Mesoscale properties of protein clusters determine the size and nature of liquid-liquid phase separation (LLPS)

Abstract

The observation of Liquid-Liquid Phase Separation (LLPS) in biological cells has dramatically shifted the paradigm that soluble proteins are uniformly dispersed in the cytoplasm or nucleoplasm. The LLPS region is preceded by a one-phase solution, where recent experiments have identified clusters in an aqueous solution with 10²-10³ proteins. Here, we theoretically consider a core-shell model with mesoscale core, surface, and bending properties of the clusters’ shell and contrast two experimental paradigms for the measured cluster size distributions of the Cytoplasmic Polyadenylation Element Binding-4 (CPEB4) and Fused in Sarcoma (FUS) proteins. The fits to the theoretical model and earlier electron paramagnetic resonance (EPR) experiments suggest that the same protein may exhibit hydrophilic, hydrophobic, and amphiphilic conformations, which act to stabilize the clusters. We find that CPEB4 clusters are much more stable compared to FUS clusters, which are less energetically favorable. This suggests that in CPEB4, LLPS consists of large-scale aggregates of clusters, while for FUS, clusters coalesce to form micron-scale LLPS domains.

Introduction

In recent years, a paradigm shift has transformed our understanding of cellular organization, with the discovery that many proteins undergo liquid-liquid phase separation (LLPS) to form condensates. This challenges the long-held view of cellular components as uniformly dispersed or structurally confined (e.g., the cytoskeleton, chromatin)¹. The LLPS of proteins is generally attributed to their stronger mutual attraction compared to their affinity for the cell’s aqueous environment². The physics of these interactions and the resulting condensates are often quantified in controlled, in vitro studies^1,3,4,5,6 with a small number of components to avoid the highly realistic but complex nature of the dynamic, active, and multi-component cellular environment^7,8,9,10.

Beyond LLPS, it has been observed that certain proteins with disordered domains can form large (30–300 nm) self-assembled clusters even within the single-phase regime where LLPS is absent^4,5,11,12,13. The observation of large mesoscale clusters is puzzling since small molecules in aqueous solution form clusters or oligomers with quite low probability in the one-phase region. In contrast, these protein clusters contain hundreds or thousands of solute monomers^4,5,11. In vitro studies have characterized this phenomenon, revealing distinct behaviors as a function of protein concentration. We focus on two representative cases: CPEB4_NTD (the intrinsically disordered N-terminal domain of Cytoplasmic Polyadenylation Element Binding protein 4, denoted as CPEB4), whose cluster size is only weakly dependent on concentration¹¹, and FUS, whose cluster size shows a strong dependence on concentration⁴.

Previous cluster formation models have been based on the sticker-spacer framework for intrinsically disordered proteins^4,14,15. Molecular dynamics simulations using this framework predicted that proteins at the cluster surface adopt more elongated conformations and align predominantly in the direction normal to the cluster’s surface¹⁵. These models also suggested that the sticker-sticker interaction energy required to form the observed large clusters is in the range of ~10 k_BT⁴. A core-shell model was proposed for FUS in coarse-grained simulations, where the proteins were modeled as a diblock copolymer, revealing an internal core-shell structure¹⁶. Core-shell clusters were computationally simulated in peptide chains containing leucine and serine. In these simulations, the hydrophobic leucine residues preferentially occupied the core, while the hydrophilic serine residues were more likely to be found in the shell¹⁷. However, these computational models neither dealt with the distinct size dependence of the different types of clusters nor addressed the specific protein properties that allow cluster formation.

We suggest that the formation of these clusters is possible since disordered proteins, in contrast to small molecules undergoing LLPS, can have several different conformations. Some stabilize the cluster core, while others stabilize the cluster surface (shell). We quantify this using a generic mesoscale core-shell model and compare the predicted cluster size distribution with experimental data to extract the core and shell energies. The mesoscale nature of these energies distinguishes the clusters from relatively small oligomers whose properties are very sensitive to molecular details^18,19. In contrast to molecular simulations^{4,14,15,16,17}, the mesoscale nature of our approach bypasses the detailed molecular calculation of protein conformations. Instead, we characterize those conformations by the contributions to the energies relevant to cluster self-assembly using only a few parameters extracted from the experiment.

We used a similar approach to investigate the temperature dependence of the core and shell energies of CPEB4 clusters as they approach LLPS¹¹. In this work, we extend the model to derive a scaling law for the dependence of the clusters’ mean size on the total protein concentration²⁰, as well as the size distribution around the mean value. This concentration dependence provides estimates for the bending energies of the clusters, as well as convincing evidence confirming that FUS clusters that precede LLPS lie below their Critical Aggregation Concentration²¹ (CAC). In contrast, CPEB4 clusters lie above their CAC²². Furthermore, we extract the energetic parameters of Ddx4n1 and α-synuclein clusters by fitting their concentration-dependent size distributions to our theoretical model¹³.

