi }_i/{\varphi }_0\right){\mathit{f}}_i$. Therefore, the mobility function has the form $M_P\left({\varphi }_P\right)\propto {\varphi }_P\left(1-{\varphi }_P\right)$ if the solution is near binary, ${\varphi }_0\approx 1$. Since the volume fraction of RNA is much lower than that of FC proteins, we can also use this mobility function in the presence of RNA. Taken together, the dynamics of the FC proteins is given by:

$$\begin{array}{ll}\frac{\partial {\varphi }_P(\mathit{x},t)}{\partial t}& =-\mathbf{\nabla }\cdot \underset{{\mathit{j}}_P}{\underbrace{\left[-\frac{M_P\left({\varphi }_P\right)}{k_{B}T}\mathbf{\nabla }\left(V_m\frac{\delta ℱ}{\delta {\varphi }_P}\right)\right]}}\\ & =D_P\mathbf{\nabla }\cdot \left[\left(1+\frac{1-l_P}{l_P}{\varphi }_P-2\beta {\varphi }_P\left(1-{\varphi }_P\right)-2\beta {\varphi }_P\left(1-{\varphi }_P\right)w^2{\mathbf{\nabla }}^2\right)\mathbf{\nabla }{\varphi }_P+\mathrm{\Delta }\chi {\varphi }_P\left(1-{\varphi }_P\right)\mathbf{\nabla }{\varphi }_R\right].\end{array}$$

For better numerical performance and stability of our simulations, we further linearize the higher order gradient term (that is, the interfacial term) by expanding around the critical point ${\varphi }_P={\varphi }_c$ and using ${\varphi }_P\left(1-{\varphi }_P\right)w^2\approx {\varphi }_c\left(1-{\varphi }_c\right)w^2$. With these approximations, one has:

$$\frac{\partial {\varphi }_P(\mathit{x},t)}{\partial t}\approx D_P\mathbf{\nabla }\cdot \left[\left(1+\frac{1-l_P}{l_P}{\varphi }_P-2\beta {\varphi }_P\left(1-{\varphi }_P\right)-2\beta w^2{\varphi }_c\left(1-{\varphi }_c\right){\mathbf{\nabla }}^2\right)\mathbf{\nabla }{\varphi }_P+\mathrm{\Delta }\chi {\varphi }_P\left(1-{\varphi }_P\right)\mathbf{\nabla }{\varphi }_R\right].$$

Note that, in our simulations, all the molecular species consist of coarse-grained monomers. The size of each coarse-grained monomer corresponds to a FC protein. Consequently, the monomer diffusion coefficient and the center-of-mass diffusion coefficient of FC proteins are identical, $D_P={\bar{D}}_P$. The FC proteins spontaneously phase separate when the coefficient of ${\varphi }_P$ in the parentheses changes sign, which is the case when $\beta >{\beta }_c$ (cf. Eq. S19).

Dynamics of rRNA

The dynamics of nascent rRNAs are qualitatively different from the FC protein dynamics discussed above. In addition to mass-conserving fluxes ${\mathit{j}}_R=-M_R\left({\varphi }_R\right)\mathbf{\nabla }{\mu }_R/k_{B}T$ with the chemical potential given by

$$\frac{{\mu }_R}{k_{B}T}=\frac{v_R}{k_{B}T}\frac{\delta ℱ}{\delta {\varphi }_R}=1+\log {\varphi }_R+\mathrm{\Delta }\chi l_R{\varphi }_P,$$ (S22)

where $v_R=V_m{l}_R$, non-equilibrium reactions produce and turn over rRNAs. The FC proteins catalyze (RNA Pol I) and regulate (UBTF) transcription of rRNA through a nonequilibrium reaction which consumes ATP, GTP, CTP, and UTP. The production rate of nascent rRNA should therefore depend on the volume fraction of the FC proteins ${\varphi }_P$. Following mass-action kinetics, we assume a simple first-order dependence, $k_t{\varphi }_P$, where $k_t$ is the transcription rate constant. The subsequent biogenesis of ribosomal subunits involves successive chemical modifications and processing of the nascent rRNA, which together with RNA-binding proteins assembles into ribosomal subunits. Here, we idealize these reactions by a net processing rate of the nascent rRNA, $k_p{\varphi }_R$, which is first order in the volume fraction of nascent rRNA ${\varphi }_R$. Combining the above terms, the dynamics of the nascent rRNA in the limit ${\varphi }_R\ll 1$ is given by:

$$\frac{\partial {\varphi }_R(\mathit{x},t)}{\partial t}=-\mathbf{\nabla }\cdot \underset{{\mathit{j}}_R}{\underbrace{\left[-\frac{M_R\left({\varphi }_R\right)}{k_{B}T}\mathbf{\nabla }\left(V_m{l}_R\frac{\delta ℱ}{\delta {\varphi }_R}\right)\right]}}+k_t{\varphi }_P-k_p{\varphi }_R\approx D_R{\mathbf{\nabla }}^2{\varphi }_R+k_t{\varphi }_P-k_p{\varphi }_R.$$ (S23)

Here, we have used $M_R\left({\varphi }_R\right)\approx D_R{\varphi }_R$ for the mobility of rRNA. This can be intuitively understood as the thermodynamic force $-\mathbf{\nabla }{\mu }_R$ inducing an effective drift velocity ${\mathit{v}}_R=-D_R\mathbf{\nabla }{\mu }_R/k_{B}T$, and thus a net current ${\mathit{j}}_R={\mathit{v}}_R{\varphi }_R$. Note that the expression for the rRNA flux is reminiscent of a Smoluchowsky diffusion equation with diffusivity $D_R$. Moreover, also note that $D_R$ refers to the diffusion coefficient (hopping rate) of a single coarse-grained monomer of nascent rRNA in a lattice gas description. To compute the diffusivity of a single monomer, we start with the measured diffusivity of a full nascent RNA molecule, $\overline{D_R}$, and use the Rouse model to get $D_R=\overline{D_R}{l}_R\approx 20\overline{D_R}$.

In the above equation, we have omitted the cross-diffusion term, $\mathbf{\nabla }\cdot \left(\mathrm{\Delta }\chi D_R{l}_R{\varphi }_R\mathbf{\nabla }{\varphi }_P\right)$, for numerical stability of our simulations and because ${\varphi }_R\ll 1$. This cross-diffusion term leads to an RNA concentration jump at the FC interface which is difficult to resolve numerically and which, for $\mathrm{\Delta }\chi >0$, depletes RNA from the FCs. This can also be understood by linearizing the cross-diffusion term around the typical RNA volume fraction ${\varphi }_R\approx {\varphi }_c{k}_t/k_p$, an approximation that does not guarantee positivity of ${\varphi }_R$ but explains that RNA will be pushed out of the FCs for $\mathrm{\Delta }\chi >0$. We included cross-diffusion only in Supplementary Movie 4, which considers an equilibrium scenario without chemical reactions.

