Nascent ribosomal RNA act as surfactant that suppresses growth of fibrillar centers in nucleolus

Liquid-liquid phase separation (LLPS) has been thought to be the biophysical principle governing the assembly of the multiphase structures of nucleoli, the site of ribosomal biogenesis. Condensates assembled through LLPS increase their sizes to minimize the surface energy as far as their components are available. However, multiple microphases, fibrillar centers (FCs), dispersed in a nucleolus are stable and their sizes do not grow unless the transcription of pre-ribosomal RNA (pre-rRNA) is inhibited. To understand the mechanism of the suppression of the FC growth, we here construct a minimal theoretical model by taking into account nascent pre-rRNAs tethered to FC surfaces by RNA polymerase I. The prediction of this theory was supported by our experiments that quantitatively measure the dependence of the size of FCs on the transcription level. This work sheds light on the role of nascent RNAs in controlling the size of nuclear bodies.

The free energy of the system has the form F d is the free energy of a DFC layer and is a functional of the occupancy α p of the fibrillarinbinding region (FBR) of nascent pre-rRNA, the volume fraction ϕ p of fibrillarin (FBL) RNA-binding proteins (RBPs), and the volume fraction ϕ r of the nascent pre-rRNA units (see below).γ in is the interfacial tensions at the FC-DFC interface and γ ex is the interfacial tension at the DFC-GC interface.r in is the distance between the center of FC and the FC-DFC interface.r ex is the distance between the center of FC and the DFC-GC interface.
The free energy of a DFC layer is written in the form where f d is the free energy density in the DFC layer and r is the radical coordinate from the center of the FC.b is the length of a pre-rRNA unit.The free energy density f d is composed of 4 contributions where f ela is the elastic free energy density of nascent RNA transcripts, f mix is the free energy density due to the mixing entropy of RBPs and solvent molecules, f int is the free energy density due to the interactions between RBPs, and f bnd is the free energy density due to the binding of RBPs to nascent RNA transcripts.µ p is the chemical potential of RBPs and Π ex is the external pressure.The external radius r ex and the volume fraction ϕ r of nascent pre-rRNA have the relationship The elastic free energy density has the form where σ(r) = σ in r 2 in /r 2 for the spherical geometry and σ(r) = σ in for the planer geometry.k B is the Boltzmann constant and T is the absolute temperature.The derivation of Supplementary Equation (S5) is shown in Supplementary Note 5.
The free energy due to the mixing entropy of RBPs and solvent molecules has the form The free energy due to the interactions between RBPs has the form where χ is the interaction parameter that accounts for the attractive interaction between RBPs.For simplicity, we assumed that RBPs that are bound to nascent RNA transcripts are equivalent to RBPs that are freely diffusing in the DFC layer.The solvent molecules (water) have affinity to nascent RNA units rather than RPBs.We thus assume that solvent molecules and nascent pre-rRNA units are equal in terms of the interaction.
The free energy due to the binding of RBPs to nascent RNA transcripts has the form where −ϵk B T is the energy increase due to the binding of RBPs to nascent RNA transcripts.
For simplicity, we assumed that each nascent RNA unit has one binding site of RBPs.