In summary, our comparison of experiment and theory leads us to conclude that these proteins (CPEB4 vs FUS and the others, which we show behave like FUS, such as Ddx4n1, and α-synuclein) represent two different paradigms for cluster formation, leading to LLPS: CPEB4 clusters are more energetically stable and favorable than those in FUS, which are essentially large concentration fluctuations. The cluster energetics and stability have important implications for the nature of the large, micron-scale domains in the LLPS region of the phase diagram. For CPEB4, the analysis, as well as EPR measurements^5,11, strongly suggest that LLPS is induced by the aggregation of relatively stable clusters. In contrast, the less stable FUS clusters likely coalesce in the LLPS phase, resulting in large domains with proteins in their hydrophobic conformation.

Methods

Theoretical model of cluster structure and protein configurations

The theoretical model aims to describe the cluster size distribution within the framework of continuum mesoscopic theory, drawing parallels to other mesoscale structures observed in systems containing self-assembling amphiphiles. Specifically, we propose that these clusters resemble microemulsions (swollen micelles), where an amphiphilic surface layer separates a hydrophobic core from the surrounding aqueous solvent^21,22. We show below that this is consistent with the measured, mesoscale cluster equilibrium distribution and the concentration dependence of the average cluster size. The distinct behavior of these quantities leads us to delineate two paradigms (CPEB4 with low-energy clusters and FUS with high-energy clusters) for cluster formation and their relationship to LLPS; these are the main conclusions of our paper.

While traditional microemulsions require “oil” and “amphiphiles,” the protein-water-salt solutions of interest here achieve similar behavior without these additives. Instead, the disordered protein domains, composed of sequences with both hydrophobic and hydrophilic regions, enable the same protein species to adopt distinct configurations based on their local environment. This adaptability gives rise to different ensembles of protein states in the dilute aqueous phase, the dense cluster core, and the interfacial cluster shell.

The clusters are stabilized when protein configurations in the shell minimize the interfacial energy cost. This stabilization is achieved when proteins in the shell behave like amphiphiles, adopting configurations akin to the role of surfactants in microemulsions, in which their hydrophobic amino acids are oriented (but still internally disordered as for block copolymers²³) toward the dense core. In contrast, their hydrophilic amino acids face the aqueous phase. Molecular-scale evidence for this is provided in Supplementary Table S1 of the Supplementary Information (SI) and its accompanying explanatory caption.

Based on this, along with the aforementioned computational and experimental evidence, we hypothesize that proteins adopt three distinct configurations corresponding to the different regions of the system, as illustrated in Fig.1:

• Water-soluble configuration: In the dilute aqueous solution, hydrophobic amino acids are collapsed into a “blob” (in the sense used in polymer physics²⁴) shielded by the hydrophilic ones. In this configuration, interactions between amino acids within the same protein chain are in a minimal energy configuration, although disordered.

• Hydrophobic configuration: In the dense cluster core, hydrophobic amino acids are exposed and interact with those of other proteins, while the hydrophilic blobs are collapsed, driving cluster formation. We suggest that the hydrophobic blob maximizes the interaction with the neighboring proteins’ hydrophobic blobs at the expense of water, which is partially depleted.

• Amphiphilic configuration: At the cluster surface, proteins orient their hydrophobic amino acids toward the core (still disordered as in a polymer blob) and their hydrophilic amino acids toward the aqueous phase (also disordered as in a blob). In this configuration, proteins are more extended due to their two-blob nature, compared to the previous cases¹⁵. Although a detailed molecular analysis, such as determining the size of hydrophobic and hydrophilic regions or their relative solubility energies, is beyond the scope of this work, we note that both CPEB4 and the FUS protein family (including hnRNPA3, EWSR1, and TAF15) can be modeled as comprising two distinct blobs: a relatively ordered RNA-binding domain and one or two intrinsically disordered tails (Fig. S1A). These two regions display distinct hydrophobicity indices²⁵ (Table S1), supporting the hypothesis that proteins within the cluster shell can behave like amphiphiles.

  Fig. 1: Protein conformations.

  

  A Illustration of conformations of dilute protein monomers in the solvent and those packed in the suggested core-shell structure of the clusters. The blue color denotes more hydrophobic parts, but does not imply that these segments are folded. The black, hydrophilic domain of the proteins in the cluster core is more collapsed. B The interfacial layer consists of more hydrophilic protein segments that face the solvent and more hydrophobic segments that face the core.