Model parameters

Table 1 shows the model parameters which were used in this paper for all numerical simulations and calculations, unless stated otherwise.

Diffusivity of FC proteins.

Previous studies show that binding of FC proteins to rDNA is necessary to form FCs at physiologically relevant concentrations⁸¹. Therefore, we consider chromatin-bound FC proteins as the relevant population and estimate the diffusivity of this population based on experimental measurements relating to chromatin-bound UBTF by Ide et al.⁸⁸. Ide et al report the mean-squared displacement (MSD or $⟨r^2⟩$) vs. time plot for chromatin-bound UBTF in the nucleolus of mammalian cells in their figure S4G. This plot grows to $⟨r^2⟩=0.025\mathrm{\mu }{\mathrm{m}}^2$ over a time period of $t=1\mathrm{s}$, giving an estimate of the diffusivity $D_P=⟨r^2⟩/\left(6t\right)\approx 4\times {10}^{-3}\mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$. Note that this value should only be seen as a rough estimate, because it (like our model in general) neglects that chromatin-bound proteins undergo sub-diffusive motion together with the chain that they are bound to.

Reaction rates.

Next, we will determine the transcription rate $k_t$ and the processing rate $k_p$ of rRNA. As we have discussed in section “The estimated volume fraction of nascent rRNA is small”, there are about $N_R{N}_D=2\times {10}^4$ copies of nascent rRNA per cell. From experimental measurements^9,127, we know that rRNAs are transcribed at a rate of around $K_t={10}^3-{10}^4$ nascent rRNA/min per nucleus. We will assume a rate of $K_t=1.0\times {10}^4$ nascent rRNA/min per nucleus. The transcription rate constant is given by:

$$k_t=\frac{K_t}{V}\frac{v_R}{{\varphi }_P}=\frac{K_t}{N_R{N}_D}\frac{N_R{N}_D{v}_R}{V}\frac{1}{{\varphi }_P}=\frac{K_t}{N_R{N}_D}\frac{{\varphi }_R}{{\varphi }_P},$$ (S24)

where ${\varphi }_P\sim 0.1$ is the typical volume fraction of FC proteins and ${\varphi }_R\sim 0.01$ is the typical volume fraction of rRNA. As discussed in section “The estimated volume fraction of nascent rRNA is small”, we chose this value of the rRNA volume fraction as a compromise between the larger value expected when distributing $N_R{N}_D$ rRNA copies over the volume of all fibrillar centers $V=N{N}_{FC}\pi d_{FC}^3/6$ and the smaller value expected when distributing $N_R{N}_D$ rRNA copies over the volume of all nucleoli $V=N\pi d^3/6$. This volume fraction of ${\varphi }_R\sim 0.01$ corresponds to a rRNA concentration of 3000 nascent rRNA/μm³ as reported by Riback et al.⁶⁸. Thus, the transcription rate constant is roughly given by

$$k_t=0.05/\min .$$ (S25)

Consistent with our model, we have here assumed that rDNA genes are distributed relatively uniformly in space, so that the FC protein distribution is the main predictor of rRNA transcription.

In steady state, and neglecting degradation of rRNA, the total rate of rRNA transcription must be matched by rRNA processing, $K_p=K_t$. With this, we can estimate the processing rate to roughly be

$$k_p=\frac{K_p}{N_R{N}_D}=0.5/\mathrm{m}\mathrm{i}\mathrm{n}.$$ (S26)

This rate should be higher than the inverse residence time of rRNA in the dense fibrillar component to ensure that most of the rRNA is processed within the DFC under homeostatic conditions.

Details of numerical simulations

The model (Equations 2 and 3) was simulated with a custom code (https://github.com/npradeep96/Nucleolus_arrested_coarsening, archived on Zenodo: https://doi.org/10.5281/zenodo.18892655) written in Python using finite-volume methods implemented in the FiPy¹²⁸ library. In all simulations, we used no-flux boundary conditions for all species and started with a constant volume fraction of FC proteins, ${\varphi }_P(\mathit{x},t=0)=\overline{{\varphi }_P}=0.09$, and no rRNA, ${\varphi }_R(\mathit{x},t=0)=0$, unless mentioned otherwise. Surface tension effects operate over an interface width proportional to the interaction range $w=6\mathrm{n}\mathrm{m}$, and set the smallest length scale in our system. We used a mesh size of $\mathrm{\Delta }x=20\mathrm{n}\mathrm{m}$ for our simulations which ensures enough resolution to account for the interface while also not making the numerical simulations too costly by having too many mesh points. The size of the domain was set to the average size of the nucleolus, which was about 3 μm for our experimental system of mESCs. We used adaptive time stepping to increase or decrease the time step $\mathrm{\Delta }t$. The time step was decreased if the residuals when solving the non-linear equations at each time step were larger than 10⁻² and increased if the residuals were smaller. The maximum size of the time step was capped at $\mathrm{\Delta }{t}_{\text{max}}=0.01$ minutes. Since the equations were stiff with the FC protein dynamics faster than the dynamics of rRNA, we used Strang splitting to first integrate the FC protein equations over a time step of $\mathrm{\Delta }t/2$, and used the resulting FC protein profile as the input to integrate the rRNA dynamics over a time step of $\mathrm{\Delta }t$.

Effective interaction energy of FC proteins due to non-equilibrium rRNA dynamics

First, we note that Equation 3 is linear in the volume fractions ${\varphi }_P(\mathit{x},t)$ and ${\varphi }_R(\mathit{x},t)$. Hence, the Fourier transform of these fields ${\varphi }_P(\mathit{x},t)\to {\mathrm{\Phi }}_P(\mathit{q},t)$ and ${\varphi }_R(\mathit{x},t)\to {\mathrm{\Phi }}_R(\mathit{q},t)$ can diagonalize Equation 3 into a system of independent ordinary differential equations. Knowing that the dynamics of this system should reach a nonequilibrium steady state based on our simulations, we first solve for the profile of rRNA in steady state. This is analogous to making an adiabatic (quasi-steady state) approximation for the rRNA dynamics. This approximation leads to the following linear equation for ${\mathrm{\Phi }}_R(\mathit{q},t)$ in response to a distribution of FC proteins characterized by ${\mathrm{\Phi }}_P(\mathit{q},t)$:

$$\frac{\partial {\mathrm{\Phi }}_R(\mathit{q},t)}{\partial t}=-\underset{\text{Diffusion}}{\underbrace{D_R{\left|\mathit{q}\right|}^2{\mathrm{\Phi }}_R}}+\underset{\text{Transcription}}{\underbrace{k_t{\mathrm{\Phi }}_P}}-\underset{\text{Processing}}{\underbrace{k_p{\mathrm{\Phi }}_R}}=0\Rightarrow {\mathrm{\Phi }}_R(\mathit{q},t)=\underset{G(\mathit{q})}{\underbrace{\frac{k_t}{D_R{\left|\mathit{q}\right|}^2+k_p}}}{\mathrm{\Phi }}_P(\mathit{q},t).$$ (S27)