Local equilibrium
With the variations α p (r) → α p (r) + δα p (r), ϕ p (r) → ϕ p (r) + δϕ p (r), and ϕ r (r) → ϕ r (r) + δϕ r (r), the free energy changes as Because the free energy density f d does not include spatial derivatives of α p (r), ϕ p (r), and ϕ r (r), the functional derivatives of f d with respect to α pol (r), ϕ pol (r), and ϕ r (r) can be replaced to the partial derivatives of f d with respect to α p , ϕ p , and ϕ r .The variation δr ex of the external radius results from the variation δϕ r (r) with the condition of eq.S4: At the minimum of the free energy (the local equilibrium), δF = 0 for any functions of δα pol (r), δϕ pol (r), and δϕ r (r): By using Supplementary Equations (S3) − (S8), Supplementary equation (S12) is rewritten in the form Supplementary Equation (S15) suggests that the chemical potential of RBPs is uniform in the DFC layer.By using Supplementary Equations ( S3) -(S8), Supplementary Equation We eliminated µ p in Supplementary Equation (S11) by using Supplementary Equation (S15) to derive the form of Supplementary Equation (S16).By using Supplementary Equations (S3) -(S8), Supplementary Equation (S13) is rewritten in the form Supplementary Equation (S13) is integrated with respect to ϕ r as where −Πb 3 is the integral constant.By eliminating µ r from Supplementary Equation (S18) by using Supplementary Equation (S13), Supplementary Equation (S18) is rewritten in the form The integral constant thus turns out to be the osmotic pressure.By using Supplementary Equations (S3) -(S8), Supplementary Equation (S19) is rewritten in the form with We eliminated µ p in Supplementary Equation (S19) by using Supplementary Equation (S15) to derive the form of Supplementary Equation (S20).So far, we derived the fact that the osmotic pressure is uniform in the DFC layer.The value of the osmotic pressure is determined by taking the limit r → r ex to Supplementary Equation (S18) as By substituting Supplementary Equation (S24) into Supplementary Equation (S16), the occupancy of the FBRs of nascent pre-rRNAs is derived in the form For ϕ r < 1 and ϕ p < 1, Supplementary Equation (S20) has an approximate form with Supplementary Equation (S26) is derived by expanding Supplementary Equation (S20) in a power series of ϕ p and ϕ r and neglecting higher order terms of these volume fractions.
We assumed that ϕ p is very small and neglected even the terms linear to ϕ p .We also used the boundary condition Π = Π ex for the planer geometry, see Supplementary Equation (S22) for r ex → ∞.The osmotic pressure has three contributions: the contribution from the entropic elasticity of the RNP complexes (the first term of Suppplementary Equation (S26)), the contribution from the two-body interaction between the units of RNP complexes (the second term of Supplementary Equation (S26)), and the contribution from the three-body interactions between these units (the third term of Supplementary Equation (S26)).The two-body interaction between the units of RNA complexes is repulsive for χ < χ s (the good solvent regime) and is attractive for χ > χ s (the poor solvent regime).In the good solvent regime, χ < χ s , the first and second terms of Supplementary Equation (S26) dominate the third term of this equation and the volume fraction of the FBRs of nascent pre-rRNA thus has the form if there are not applied pressure to the DFC layer.In the poor solvent regime, χ > χ s , the second and third terms of Supplementary Equation (S26) dominates the third term of this equation and the volume fraction of the FBRs of nascent pre-rRNA has the form Eqs. S28 and S29 correspond to the volume fraction of polymer units for the good and poor solvent conditions, as predicted by the Alexander model, respectively.