Direct experimental evidence for the coexistence of the three conformations comes from EPR measurements of the molecular rotational diffusion times of spin-labeled CPEB4 in which three distinct times, separated by two orders of magnitude, were observed^5,11. The shortest time is identified with the soluble conformation, the longest with the amphiphilic configuration, and an intermediate time with the proteins in the core. These are all the same chemical species, which can assume different conformations at a given temperature and concentration due to their disordered nature. Three typical sets of conformations were also shown using different molecular dynamics approaches in other proteins with intrinsically disordered domains^15,26, suggesting that proteins at the surface of condensates assume different configurations than those at their core^14,15.

Theoretical model for the formation energies of the core and shell

To account for the self-assembly energy of the clusters, our mesoscopic model considers the energy of the cluster core and shell. The cluster geometry is taken to be spherical, consistent with atomic force microscopy imaging¹¹ and electron microscopy²⁷ of CPEB4, and transmission electron microscopy of FUS clusters⁴ in the salinity and temperature regime studied.

We note that long (100–200 nm) FUS fiber-like structures have been reported under different experimental conditions²⁸, which are outside the scope of the present work. The formation of the clusters we consider here is reversible upon change of temperature (CPEB4¹¹), dilution (α-synuclein¹³), and protein concentration (FUS⁴). Therefore, the theoretical model we developed here assumes the system is in equilibrium.

The two-blob amino acid structure of the proteins and the computationally predicted elongated structure at the cluster shell¹⁵ indicate a nematic-like ordering of the cluster shell with non-zero bending modulus²². Therefore, we model the surface energy of the cluster in the spirit of an amphiphilic layer with interfacial tension, γ, bending modulus $\bar{\kappa }$, and spontaneous curvature J_s²¹. For a spherical cluster, this results in the following surface energy^11,22,29:

$${U}_{{shell}}=4\pi \gamma {R}^{2}-8\pi {\kappa }_{B}{J}_{s}R+8\pi \bar{\kappa }.$$

(1)

where $\bar{\kappa }\equiv {\kappa }_{B}+\frac{{\kappa }_{G}}{2},$ since for spherical clusters, the contributions of the bending, ${\kappa }_{B}$, and saddle-splay moduli, ${\kappa }_{G}$, cannot be distinguished.

The core free energy relative to that of a protein molecule in the solution is given by the product of the number of proteins in the cluster, $N$, and the difference between solubility energy, ${\epsilon }_{B}$, and the protein chemical potential, $\mu$,

$${U}_{{core}}=N\cdot \left({\epsilon }_{B}-\mu \right).$$

(2)

The chemical potential of a protein molecule in a dilute solution (all the experimental systems considered here are very dilute with concentrations of 0.1–100 μM) is the logarithm of the dispersed (monomeric) protein volume fraction,$\,\mu =k_B T \; {\mathrm{ln}}({\phi }_{m})$, as seen in Supplementary Note S1, Sec. I. 2. The solubility energy, ${\epsilon }_{B}$, represents the combined effects of two factors: (a) the favorable interaction energy between amino acids in the dense cluster core (per protein) and (b) the entropic cost associated with the reduced conformational space of the protein in the “hydrophobic” state compared to the ‘water-soluble’ configurations. Since cluster formation is favorable in the salinity and temperature regime studied, the solubility energy is negative (${\epsilon }_{B} < 0$).

The core energy is a function of the number of proteins in the cluster $N$, while the shell energy is a function of its radius $R$. To present the cluster formation energy as a function of one geometrical parameter, we link these two with

$$N=\frac{4\pi }{3}\frac{{R}^{3}}{v}.$$

(3)

The protein volume, $v$, is assumed to be the same in the core, shell, and solution. This simplification avoids introducing additional fitting parameters. With this simplification, the relation between the protein volume fraction, $\phi$, and the protein concentration, $C$, is given by

$$\phi =v\cdot C.$$

(4)

For the subsequent calculations, we estimate $v$ in all configurations by modeling the protein as an ideal polymer. The radius of gyration, ${R}_{G}$ is given by ${R}_{G}={l}_{{aa}}{\left({N}_{{aa}}/6\right)}^{1/2}$, where ${N}_{{aa}}$ is the number of amino acids and ${l}_{{aa}}$ is a typical length of each amino acid (0.36 nm). The protein volume is then expressed as $v=\frac{4\pi }{3}{R}_{G}^{3}$. However, it is important to note that this is a rough approximation. The actual ${R}_{G}$ in solution is likely larger, as the proteins in question have a large intrinsically disordered domain that typically follows the scaling law ${R}_{G} \sim {N}^{0.57}$³⁰. In contrast, proteins in the core are “collapsed” (in the sense of polymers) and occupy a smaller volume, in the extreme case $v \sim {N}$ (instead of $v \sim {N}^{\frac{3}{2}}$ in the solution). The proteins in the shell are more extended and occupy an intermediate volume, situated between the extremes of the core and soluble configurations.