Using the above expression to adiabatically eliminate ${\varphi }_R(\mathit{x},t)$, one can derive an expression for the effective interaction energy between FC proteins and rRNA:

$$\begin{array}{ll}\frac{u}{k_{B}T}& =\mathrm{\Delta }\chi \int \frac{d^3\mathit{x}}{V_m}{\varphi }_P(\mathit{x},t){\varphi }_R(\mathit{x},t)=\mathrm{\Delta }\chi \int \frac{d^3\mathit{x}}{V_m}{\varphi }_P(\mathit{x},t)\int \frac{d^3\mathit{q}}{(2\pi {)}^3}{e}^{i\mathit{q}\cdot \mathit{x}}{\mathrm{\Phi }}_R(\mathit{q},t)\\ & =\mathrm{\Delta }\chi \int \frac{d^3\mathit{x}}{V_m}{\varphi }_P(\mathit{x},t)\int \frac{d^3\mathit{q}}{(2\pi {)}^3}{e}^{i\mathit{q}\cdot \mathit{x}}\frac{k_t}{D_R{\left|\mathit{q}\right|}^2+k_p}{\mathrm{\Phi }}_P(\mathit{q},t)\\ & =\mathrm{\Delta }\chi \int \frac{d^3\mathit{x}}{V_m}{\varphi }_P(\mathit{x},t)\int \frac{d^3\mathit{q}}{(2\pi {)}^3}{e}^{i\mathit{q}\cdot \mathit{x}}\frac{k_t}{D_R{\left|\mathit{q}\right|}^2+k_p}\int d^3{\mathit{x}}^{\prime }{e}^{-i\mathit{q}\cdot {\mathit{x}}^{\prime }}{\varphi }_P\left({\mathit{x}}^{\prime },t\right)\\ & =\mathrm{\Delta }\chi \iint \frac{d^3\mathit{x}}{V_m}{d}^3{\mathit{x}}^{\prime }{\varphi }_P(\mathit{x},t)\left\{\int \frac{d^3\mathit{q}}{(2\pi {)}^3}{e}^{i\mathit{q}\cdot \left(\mathit{x}-{\mathit{x}}^{\prime }\right)}\underset{G(\mathit{q})}{\underbrace{\frac{k_t}{D_R{\left|\mathit{q}\right|}^2+k_p}}}\right\}{\varphi }_P\left({\mathit{x}}^{\prime },t\right).\end{array}$$

Now consider the term in the braces. It is well-known that the inverse Fourier Transform of $G(\mathit{q})$ is a Yukawa-like screened electrostatic interaction:

$$\int \frac{d^3\mathit{q}}{(2\pi {)}^3}{e}^{i\mathit{q}\cdot \left(\mathit{x}-{\mathit{x}}^{\prime }\right)}\frac{k_t}{D_R{\left|\mathit{q}\right|}^2+k_p}=\frac{k_t}{k_p{l}^3}\underset{\text{Yukawa-like}}{\underbrace{\frac{e^{\frac{-\left|\mathit{x}-{\mathit{x}}^{\prime }\right|}{l}}}{4\pi \frac{\left|\mathit{x}-{\mathit{x}}^{\prime }\right|}{l}}}}:=\frac{k_t}{k_p{l}^3}g\left(\frac{\left|\mathit{x}-{\mathit{x}}^{\prime }\right|}{l}\right),$$ (S28)

where $l=\sqrt{D_R/k_p}$ is the rRNA diffusion length. Using the above expression, the effective interaction energy between FC proteins and rRNA is given by:

$$\frac{U}{k_{B}T}=\mathrm{\Delta }\chi \frac{k_t}{k_p}\int \frac{d^3\mathit{x}}{V_m}\int \frac{d^3{\mathit{x}}^{\prime }}{l^3}{\varphi }_P(\mathit{x},t)g\left(\frac{\mathit{x}-{\mathit{x}}^{\prime }}{l}\right){\varphi }_P\left({\mathit{x}}^{\prime },t\right).$$ (S29)

This result is reminiscent of the Yukawa potential for screened electrostatic interactions¹²⁹. Thus, our analysis shows that the non-equilibrium process of FC protein-dependent rRNA transcription, rRNA processing, and diffusion sets up rRNA gradients which mediate emergent interactions between FC proteins that are exactly analogous to screened electrostatic interactions. However, in contrast to the bona fide screened electrostatic interactions in living cells which have Debye length of around 1 nm¹³⁰ at physiological salt concentrations, these emergent interactions are much longer-ranged and act over a length scale $l=\sqrt{D_R/k_p}\approx 290\mathrm{n}\mathrm{m}$ which is larger than the size (< 200 nm) of a single nascent rRNA molecule⁶⁸.

Lyapunov functional for FC protein dynamics

Our simulations show that the non-equilibrium dynamics of our model (Equations 2 and 3) lead to a steady state. By using an adiabatic approximation for the distribution of rRNA as explained above, one can map these dynamics to a pseudo-equilibrium system with the FC proteins as only degree of freedom,

$$\frac{\partial {\varphi }_P(\mathit{x},t)}{\partial t}=-\mathbf{\nabla }\cdot \left[-\frac{M_P\left({\varphi }_P\right)}{k_{B}T}\mathbf{\nabla }\left(V_m\frac{\delta ℒ}{\delta {\varphi }_P}\right)\right],\text{where}\text{the}\text{Lyapunov}\text{functional,}ℒ={ℱ}_0+\frac{𝒰}{2},$$ (S30)

is gradually minimized. Here, we have defined ${ℱ}_0\left[{\varphi }_P\right]:=ℱ\left[{\varphi }_P,{\varphi }_R=0\right]$; cf. Eqs. S18 and S29. A factor of ½ multiplies $𝒰$ in the Lyapunov functional to avoid the double-counting when calculating the effective self-interaction energy between FC proteins. It is easy to test that the chemical potential of FC proteins is invariant to this transformation,

$$\frac{\delta ℒ}{\delta {\varphi }_P}={\left.\frac{\delta ℱ}{\delta {\varphi }_P}\right|}_{{\varphi }_R={\varphi }_R^{\star }}.$$ (S31)

The above optimization principle makes the Lyapunov functional $ℒ$ a powerful tool to characterize the steady state pattern of the FCs and to extract the dominant wavelength. It is also useful to analyze scenarios where the steady state distribution of rRNA can not be resolved via a Fourier decomposition, as will be discussed in the next section.