Melt regime
The volume fraction ϕ r of the FBRs of nascent pre-rRNAs increases with increasing the interaction parameter χ and, eventually, the RNP complexes occupy most volume of the DFC layer, ϕ r ≈ 1/(1 + α pr ) (the melt regime), see Fig. 3 in the main article.For simplicity, we here derive the asymptotic forms of the volume fractions ϕ r and ϕ p for the case of α pr ≈ 1 (which is the case of ϵ < −1).We derive the solution for with δϕ rm ≪ 1, δα pm ≪ 1, and ϕ p ≪ 1.With this approximation, eq.S20 is rewritten in the form )  the relationship between the volume fraction ϕ r and the external pressure Π ex is derived in the form The volume fraction of the FBRs of nascent pre-rRNAs thus has the form where δϕ p ≪ 1 and ϕ r ≪ 1.In this case, Supplementary Equation (S15) has an approximate form Supplementary Equation (S45) is derived by substituting Supplementary Equation (S44), expanding it in a power series of δϕ p (r) and ϕ r , and neglecting the higher order terms with respect to δϕ p (r) and ϕ r .By using the same expansion and approximation, Supplementary Equation (S17) is rewritten in the form The last form of Supplementary Equation (S46) is derived by eliminating log(δϕ p − 2ϕ r ) by using Supplementary Equation (S45).By solving Supplementary Equation (S46) with respect to ϕ r (r), the volume fraction ϕ r (r) of the FBRs of nascent pre-rRNAs is derived as where we introduced a parameter to simplify the expression of Supplementary Equation (S47).
The osmotic pressure has an approximate form where it is derived by substituting Supplementary Equation (S44) into Supplementary Equation (S20), expanding it in a power series of δϕ p (r) and ϕ r (r), and neglecting the higher order terms with respect to δϕ p (r) and ϕ r (r).The last form of Supplementary Equation (S49) is derived by eliminating log(δϕ p (r) − 2ϕ r (r)) by using Supplementary Equation (S46).By using Supplementary Equation (S22), the volume fraction ϕ rex (= ϕ r (r ex )) of the FBRs of nascent pre-rRNAs at r = r ex is derived as (S50) By using the boundary condition, Supplementary Equation (S50), the local volume fraction ϕ r (r) of the FBRs of nascent pre-rRNA has the form The chemical potential µ r has the form Supplementary Equations (S50), (S51), and (S52) are effective for cases in which the layer of the FBRs of nascent pre-rRNAs is a double layer of the DFC and melt phases if the DFC layer interfaces with GC at r = r ex .
The external radius r ex is derived by using the relationship where it is derived by substituting Supplementary Equation (S51) (and Supplementary Equation (S50)) into Supplementary Equation (S4).Indeed, Supplementary Equation (S53) is a quadratic equation and the ratio r ex /r in can be derived analytically.
The free energy density of the layer of the FBRs of nascent pre-rRNAs has an approximate form Supplementary Equation (S54) is derived by substituting Supplementary Equations (S44), expanding it in a power series of δϕ p and ϕ r , and neglecting the higher order terms of δϕ p and ϕ r .We also used Supplementary Equation (S51) to derive the last form of Supplementary Equation (S54).By using Supplementary Equation (S54), the free energy of the system is derived as where we used ζ is proportional to the transcription rate.The free energy, Supplementary Equation (S55), depends on the size of FCs via the ratio r ex /r in and σ in (∝ r in ), see also Supplementary Equation (S53). For dominates the second term of this equation.In this case, the first derivative of the free energy with respect to the raio r ex /in is zero at the minimum of the free energy, This leads to By substituting Supplementary Equation (S58) into Supplementary Equation (S53), the radius of FCs is derived as implying that the radius of FCs is proportional to the inverse of the square root of the transcription rate.

Melt layer
For cases in which the FBRs of pre-rRNAs form a uniform melt layer, the volume fraction ϕ r of the FBRs of pre-rRNAs is ≈ 1/2.We therefore derive the volume fractions in the forms where ϕ p ≪ 1 and δϕ r ≪ 1.We substitute Supplementary Equation (S60) into Supplementary Equations (S15), (S20), (S17), expanding these equations in a power series of ϕ p and δϕ r , and neglecting the higher order terms with respect to ϕ p and δϕ r , The osmotic pressure Π is derived by eliminating log(δϕ r − ϕ p ) in Supplementary Equation (S63) by using Supplementary Equation (S62).This leads to the form By substituting Supplementary Equation (S62) into Supplementary Equation (S4), the ratio r ex /r in is derived as where ζ is given by Supplementary Equation (S56).The ratio r ex /r in thus does not depend on the radius of FCs.The free energy has an approximate form where it is derived by using ϕ r ≈ 1/2 and ϕ p ≃ 1 − 2ϕ r ≪ 1 to Supplementary Equation (S1).The derivative of the free energy with respect to σ in b 2 is zero at the minimum of the free energy, The radius of FCs thus has the form see Supplementary Equation (S65) for the dependence of r ex /r in on ζ.For ζ ≪ 1, eq.S68 has an approximate form

Double layer of DFC and melt
In some cases, the layer occupied by the FBRs of nascent pre-rRNAs is composed of a double layer of a melt phase and a DFC phase.The chemical potentials, µ p and µ r , and the osmotic pressure Π is continuous at the interface between the two layers.The osmotic pressure increases with increasing the radial coordinate r in the DFC layer, see Supplementary Equation (S49), while the osmotic pressure is almost constant in the melt layer, see Supplementary Equation (S64) (where the third term of eq.S64 is smaller than the other terms in this equation and is neglected).This implies that the DFC layer is at the interface with the GC and the melt layer at the interface with the FC.The osmotic pressure in the DFC layer has the form where it is derived by substituting Supplementary Equation (S51) into Supplementary Equation (S49).The osmotic pressure in the melt layer has the form where it is derived by substituting the chemical potential in Supplementary Equation (S64) and neglected the third term of Supplementary Equation (S64).Supplementary Equation (S72) is derived by substituting Supplementary Equation (S51) into Supplementary Equation (S46) and is effective for cases in which the DFC layer interfaces with the GC (because Supplementary Equation (S64) is imposed at r = r ex ).The radical coordinate r i at the interface between the DFC and melt layer is derived by equating nascent pre-rRNA units is uniform for nascent pre-rRNA transcripts on a planer surface. 9,10th this approximation, the free energy per unit area of nascent pre-rRNA transcripts on a planer surface has the form