An advantage of our mesoscale approach is that these details do not affect the scaling laws that relate the most likely cluster size to the total protein concentration. However, concentration gradients (for example, between the core and shell) will introduce corrections to the cluster formation energy. A comprehensive analysis of these corrections lies beyond the scope of the present work.

The energy of forming a cluster of radius $R$ is given by the sum of Eqs. 1 and 2

$$U\left(R\right)={U}_{{core}}+{U}_{{shell}}=\frac{4\pi }{3v}\left({\epsilon }_{B}-\mu \right){R}^{3}+4\pi \gamma {R}^{2}-8\pi {\kappa }_{B}{J}_{s}R+8\pi \bar{\kappa }.$$

(5)

The balance between the minimum of $U\left(R\right)$, which favors a specific radius, and entropy that favors monomers as they maximize the translational entropy, gives the cluster number concentration (made dimensionless by multiplication by the protein molecular volume $v$)²²

$$P\left(R\right)=\frac{v\cdot {n}_{R}}{V}=\exp \left[-\frac{U\left(R\right)}{{k}_{B}T}\right].$$

(6)

$V$ is the system volume and ${n}_{R}$ is the number of clusters of size $R$. We note that this theory does not apply to clusters whose radius is comparable to the monomer size, as our mesoscopic approach used here fails in that regime. In fact, no clusters with a radius smaller than 15 nm are observed in all protein types and experimental methods discussed here (typical protein radius is 2.5–5 nm), indicating that the formation of clusters smaller than a critical radius is energetically unfavorable. This is possibly due to the high bending energy cost per cluster. This observation highlights the contrast to classical self-assembly (Supplementary Note S1 Sec. I 5).

We use this theoretical model to derive an analytical description of the size distribution of clusters and the mean size, ${R}^{* }$, as a function of the core and surface properties, and the total protein concentration in the system. This theory is compared to the experimental observations.

Results

Cluster mean radius as a function of protein concentration

We first address the most common cluster radius, ${R}^{* }$, of CPEB4 and FUS clusters as a function of total protein concentration, $C$. These are found based on the peaks of size distribution derived from dynamic light scattering (DLS) for CPEB4, which is presented in ref. ¹¹, and FUS clusters nanoparticle tracking analysis (NTA) presented in ref. ⁴. Based on these experimental results, FUS and CPEB4 obey two different paradigms: ${R}^{* }$ of FUS clusters increases significantly with $C$ (Fig. 2A, open circles), and most (~99%) of the proteins are dispersed in solution as monomers⁴. In contrast, ${R}^{* }$ of CPEB4 clusters is almost unchanged as $C$ increases (Fig. 2B, open circles), and the clusters contain about 90% of the total protein concentration¹¹.

Fig. 2: Scaling law for the most common cluster size.

The most common cluster size plotted as a function of total protein concentration, $C$. Theoretical fits are represented by dashed lines, and experimental data by circles. A FUS cluster radius obtained from the NTA measurements, which are most sensitive to the smaller clusters. Theoretical fit to ${R}^{* }=a\cdot {C}^{\frac{1}{3}}$ (corresponding to Eq. 12), the fitting parameter $a$ is found to be 5.25 ∙ 10³ nm². Data for $C$ and ${R}^{* }$ are as listed in Table S3. The inset is the same data however the y-axis was rescaled to show the linear relation between ${N}^{* }$ (estimated as $\frac{4\pi }{3}\frac{{{R}^{* }}^{3}}{v}$, with $v$ the estimated volume of FUS monomer, taken as 180 nm³) and $C$ as indicated in Eq. 13. The solid black lines are the ${R}^{* }$ (or ${N}^{* }$ in the insert) values taken from three different measurements, these are not error bars. The mean value of these is shown by the blue circles. B CPEB4: Data for $C$ and ${R}^{* }$ are as listed in Table S2. The theoretical curve is derived using a linear fit to ${{R}^{* }}^{-3}=a+b\cdot C$, as described in Eq. 14. $a$ and $b$ were found to be 2 ∙ 10⁻⁴ nm⁻³ and −0.018 nm⁻³ μM⁻¹. The curve is generated by inverting the relationship, ${R}^{* }={\left(a+b\cdot C\right)}^{-\frac{1}{3}}$. C, D Ddx4n1 and α-synuclein: cluster mass data taken from ref. ¹³. This measurement is also most accurate for the smaller clusters (in the range of 15 kDa to 5 MDa, corresponding to $N$ in the range of 10⁰–10²). ${N}^{* }$ is the number of monomers in a cluster calculated as ${N}^{* }={M}_{{cluster}}/{M}_{{mono}}$, where ${M}_{cluster}$ is the mass of the cluster obtained from mass photometry¹³ and ${M}_{mono}$ is the mass of a single protein based on its amino acid sequence. The fitting follows a linear relationship, ${N}^{* }=a\cdot v\cdot C$ (Eq. 13), where $a$ is the fitting parameter and $v$ is the monomer volume (48 nm³ for Ddx4n1 and 22 nm³ for α-synuclein). The values of $a$ were found to be 4.63 ∙ 10⁴ for Ddx4n1, 1.72 ∙ 10⁴ for α-synuclein at 20% PEG, and 8.59 ∙ 10⁴ at 5% PEG. D Two sets of measurements were done for α-synuclein, represented by the solid lines (that are not error bars). The average is shown by the open circles.