To extract the dominant wavelength, we expand the Flory-Huggins bulk free energy density $f_P\left({\varphi }_P\right)$, cf. Eq. S18, to second order around the critical point ${\varphi }_c$ (Eq. S20) and drop the thermodynamically irrelevant constant and linear terms, so that $f_P\left({\varphi }_P\right)V_m/k_{B}T\approx -\left(\beta -{\beta }_c+𝒪\left({\beta }^2\right)\right){\left({\varphi }_P-{\varphi }_c\right)}^2$ where ${\beta }_c$ is given by Eq. S19. We then write the Lyapunov functional $ℒ$ in terms of the Fourier modes ${\mathrm{\Phi }}_P(\mathit{q})$:

$$\frac{ℒ}{k_{B}T}\approx \frac{1}{V_m}\int \frac{d^3\mathit{q}}{(2\pi {)}^3}\underset{K({\left|\mathit{q}\right|}^2)}{\underbrace{\left[{\beta }_c-\beta +\beta w^2{\left|\mathit{q}\right|}^2+\frac{\mathrm{\Delta }\chi k_t}{2k_p}\frac{1}{1+l^2{\left|\mathit{q}\right|}^2}\right]}}{\left|{\mathrm{\Phi }}_P(\mathit{q})\right|}^2+𝒪({\varphi }_P^4).$$ (S32)

The coefficient $K({\left|\mathit{q}\right|}^2)$ can be interpreted as a potential well which is deepest for a specific wavenumber $\left|\mathit{q}\right|=q^{\star }$ which is related to the preferred FC size and spacing. Specifically, the length scale $2\pi /q^{\star }$ is the spacing between the centers of neighboring FCs. However, the leading term in the above Lyapunov functional does not distinguish between FC and non-FC regions. The dominant contribution to the profile of FC proteins, ${\varphi }_P$, is a sinusoid where both FC and non-FC regions have the same characteristic wavelength. In agreement with this simple expectation, our experiments (Figure 1C) show that the distance between FCs is roughly twice the diameter of FCs, or four times the FC radius. To determine the dominant wavenumber $q^{\star }$ in the non-equilibrium steady state, we minimize $K({\left|\mathit{q}\right|}^2)$ with respect to ${\left|\mathit{q}\right|}^2$. The corresponding preferred FC radius ($R^{\star }=\pi /2q^{\star }$) is given by:

$$R^{\star }=\frac{\pi }{2}{\left(\frac{D_R}{k_p}\right)}^{1/2}{\left[{\left(\frac{\mathrm{\Delta }\chi k_t{D}_R}{2\beta w^2{k}_p^2}\right)}^{1/2}-1\right]}^{-1/2}.$$ (S33)

To observe size selection and arrested coarsening, the dominant wavenumber $q^{\star }$ and the corresponding FC radius $R^{\star }$ must be real numbers. This criterion can only be satisfied if $\mathrm{\Delta }\chi >0$, since all other parameters in the above equation are positive. Therefore, this analysis predicts that non-FC proteins should have a higher affinity towards the rRNAs for coarsening to be arrested. Moreover, it predicts a critical rRNA transcription rate $k_t>2\beta w^2{k}_p^2/\left(D_R\mathrm{\Delta }\chi \right)$ above which coarsening will be arrested and can even be reversed. Our analysis sheds light on how the typical radius of fibrillar centers is controlled by a set of biophysical parameters which can be perturbed in experiment, such as the rRNA transcription rate $k_t$ and the differential rRNA-FC protein interaction $\mathrm{\Delta }\chi$. It predicts that the FC radius should decrease with the rate $k_t$ of rRNA transcription, which is consistent with observations from numerical simulations (Figure 3A).

Relative sensitivity of the pattern lengthscale

We will now determine how the lengthscale of the pattern responds to changing the parameters in our model. Based on Eq. S33, we have three different sensitivities. First,

$$-\frac{d\mathrm{l}\mathrm{o}\mathrm{g}{R}^{\star }}{d\mathrm{l}\mathrm{o}\mathrm{g}{k}_t}=-\frac{d\mathrm{l}\mathrm{o}\mathrm{g}{R}^{\star }}{d\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{\Delta }\chi }=\frac{d\mathrm{l}\mathrm{o}\mathrm{g}{R}^{\star }}{d\mathrm{l}\mathrm{o}\mathrm{g}\beta }=\frac{1}{4}+\frac{{\left(R^{\star }/l\right)}^2}{{\pi }^2},$$ (S34)

indicates that larger condensates are more sensitive to an increase in the transcription rate $k_t$, or the interaction parameters $\mathrm{\Delta }\chi$ and $\beta$. Equation S34 shows that an increase in the transcription rate $k_t$ or the differential interactions $\mathrm{\Delta }\chi$ of rRNA with other molecules will decrease FC size, whereas stronger FC-FC interactions $\beta$ will increase FC size. The scaling of the FC size with these parameters is identical (cf. first two equalities in Eq. S34), because they are part of the same parameter group in Eq. S33. Similarly,

$$\frac{d\mathrm{l}\mathrm{o}\mathrm{g}{R}^{\star }}{d\mathrm{l}\mathrm{o}\mathrm{g}{k}_p}=2\frac{{\left(R^{\star }/l\right)}^2}{{\pi }^2},$$ (S35)

indicates that a faster processing rate will increase FC size and cause loss of size control. Finally,

$$\frac{d\mathrm{l}\mathrm{o}\mathrm{g}{R}^{\star }}{d\mathrm{l}\mathrm{o}\mathrm{g}{D}_R}=\frac{1}{4}-\frac{{\left(R^{\star }/l\right)}^2}{{\pi }^2},$$ (S36)

shows that small condensates ($R^{\star }<\pi l/2$) grow whereas large condensates shrink when increasing the diffusion coefficient of rRNA. The above relations can also be used to make trade-offs between the different parameters of the model while keeping the FC radius $R^{\star }$ constant.