Relationship with conventional picture of ribosome biogenesis
The key assumption of our theory is that the terminal regions of nascent pre-rRNAs span between the top to the bottom of the DFC layer.This is motivated by a recent experiment that the upstream half of ETSs are localized at the DFC layer and the downstream half is localized at the FC. 14 The stretching of the terminal regions results from the multivalent interactions between FBLs and thus our theory should be tested in the condition of high FBL volume fraction.In the conventional picture, the DFC layer is composed of 5' ETSs that are already cleaved and folded by RNA processing.Such diffusive molecules tend to make the interaction between condensates attractive (the depletion interaction), 4 however, the non-equilibrium nature of the transcription may change the situation.We envisage a future theoretical study to predict the stable size of FCs with the conventional picture to check if the latter agrees with existing experiments, including ours.

Rheology of DFC layer
In our model, we focused on FBLs because these RBPs contribute most significantly to the localization of the terminal regions of pre-rRNAs to the DFC layer. 14FRAP experiments on the condensates of FBLs reconstituted in vitro suggest that these condensates are thixotropic (and probably also be viscoelastic) and become gels in a long timescale. 5Indeed, recent experiments showed that long non-coding RNA SLERT facilitates the transition of the RNA helicase DDX21 to the closed conformation and ensures the fluidity of proteins in the DFC layer. 15This mechanism ensures the validity of our assumption that unbound FBLs can diffuse freely in the DFC layer, although we did not take into account these factors explicitly.

5 FigureFigure S2 8 Figure 1 .
Figure S1 BMH-21 and CX-5461 treatment reduce pre-rRNA expression levels in a dose-dependent manner.a.Quantification of pre-rRNA expression levels by RT-qPCR in BMH-21-untreated and -treated conditions.Data are represented as mean ± SD (n =3).b.Quantification of pre-rRNA expression levels by RT-qPCR in CX-5461-untreated and -treated conditions.Data are represented as mean ± SD (n =3).

) 2 b 4 σ 3 N r ϕ 2 r + v b 3 1 / 3 .
This free energy is a function of the height h of the nascent pre-rRNA transcripts.The first term of Supplementary Equation (S77) is the elastic free energy of nascent pre-rRNA transcripts and the second term of Supplementary Equation (S77) is the free energy due to the interactions between nascent pre-rRNA units.k B is the Boltzmann constant and T is the absolute temperature.N r is the number of units in a nascent pre-rRNA transcript and b is the length of each unit.σ is the surface density of nascent pre-rRNA transcripts.v is the excluded volume that represents the magnitude of the interactions between nascent pre-rRNA transcripts and has a relationship v = b 3 (2 − χ) with the interaction parameter χ introduced in eq.(16) in the main article (the difference from the usual relationship b 3 (1 − 2χ)/2 is due to the volume of RBPs bound to nascent pre-rRNA transcripts.Supplementary Equation (S77) is rewritten in the formF bru k B T = 3 σN r ϕ r (S78)by using the form of the volume fraction of nascent pre-rRNA unitsϕ r = b 3 σN r h .(S79)Minimizing Supplementary Equation (S77) with respect to h leads to h = N r b σv 3b

Table 2 :
The values of the parameters used to estimate the diffusion length λ d .The diffusion constant of a RNA unit is estimated as D s ∼ (k B T )/(4πϵ w b) ∼ 8 × 10 −11 m 2 /s and the time to synthesize a RNA unit is estimated as τ s = b/v e ∼ 0.12 s by using the values listed in this table.We estimated b from the persistence length of singlestranded DNA.