These behaviors agree with those of self-assembly above the CAC for CPEB4 and below the CAC for FUS: below the CAC, the fraction of protein in clusters is small, and the most common cluster radius, ${R}^{* }$, varies strongly as a power law of the protein concentration, $C$. Above the CAC, most of the proteins are found in clusters and ${R}^{* }$ varies only slightly with $C$.

We use self-assembly theory (Eqs. 1–3 and Supplementary Note S1) to predict the properties of the cluster distribution both below and above the CAC and compare them with the experiments. From Eq. 6 (Supplementary Note S1 section I.1), we write the cluster number concentration in terms of the cluster aggregation number, $N$,

$$P(N)=\exp \left[-N \frac{{\epsilon }_{N}}{k_{B} T}+N \; {\mathrm{ln}}\left({\phi }_{m}\right)\right].$$

(7)

${\phi }_{m}$ is the protein monomer volume fraction, and ${\epsilon }_{N}$ is the energy per protein monomer in a cluster of size $N$. This can be related to the parameters introduced in Eqs. 1 and 2 by considering the total number of proteins in a cluster (Eq. 3),

$${\epsilon }_{N}={\epsilon }_{B}+4\pi \gamma {\left(\frac{3v}{4\pi }\right)}^{\frac{2}{3}}{N}^{-\frac{1}{3}}-8\pi \kappa {J}_{s}{\left(\frac{3v}{4\pi }\right)}^{\frac{1}{3}}{N}^{-\frac{2}{3}}+8\pi \bar{\kappa }\cdot {N}^{-1}.$$

(8)

The conservation of protein number in the system constrains ${\phi }_{m}$ and the total protein volume fraction ϕ via the volume fraction of proteins in clusters ${\phi }_{c}$:

$$\begin{array}{cc}(a)\phi ={\phi }_{m}+{\phi }_{c},\, & (b){\phi }_{c}=\mathop{\sum }\limits_{N=2}^{N=\infty }N\cdot P(N)\end{array}$$

(9)

Equation 7 can then be written as a self-consistent equation for the cluster number distribution (Supplementary Note S1 section I.1),

$$P(N)=\exp \left[-N \frac{{\epsilon }_{N}}{k_{B} T}+N \; {\mathrm{ln}}(\phi) \right]\cdot {\left(1-\mathop{\sum }\limits_{j=2}^{j=\infty }\frac{j\cdot P(j)}{\phi }\right)}^{N}.$$

(10)

We solve Eq. 10 for the most probable value of $N={N}^{* }$ using a saddle point approximation of the cluster energy, which predicts the number of proteins in the most probable clusters (see Supplementary Note S1 section I.3). This approximation is valid as long as the distribution width is relatively narrow.

In addition, from our fits below, we estimate that the surface contributions of the tension and spontaneous curvature are much smaller than the bending contributions, $8\pi \bar{\kappa }$. Thus, in estimating ${N}^{* },$ we neglect the first two terms in Eq. 1 in a zeroth-order approximation. We note that this applies to the estimation of ${N}^{* }$ while the fits of the entire normalized distribution shown in the next section are insensitive to the bending contributions.

Taking the limit of large clusters $(N\gg 1)$, we find that the concentration dependence of ${N}^{* }\left(\phi \right)$ below the CAC varies linearly with the protein volume fraction $\phi$ (see Eq. S45 in Supplementary Note S1):

$${\mathrm{ln}}\,{N}^{*}=\frac{8\pi {\bar{\kappa }}}{{k}_{B}T}+1+\log \phi .$$

(11)

With Eqs. 3 and 4, Eq. 11 can be used to fit the theory to experimental data

$${R}^{*} ={\left(\frac{3}{4\pi }\right)}^{\frac{1}{3}}\exp \left[\frac{8\pi \bar{\kappa }}{3{k}_{B}T}+\frac{1}{3}\right]{v}^{\frac{2}{3}}\cdot {C}^{\frac{1}{3}}.$$

(12)