Analysis in the spirit of classical nucleation theory

The main purpose of this section is to show that considering the rRNA to be transcribed only at the FC boundary should not affect the qualitative results described in the main text. The analysis so far shows that, because interactions between rRNA and non-FC molecules are energetically favored over interactions between rRNA and FC proteins, rRNA concentration gradients generated by reactions (transcription and processing) and diffusion counteract FC phase separation and mediate FC-FC repulsion over the diffusion length of rRNA. To illustrate how this effective repulsion among FC proteins leads to arrested coarsening, we will now consider a scenario reminiscent of classical nucleation theory, by focusing on an isolated FC which is far away from any of the other FCs. In classical nucleation theory, a lower bulk free energy density in the condensate compared to its surroundings, i.e., $\mathrm{\Delta }{\mathrm{\Pi }}_0:=f_P\left({\left.{\varphi }_P\right|}_{\text{in}}\right)-f_P\left({\left.{\varphi }_P\right|}_{\text{out}}\right)<0$, drives condensate growth. This growth is counteracted by the Laplace pressure, $2\gamma /R$, with the surface tension $\gamma$ given by

$$\frac{\gamma V_m}{k_{B}T}=\frac{w\beta {\left(2{\varphi }_c^2-1/\beta \right)}^{3/2}}{\sqrt{\left(1-{\varphi }_c\right){\varphi }_c}}.$$ (S37)

Since the Laplace pressure shrinks with increasing droplet radius, there is an inherent instability which causes large droplets, $R>R_c$, where the critical radius is defined by $\mathrm{\Delta }{\mathrm{\Pi }}_0+2\gamma /R_c=0$, to grow and small droplets to shrink. This picture is changed by the effective Yukawa-like long-ranged repulsion between FC proteins, as can be seen by calculating the effective self-interaction energy $𝒰$ (Eq. S29) and the Lyapunov functional $ℒ$ (Eq. S30) of a single FC. In this section, we will use the existence of the Lyapunov functional, when the concentration profile of rRNA shows a linear response to the distribution of FC proteins, to analyze our model (Equations 2 and 3). Moreover, we also consider a more realistic scenario where the dynamics of rRNA is given by

$$\frac{\partial {\varphi }_R(\mathit{x},t)}{\partial t}=\mathbf{\nabla }\cdot \left[D_R\mathbf{\nabla }{\varphi }_R+D_R\mathrm{\Delta }\chi l_R{\varphi }_R\mathbf{\nabla }{\varphi }_P\right]+k_t\left({\varphi }_P,\mathbf{\nabla }{\varphi }_P\right)-k_p\left({\varphi }_P\right){\varphi }_R=0,$$ (S38)

which we assume to adiabatically reach a pseudo-steady state. Here, we have also included the reciprocal part of the interactions, which will push rRNA out of FCs. Note that, although the generalized rRNA dynamics (Eq. S38) is still linear in the profile of rRNA, ${\varphi }_R$, it no longer shows a linear response to the distribution of FC proteins described by ${\varphi }_P$. In other words, there is no Green’s function solution that linearly maps ${\varphi }_P$ to ${\varphi }_R\left[{\varphi }_P\right]$. However, it is possible to use perturbative methods to generate a series in powers of ${\varphi }_P$, which then leads to a generalized Lyapunov functional that accurately describes the final steady state of the system. To leading order, this steady state will be determined by the FC self-interaction energy $𝒰$ and the corresponding pressure.

To simplify the profile of FC proteins, ${\varphi }_P$, we consider a sharp interface approximation where ${\varphi }_P\approx \mathrm{\Delta }{\varphi }_P[1-\mathrm{\Theta }(r-R)]$ with $\mathrm{\Theta }$ the Heaviside step function and $R$ the FC radius. Here, $\mathrm{\Delta }{\varphi }_P$ defines the difference in FC protein abundancies between the FC and the non-FC region. Thus, we assumed for simplicity that the volume fraction of FC proteins outside of FCs is negligible. We consider a scenario where transcription happens either in the bulk, as assumed in the previous sections, or a more realistic scenario where rRNA is transcribed at the boundary of FCs⁸³:

$$k_t\left({\varphi }_P,\mathbf{\nabla }{\varphi }_P\right)\equiv k_{t,blk}\mathrm{\Delta }{\varphi }_P[1-\mathrm{\Theta }(r-R)]+k_{t,bnd}\delta (r-R).$$ (S39)

For the processing of rRNA, we consider a scenario where the processing rate is uniform, as assumed in the previous sections, or a more realistic scenario where rRNA is processed outside of FCs⁸³:

$$k_p\left({\varphi }_P\right)\equiv k_p+k_{p,DFC}\mathrm{\Theta }(r-R).$$ (S40)

For the distribution of rRNA inside and outside of the FC, one thus has

$$\begin{gathered} \mathbf{\nabla }\cdot \left[D_R\mathbf{\nabla }{\varphi }_R\right]+k_{t,blk}\mathrm{\Delta }{\varphi }_P-k_p{\varphi }_R=0,r<R \\ \mathbf{\nabla }\cdot \left[D_R\mathbf{\nabla }{\varphi }_R\right]-\left(k_p+k_{p,DFC}\right){\varphi }_R=0,r>R \end{gathered}$$ (S41)

The boundary conditions at $r=R$ can be determined by requiring that the currents of rRNA are finite and by integrating around the FC boundary:

$$\begin{array}{c}\frac{{\left.{\varphi }_R\right|}_{out}}{{\left.{\varphi }_R\right|}_{in}}=e^{\mathrm{\Delta }\chi l_R\mathrm{\Delta }{\varphi }_P},\\ {\left.D_R{\partial }_r{\varphi }_R\right|}_{out}-{\left.D_R{\partial }_r{\varphi }_R\right|}_{in}+k_{t,bnd}=0.\end{array}$$ (S42)

The first boundary condition indicates that, for $\mathrm{\Delta }\chi >0$, rRNA is depleted in the FCs compared to outside of the FCs. We solved this model analytically with Mathematica¹³¹ and determined the effective interaction energy between FC proteins and rRNA, which is given by

$$\frac{U}{k_{B}T}=\frac{\mathrm{\Delta }\chi }{V_m}\mathrm{\Delta }{\varphi }_P{\int }_{\left|\mathit{x}\right|<R}{d}^3\mathit{x}{\varphi }_R(\mathit{x},t).$$ (S43)

As discussed above, $\mathrm{\Delta }{\varphi }_P$ defines the difference in FC protein abundancies between the FC and the non-FC region and the volume fraction of FC proteins outside of FCs is assumed negligible.

We will now discuss the two scenarios mentioned above. First, we consider the scenario corresponding to Equation 3 of the main text, by assigning $k_{t,blk}\to k_t$, $l_R\to 0$, $k_{t,bnd}\to 0$, and $k_{p,DFC}\to 0$. In this case, one has for the average energy density of interactions between FC proteins and rRNA, and for the resulting pressure:

$$\frac{1}{V_R}\frac{U}{k_{B}T}=\frac{1}{2}\frac{k_t\mathrm{\Delta }\chi }{k_p{V}_m}\mathrm{\Delta }{\varphi }_P^2\left[2-3\left(\frac{l}{R}\right)+3{\left(\frac{l}{R}\right)}^3-3e^{-\frac{2R}{l}}\left(\frac{l}{R}\right){\left(\frac{l}{R}+1\right)}^2\right],$$ (S44)

$$\frac{1}{2}\frac{1}{S_R}\frac{{\partial }_{R}u}{k_{B}T}=\frac{1}{2}\frac{k_t\mathrm{\Delta }\chi }{k_p{V}_m}\mathrm{\Delta }{\varphi }_P^2\left[\left(1-\frac{l}{R}\right)+\left(1+\frac{l}{R}\right)e^{-\frac{2R}{l}}\right],$$ (S45)

where we have defined the volume of the droplet $V_R:=\frac{4\pi }{3}{R}^3$ and the surface area $S_R:=4\pi R^2$. Note that the factor $\frac{1}{2}$ prevents overcounting of the emergent interactions among FC proteins.

Next, we consider the more realistic scenario where rRNA is transcribed at the FC boundary, $k_{t,blk}\to 0$, processed outside of FCs, $k_p\to 0$, and where we redefine the rRNA diffusion length as $l=\sqrt{D_R/k_{p,\mathrm{D}\mathrm{F}\mathrm{C}}}$. One then has:

$$\frac{1}{V_R}\frac{U}{k_{B}T}=\frac{k_{t,\mathit{bnd}}\mathrm{\Delta }\chi }{k_{p,DFC}{V}_m}{e}^{-\mathrm{\Delta }\chi l_R\mathrm{\Delta }{\varphi }_P}\frac{\mathrm{\Delta }{\varphi }_P}{l}\left[\frac{R}{l+R}\right],$$ (S46)

$$\frac{1}{2}\frac{1}{S_R}\frac{{\partial }_{R}U}{k_{B}T}=\frac{1}{2}\frac{k_{t,\mathit{bnd}}\mathrm{\Delta }\chi }{k_{p,DFC}{V}_m}{e}^{-\mathrm{\Delta }\chi l_R\mathrm{\Delta }{\varphi }_P}\frac{\mathrm{\Delta }{\varphi }_P}{l}\left[\frac{R(4l+3R)}{3(l+R{)}^2}\right].$$ (S47)

Importantly, in both scenarios the free energy density and the effective pressure due to interactions between FC proteins and rRNA are monotonically increasing functions of the FC radius $R$. Thus, increasing the detail of the model does not affect the phenomenology, the predictions, or the conclusions made in the main text.

To calculate the FC size based on these arguments, one can minimize the total free energy of the system by summing up the free energies of each FC, as discussed in the main text (Figure 2G). Alternatively, one can also determine the roots of the osmotic pressure jump, $\mathrm{\Delta }\mathrm{\Pi }=\mathrm{\Delta }{\mathrm{\Pi }}_0+2\gamma /R+{\partial }_{R}u/\left(2S_R\right)$. The additional stress ${\partial }_{R}u/\left(2S_R\right)$, Eq. S45 or S47, opposes condensate growth and increases monotonically with the FC radius. For small condensates, the Laplace pressure dominates the dynamics. For large condensates, the additional stress monotonically approaches a saturation value. Since the stress opposing FC growth increases, there is naturally a stable fixed point for the dynamics. The shortcoming of this intuitive analysis is that, in our experiments and simulations, FCs are not isolated. On the contrary, the inter-FC distance is about twice the FC diameter (Figure 1C) so that the effect of nearby FCs becomes important when trying to quantitatively predict condensate size. We have already addressed this shortcoming in the previous section by (i) deriving a Lyapunov functional for the dynamics of the system in terms of the Fourier modes $\mathit{q}$ and (ii) using it to extract the dominant wavelength selected by the system.

Distributed versus consolidated FCs: a mean first-passage time analysis

In this section, we will use a simple model to analyze the residence time of rRNA in the DFC in wild-type cells (distributed FC pattern) compared to TCOF1-overexpressed cells (consolidated FCs). The mean first-passage time $⟨\tau ⟩$ of a single copy of rRNA, for moving from the shell $r$ to the shell $R$ in spherical coordinates, is given by the solution of¹³²

$$\frac{D}{r^2}{\partial }_r\left[r^2{\partial }_r⟨\tau ⟩(r)\right]=-1,$$ (S48)

with $D$ the diffusion coefficient and $⟨\tau ⟩(R)=0$. A reflective boundary at the inner shell $r_0$ corresponds to ${\partial }_r⟨\tau ⟩\left(r_0\right)=0$. Using this simple framework, we will now consider two competing scenarios.

The first scenario corresponds to TCOF1-overexpressed cells in which the size control of FCs is perturbed. Hence, we consider a single consolidated, spherical FC with radius $r_0$ located at the center of a spherical DFC with radius $R$. Considering that the volume of the FC is $\rho$ times the volume of the DFC, one has $r_0={\rho }^{1/3}R$. Assuming that the FC is impermeable, the corresponding mean-first passage time is given by the solution of Eq. S48:

$$⟨{\tau }_1⟩(r)=\frac{r{R}^3-r^{3}R+2r{\mathrm{r}}_0^3-2R{\mathrm{r}}_0^3}{6DrR}.$$ (S49)

In this case, the starting location of the rRNA will be at the FC-DFC interface, $r=r_0$.

As second, competing scenario, we consider a caricature of wild-type cells with $N\to \infty$ distributed FCs inside of a single spherical DFC. We assume that the FCs are distributed uniformly and that their total volume is $\rho$ times the volume of the DFC. This geometrical arrangement is supported by our experimental data, which does not suggest that DFCs are separated (Fig. S4A). Since the total mass of FCs and DFCs in TCOF1-overexpressed cells is the same as in wild-type cells (Fig. S4B), this indicates that the DFC should have roughly the same radius as in wild-type cells, 𝑅. If the distributed FCs do not significantly hinder rRNA diffusion, then the resulting mean first-passage time, after averaging over the starting locations and setting $r_0\to 0$, is given by

$$⟨{\tau }_{\mathrm{\infty }}⟩=\frac{R^2}{15D}.$$ (S50a)

Thus, in the case of a consolidated FC, the residence time of rRNA in the DFC changes to a relative value of

$$\frac{⟨{\tau }_1⟩}{⟨{\tau }_{\mathrm{\infty }}⟩}=\frac{5}{2}\left(1+2\rho -3{\rho }^{2/3}\right).$$ (S51a)

For $\rho \approx 0.3$ based on our experimental data (Fig. S4B), this simple analysis suggests that the residence time of rRNA in the DFC is reduced to 64%. If, in addition, the FCs are almost impermeable for the rRNA, and thereby act as obstacles, this will also effectively reduce the diffusion coefficient to¹³³ $D(1-1.83\rho )$. In that case, the residence time of rRNA in the DFC in TCOF1-overexpressed cells is further reduced to 29% of the residence time in the DFC in wild-type cells. Assuming a constant processing rate, this analysis is consistent with our observation of a drastic decrease in processing efficiency.