In Fig. 2A, we show ${R}^{* }$, which was obtained from FUS NTA measurements (Fig. 4A of ref. ⁴) versus the FUS concentration $C$ (Fig. 2A, open circles). The fit (with one fitting parameter) agrees well with the theory. The inset shows the consistency with the power law ${N}^{* } \sim {{R}^{* }}^{3}\sim C \sim \phi$ (Eq. 11). We note that the largest deviation from the theoretical fit is at 0.125 μM, where the width of the distribution of cluster sizes is maximal, and our approximations are not accurate. From the fitting parameter (the slope in Eq. 12, ${\left(\frac{3}{4\pi }\right)}^{\frac{1}{3}}\exp \left[\frac{8\pi \bar{\kappa }}{3{k}_{B}T}+\frac{1}{3}\right]{v}^{\frac{2}{3}}$), we found that $8\pi \bar{\kappa }$ is roughly 16 k_BT for FUS.

In addition to FUS, α-synuclein, and Ddx4n1 proteins were recently reported to form clusters whose size strongly depends on the protein concentration and contain only a small fraction of the total protein in the system¹³. This behavior is indicative of a system below the CAC and aligns with the scaling relationship presented in Eq. 11, ${N}^{* } \sim \phi$. We fitted the mass of these clusters (${M}_{cluster}$, measured using mass photometry¹³), proportional to ${N}^{* }$, to the linear relation obtained from Eq. 11

$${N}^{* }=\frac{{M}_{{cluster}}}{{M}_{{mono}}}=\exp \left[\frac{8\pi \bar{\kappa }}{{k}_{B}T}+1\right]\cdot v\cdot C.$$

(13)

Here ${M}_{{mono}}$ is the monomer mass, 14.46 kDa for α-synuclein and 25 kDa for Ddx4n1. The results are presented in Fig. 2C for Ddx4n1 and 2D for α-synuclein at 5 and 20% PEG concentrations and show a good agreement with the model. From the slope ($\exp \left[\frac{8\pi \bar{\kappa }}{{k}_{B}T}+1\right]$, 4.63 ∙ 10⁴ for Ddx4n1, 1.72 ∙ 10⁴ for α-synuclein at 20% PEG, and 8.59 ∙ 10⁴ at 5% PEG) and the estimated volume of the protein monomers (48 nm³ for Ddx4n1 and 22 nm³ for α-synuclein), we find that the contribution of bending energy to the formation energy of these clusters ($8\pi \bar{\kappa }$) is 9.7 k_BT for Ddx4n1, 10.4 k_BT for α-synuclein with 20% PEG, and 8.8 k_BT at 5% PEG. These are comparable to the bending energy of FUS.

We note that the number of monomers in the Ddx4n1 and α-synuclein clusters (a few dozen, Fig. 2C, D) is significantly lower than those in FUS or CPEB4 clusters (10²–10³ monomers). This difference makes the continuum core-shell model presented here less applicable and could explain the observed deviation from linear behavior for small clusters. Nevertheless, our results indicate that these clusters behave as swollen micelles (microemulsions) below the CAC with a significant bending rigidity, supporting the suggestion by ref. ¹³ that they exhibit a micelle-like structure¹³.

In contrast to FUS, Ddx4n1 α-synuclein and other proteins of the FET family (presented in the Supplementary Note S2 and Supplementary Fig. S4 and Tables S5–S7), the value of ${R}^{* }$ for CPEB4 clusters increases slowly with protein concentration, indicative of a system above the CAC where most of the added protein forms additional clusters. In Supplementary Note S1, of section I. 6), we show that under such conditions and near the CAC, the most probable cluster radius varies with the protein concentration as (Eq. S65):

$$\frac{{R}_{{CAC}}^{3}}{{{R}^{* }}^{3}}=1+\frac{1}{{f}_{{CAC}}}\left(1-\frac{C}{{C}_{{CAC}}}\right).$$

(14)

with ${C}_{{CAC}}$, ${R}_{{CAC}}$ and ${f}_{{CAC}}$ the concentration, the most probable cluster size, and the protein fraction in clusters at the CAC, respectively.