Analysis of the total volume of FCs

Assuming that the total mass of FC molecules is constant, the total added volume of all FCs will be determined by the difference $\mathrm{\Delta }{\varphi }_P$ between the volume fractions of FC molecules inside FCs and outside of FCs. This difference $\mathrm{\Delta }{\varphi }_P$ depends on the competition between attractive pairwise and repulsive higher order interactions. The weaker the pairwise interaction, the smaller $\mathrm{\Delta }{\varphi }_P$ and larger the total added volume of all FCs. The transcription of rRNA arrests coarsening by weakening the effective pairwise interactions among FCs. According to Eq. S32, the scale-dependent pairwise interaction between FC molecules is given by

$${\beta }_c-\beta +\frac{\mathrm{\Delta }\chi k_t}{2k_p}\frac{1}{1+l^2{\left|\mathit{q}\right|}^2},$$ (S34)

where we have dropped the surface tension term $\beta w^2{\left|\mathit{q}\right|}^2$ because it only minimally contributes to the concentration. At the length scale selected by the system (cf. Eq. S33), this pairwise interaction is given by

$${\beta }_c-\beta +\beta {\left(\frac{w}{l}\right)}^2\left[{\left(\frac{2R^{\star }}{\pi l}\right)}^{-2}+1\right],$$ (S35)

For sufficiently small FC molecule interaction ranges $w$ compared to the rRNA diffusion length $l$, which reflects the biology, this is the same as the pairwise interaction of FCs in absence of rRNA, indicating that the total volume of FCs remains unchanged. This is consistent with our experimental data.

Nascent RNA capture and RT-qPCR

We used Click-iT^™ Nascent RNA Capture Kit (Invitrogen, C10365) to extract nascent RNA. Stock and working solutions were prepared according to manufacturer’s instruction. Cells were plated in 12-well plates (Corning, 07-200-82) with 5% confluency, and the 5-ethynyl Uridine (EU) labeling was performed when cells reached 70% confluency. Three biological replicates were plated for each condition.

To prepare EU-labeled spike-in RNA, E. coli were cultured in 2.5 mL LB for 12 hours, followed by adding another 2.5 mL LB containing 0.2 mM EU (leading to 5 mL LB containing 0.1 mM EU). The E. coli were cultured for another hour, followed by pelleting and RNA extraction using the RNeasy Plus Kits (QIAGEN, 74134).

To get EU-labeled nascent RNA, 500 μL prewarmed media containing 0.5 mM EU were added to each well. The treatment conditions (e.g., doxycycline) were maintained in the EU containing media. After 35-minute incubation in incubator, cells were harvested and RNA extractions were performed using the RNeasy Plus Kits (QIAGEN, 74134). The “40 reactions” scale in the manufacturer’s instruction (Invitrogen, C10365) was applied in all following reactions. For example: 250 ng EU-labeled nascent from sample, 250 ng EU-labeled spike-in RNA, and 0.25 mM Biotin Azide were combined for each Click reaction; 100 ng biotinylated RNA and 10 μL magnetic beads were combined for each binding reaction; 12 μL Click-iT^® reaction wash buffer 2 was used to resuspend RNA-bound beads.

To synthesize cDNAs, the SuperScript^® VILO^™ cDNA synthesis kit (Invitrogen, 11754-050) was used following manufacturer’s instruction, where the volumes of reaction components were scaled appropriately to make the total volume of each cDNA synthesis reaction being 24 μL.

Following cDNA synthesis, RT-qPCR was performed using the PowerTrack SYBR Green Master Mix (Applied Biosystems, A46109). The scale of 20 μL per reaction was chosen. 0.2 uL cDNA from last step was added to each reaction (diluted the cDNA from last step first before adding to RT-qPCR reaction since 0.2 uL was hard to pipette). We designed primer pairs that amplify i. one exon of rRNA (total rRNA), ii. one exon-intron joint of rRNA (unprocessed rRNA), and iii. rrsA mRNA (spike-in from E. coli). The primer sequences are as follows:

1. total rRNA (spanning 18s rRNA):

   FWD: CTCTAGATAACCTCGGGCCG. REV: GTCGGGAGTGGGTAATTTGC.

2. unprocessed rRNA (spanning pre-rRNA A1 processing site):

   FWD: ATGCACTCTCCCGTTCCG. REV: CCGTGCGTACTTAGACATGC.

3. rrsA mRNA:

   FWD: CGCAACCCTTATCCTTTGTT. REV: TAAGGGCCATGATGACTTGA.

For each combination of a primer pair and a cDNA sample, four technical replicates were prepared to account for potential variations of RT-qPCR.

Quantifications of nascent rRNA transcription and processing

To quantify the relative rRNA transcription efficiency in the TCOF1 overexpression condition (+TCOF1), ΔΔCt was calculated by first normalizing the total rRNA to rrsA mRNA (ΔCt), followed by further normalizing the ΔCt of +TCOF1 to that of −TCOF1. The transcription efficiency is defined as 2^(−ΔΔCt).

To quantify the relative rRNA processing efficiency in the TCOF1 overexpression condition (+TCOF1), ΔΔCt’ was calculated by first normalizing the total rRNA to unprocessed rRNA (ΔCt’), followed by further normalizing the ΔCt’ of +TCOF1 to that of −TCOF1. The processing efficiency is defined as 2^(−ΔΔCt’).

All Ct values of four technical replicates from three biological replicates were combined together for such quantifications to increase statistical power.

Pulse-chase nucleotide labeling combined with immunofluorescence

We used Click-iT^™ RNA Alexa Fluor^™ 594 Imaging Kit to visualize newly-synthesized RNA (Invitrogen, C10330). Working solutions were prepared and stored according to manufacturer’s instruction. Cells were plated in glass-bottom 24-well plates (Mattek Corporation P24G-1.5-13-F) with 5% confluency, and the 5-ethynyl Uridine (EU) labeling was performed when cells reached 70% confluency.

To perform the pulse-chase nucleotide labeling experiment, we first kept 250 μL media in each well, followed by adding another 250 μL prewarmed media containing 2 mM EU (leading to 0.5 mL media containing 1 mM EU in each well). Cells were incubated in EU-containing media for 10 minutes at 37°C (“pulse”) before two quick washes with prewarmed PBS, followed by incubating with 0.5 mL fresh media (“chase”) for various duration (t = 0, 1, 2, 5, 10, 15, 20 and 30 minutes) prior to fixation. The treatment conditions (e.g., doxycycline) were maintained during the pulse and chase periods. The timing of adding the EU-containing media to different wells were well-planned such that the time point of fixation of different wells with various chase duration were synchronized.

Cell fixation, permeabilization, and Click-iT reaction were done following manufacturer’s instruction (Invitrogen, C10365). After Click-iT reaction, cells were washed three times in PBS for 5min and immediately processed to immunofluorescence.

For immunofluorescence, cells were blocked with 4% IgG-free Bovine Serum Albumin, BSA, (VWR, 102643-516) for 1 hour at RT and incubated with primary antibodies (anti-Fibrillarin Abcam ab5821 Rabbit 1:500 dilution, anti-TCOF1 Sigma WH0006949M2 Mouse 1:500 dilution) in 4% IgG-free BSA at 4°C overnight. After three washes in PBS, primary antibody was recognized by secondary antibodies (Goat anti-Rabbit IgG Alexa Fluor 647 Invitrogen A21244 1:1000 dilution, Goat anti-Mouse IgG Alexa Fluor 488 Invitrogen A11029 1:1000 dilution) at RT for 1 hour in the dark. Then, cells were washed three times with PBS.

For nuclear staining, Hoechst 33342 (Thermo Scientific, 62249) was 1:2,000 diluted in PBS and stained the sample for five minutes, followed by three washes with PBS.

4-color images (Blue: Hoechst 33342; Green: TCOF1; Red: EU-RNA; Far-Red: Fibrillarin) were acquired as z-stack on ZEISS 980 with a 63x 1.47NA objective and with the Airyscan detector, and 3D-processed by ZEN Blue.

Analysis of newly-synthesized rRNA efflux

Eight different chase durations in both −/+TCOF1 were imaged, resulting in 16 conditions in total. In the multi-color image dataset, Hoechst 33342 represents nuclei, TCOF1 represents FC, EU-RNA represents newly-synthesized rRNA, and Fibrillarin represents DFC. TCOF1 and EU-RNA channels were used for efflux calculation, while Hoechst 33342 and Fibrillarin channels were used for quality reference.

3D-processed z-stack images were converted to 2D images by maximum intensity projection (MIP) along the z-axis. Puncta in the TCOF1 channel were identified as described above (in “Patterning quantifications”). Only TCOF1 puncta identified within nuclei were kept, which was around 1,200 in each condition. The local MIPs of TCOF1, EU-RNA and Fibrillarin channels containing a 1.5μm×1.5μm region centered at all TCOF1 punctum centers were cropped, leading to a pool of 3-channel cropped images for each condition.

To estimate the outer boundaries of FC and DFC in −TCOF1 conditions, all available cropped images among all −TCOF1 conditions with different chase durations were averaged together by channel. The radial distribution function (RDF) of FC and DFC intensities in the averaged result were computed, and the outer boundaries of FC ($r_{\mathrm{F}\mathrm{C}}-$) and DFC ($r_{\mathrm{D}\mathrm{F}\mathrm{C}}-$) were defined as the position from the center where the intensity drop to half of the maximum. Same calculations were done to estimate the outer boundaries of FC ($r_{\mathrm{F}\mathrm{C}}+$) and DFC DFC ($r_{\mathrm{D}\mathrm{F}\mathrm{C}}+$) in +TCOF1 conditions.

To estimate the remaining nascent rRNA signal within FC after EU removal as well as the estimation uncertainty, a quarter of the cropped images from the pool were randomly sampled (around 300) whose averages were calculated by channel, resulting in a 1.5μm×1.5μm 3-channel averaged image. Such a down-sampling was performed for 50 times. For each down-sampled averaged image, the outer boundaries of FC and DFC were computed. Valid down-sampled averaged images of all −TCOF1 conditions with different chase durations were verified by showing comparable FC outer boundary with $r_{\mathrm{F}\mathrm{C}}-$ and comparable DFC out boundary with $r_{\mathrm{D}\mathrm{F}\mathrm{C}}-$. For each valid down-sampled averaged image, the total EU-RNA signal within the $r_{\mathrm{F}\mathrm{C}}-$ was measured. The mean and standard deviation of all valid down-sampled averaged images at each chase time point were calculated, and rescaled to the mean at $\mathrm{t}=0$. Same procedures were done for all +TCOF1 conditions with different chase durations. Finally, an exponential fitting was applied to either all eight chase time points in −TCOF1 or all eight chase time points in +TCOF1.

QUANTIFICATION AND STATISTICAL ANALYSIS

To compare the statistical distributions with data, the Cramer-von Mises test in Python was applied in Figure 1E. The unpaired two-tailed student’s t-test via the ttest_ind() function in the Scipy library in Python was applied in Figure 3C, 3E, and 4B, where the raw data points in those graphs are directly visible with the distributions depicted as violin plots and the mean highlighted as bars. The unpaired two-tailed student’s t-test via Prism 9 (GraphPad) was applied in Figure 4C–D, where the raw data points in those graphs are directly visible with the mean +/− standard deviation highlighted as bargraphs with errorbars. The p-values are reported in the figure captions. All statistical results were done without randomization or stratification.

Supplementary Material

2

Supplementary Movie 1: Simulation showing that FCs coarsen in the absence of rRNA transcription.

3

Supplementary Movie 2: Simulation showing FC size selection in presence of rRNA transcription.

4

Supplementary Movie 3: Simulation showing that FCs would coarsen if nascent rRNA had higher affinity for FC proteins than for non-FC proteins.

5

Supplementary Movie 4: Simulation demonstrating that the presence of rRNA alone, in absence of active transcription, is not sufficient to arrest the coarsening of FCs.

6

Supplementary Movie 5: Simulation showing that inhibition of rRNA transcription leads to FC coarsening, and resumption of rRNA transcription leads to a recovery of a pattern of FCs with controlled size.

7

Supplementary Movie 6: Experiment demonstrating that inhibition of rRNA transcription leads to FC coarsening.

8

Supplementary Movie 7: Experiment demonstrating that resumption of rRNA transcription by refeeding starved cells leads to a recovery of a pattern of FCs with controlled size.

Highlights.

• Non-equilibrium processes contribute to biomolecular condensate patterning in cells

• RNA synthesis arrests coarsening & governs distribution of nucleolar fibrillar centers

• Theory and experiment explain key features of condensate patterning in cells

• Altered patterning of fibrillar center condensates disrupts ribosomal RNA processing