We fit this theoretical prediction to the size of CEPB4 clusters as a function of total protein concentration in Fig. 2B and found good agreement. Unfortunately, there are only three data points for CPEB4, so the details of the fit are not statistically significant. Nevertheless, we estimate the bending energy of these clusters by rearranging Eq. S55 (noting that ${\epsilon }_{T}=8\pi \bar{\kappa }$), correlating the concentration and radius at the CAC and the bending energy,

$$\frac{8\pi \bar{\kappa }}{k_BT}={{\mathrm{ln}}}\frac{4\pi {R}_{{CAC}}^{3}}{3{v}^{2}\cdot {f}_{{CAC}}\cdot {C}_{{CAC}}}-1.$$

(15)

We do not know the concentration, cluster radii, or their volume fractions at the CAC, but we can estimate them by considering the values measured at 10 μM (which is the closest measurement to CAC reported). With these values and assuming ${f}_{{CAC}}=0.9$ (90% of the protein was estimated to be in clusters¹¹), a cluster radius of 17 nm (Table S2), and a CPEB4 monomer volume of 125 nm³, the bending energy is ($8\pi \bar{\kappa }$) 11 k_BT. To compare, the bending energy per cluster of FUS was estimated above to be 16 k_BT, a significant 5 k_BT difference since it enters into the exponential that characterizes the distribution (Eq. 12).

We estimate the bending modulus of the different clusters by considering ${\kappa }_{B} \sim -{\kappa }_{G}$ as in typical amphiphilic layers²¹. With that, we find the bending modulus, ${\kappa }_{B}$, of FUS as 1.3 k_BT, CPEB4 is 0.9 k_BT, Ddx4n1 is 0.8 k_BT, α-synuclein at 20% PEG is 0.8 k_BT and 0.7 k_BT at 5% PEG. This is consistent with an experimental estimate for micron-scale LLPS domains in stress granules³¹. We do not expect ${\kappa }_{B}$ to be different for mesoscale or micron-scale clusters since it is a material property of the interface and is independent of the shape²¹.

Despite the approximate nature of these fits, the qualitative difference between CPEB4 and FUS (and the others) for the concentration dependence of the most probable aggregation number ${N}^{* }$ or radius ${R}^{* }$ is clear; CPEB4 shows a very weak dependence, characteristic of a self-assembling system above its CAC, while FUS and the others (measured by NTA or cluster mass photometry, both sensitive to the smaller clusters—in contrast to the FUS DLS data), show a strong dependence, indicative of systems below their CAC. This is striking and consistent with the experimental observation that most CPEB4 proteins are found in clusters (above the CAC), while most FUS proteins are dispersed in solution (below the CAC). These differences are primarily due to the 5 k_BT difference in their bending energies per cluster; the larger energy for FUS reduces the probability of the most probable cluster by a factor of ${e}^{-5}\, \sim \,0.007$ compared with CPEB4. This assumes that the contributions of the other energies per cluster are small compared with the bending energy contributions, which for both proteins are greater than 10 k_BT. This indeed is what we find in the following sections.

Analysis of the cluster size distribution to infer core and surface energies

Next, we analyze the cluster size distribution using our core-shell model (Eqs. 1–3) to infer the core energy, surface tension, and spontaneous curvature of the clusters. To eliminate terms that are independent of the cluster size, we normalize the cluster number distribution to its peak value at $P(R={R}^{* })$ that was derived in the previous section. With this normalization, the number distribution can be written as a function of only three parameters (full derivation in Supplementary Information of ref. ¹¹, where this approach was used to analyze the temperature dependence of size distribution)

$$\frac{P\left(R\right)}{P({R}^{* })}=\exp \left[-{A}_{3}\left(\frac{{R}^{3}}{{{R}^{* }}^{3}}-1\right)-{A}_{2}\left(\frac{{R}^{2}}{{{R}^{* }}^{2}}-1\right) -{A}_{1}\left(\frac{R}{{R}^{* }}-1\right)\right]$$

(16)

With

$$\begin{array}{ccc}(a){A}_{1}=-\frac{8\pi {{\kappa }_{B}J}_{s}{R}^{* }}{{k}_{B}T},\, & (b){A}_{2}=\frac{4\pi \gamma {{R}^{* }}^{2}}{{k}_{B}T},\, & (c)\,{A}_{3}=\frac{4\pi }{3}\frac{{{R}^{* }}^{3}}{v}\frac{\left({\epsilon }_{B}-{\mathrm{ln}}{\phi }_{m}\right)}{{k}_{B}T}.\,\end{array}$$

(17)

${A}_{1}$ is a measure of the spontaneous curvature, which we denote as “curvature tendency”, with positive values (${J}_{s} < 0$) signifying a bulkier hydrophobic blob and a smaller hydrophilic blob, and the opposite for negative values of ${A}_{1}$ (${J}_{s} > 0$)^22,32. ${A}_{2}$ is a measure of the residual interfacial tension energy that arises from less-closely packed regions in the ‘amphiphilic’ layer where there is hydrophobic (core)— hydrophilic (solvent) contact, and ${A}_{3}$ is a measure of the core energy, all normalized by the thermal energy k_BT. The bending rigidity $\bar{\kappa }$ calculated in the previous section cannot be inferred from these fits, since its contribution to the energy per cluster is independent of $R$ and is eliminated by the normalization.

We find ${A}_{1}$, ${A}_{2}$, and ${A}_{3}$ by performing a least squares minimization of ${\mathrm{ln}}\left[\frac{P\left(R\right)}{P\left({R}^{* }\right)}\right]$ since its linearity with the parameters simplifies the procedure:

$${\chi }_{\min }^{2}= \min \mathop{\sum }\limits_{i=1}^{i=M}{\left[{{\mathrm{ln}}}\left(\frac{P\left({R}_{i}\right)}{P({R}^{* })}\right) +{A}_{3}\left(\frac{{R}_{i}^{3}}{{{R}^{* }}^{3}}-1\right) +{A}_{2}\left(\frac{{R}_{i}^{2}}{{{R}^{* }}^{2}}-1\right) +{A}_{1}\left(\frac{{R}_{i}}{{R}^{* }}-1\right)\right]}^{2}.$$

(18)

Here, $M$ is the number of DLS data points, ${R}_{i}$ is the cluster radius, and $P({R}_{i})$ is the cluster number concentration of the radius ${R}_{i}$.

We fit the experimental data to the theoretical model exclusively in the region where the normalized cluster size distribution satisfies $R > {R}^{* }$. This regime was chosen because larger clusters exhibit greater scattering cross-sections, resulting in a more accurate measurement in the DLS data³³. Additionally, our mesoscopic theory is more applicable to the regime of larger radii, since we use continuum concepts such as interfacial tension and curvature energy, which assume that the cluster radius is much larger than the molecular size. Specifically, CPEB4 proteins have a typical diameter of 3–5 nm, while the cluster radius ${R}^{* }$ is ~20 nm (Table S2). Consequently, clusters with $R < {R}^{* }$ have a radius of curvature comparable to the protein diameter and are not adequately described by our model. The limitations of fitting at $R < {R}^{* }$ range and a representative example that includes the fit for smaller cluster radii is discussed in more detail in Supplementary Note S3 and Supplementary Fig. S5 in Sec. IV of the SI.

The parameter values obtained from the fits presented in Fig. 3 (and complementary Fig. S2) are found in Tables S2, S3 for the DLS number distribution of CPEB4 and FUS clusters (sensitive to the larger clusters) and Table S4 for the NTA measurements in FUS (sensitive to the smaller clusters). Due to the small number of data points and the experimental noise, the fitting results reported here should be considered indicative only of the trend with increasing protein concentration.

Fig. 3: Normalized dynamic light scattering (DLS) number distribution.

Open circles—experimental data, dashed lines—theoretical fit to Eq. 16. The protein concentration (in Molar) is indicated in the legends. A CPEB4 at 277 K and 100 mM NaCl. Data taken from ref. ¹¹. The fitting result is summarized in Table S2, and a linear-linear plot is presented in Fig. S2A. B FUS at 298 °K, 20 mM Tris, pH 7.4, and 100 mM KCl. Data taken from Kar et al. (2022). The result of the fitting is summarized in Table S3. A linear-linear plot is presented in Fig. S2B.

Analysis of the clusters’ size distributions of other proteins of the FET family (EWSR1, TAF15, and hnRNPA3), which are also well-described by our model, are presented in Supplementary Note S2 and Supplementary Fig. S4 and Supplementary Tables S5–S7 of Sec. III of the SI.

Except for points $R\approx {R}^{* }$, the CPEB4 data for the log of the normalized distribution shown in Fig. 3A is quite linear in the cluster radius, which signifies relatively small core (${A}_{3}$) and interfacial tension (${A}_{2}$) energies compared with the curvature tendency contribution (${A}_{1}$), whose contribution to ${\mathrm{ln}}\left(P\left(R\right)/P({R}^{* })\right)$ is linear in $R$. A similar trend is observed in the temperature dependence of CPEB4, as shown in Fig. 9C of ref. ¹¹. In other words, the CPEB4 cluster size distribution is dominated by the curvature tendency of the proteins in the shell. Large clusters have a decreased probability due to their actual curvature, whose sign is opposite to the preferred curvature of the shell proteins (${A}_{1} > 0$).

The numerical fits support this qualitative observation: CPEB4 clusters have negligible core energy (Table S2). The interfacial tension energy is small (~0.3 k_BT corresponding to interfacial tension of 0.2–0.3 μN/m), and the curvature tendency term is the dominant term, fitted to 1.9–2.1 k_BT. The cluster properties are almost unchanged even as the concentration is increased by an order of magnitude (from 10 to 100 μM), consistent with a system above the CAC.

Using the estimated bending rigidity of CPEB4 clusters from the previous section (1.1 k_BT) and the fitted values of ${A}_{1}$ (Table S2), we estimate the spontaneous curvature, ${J}_{s}$ (Eq. 17A), of CPEB4 proteins on the shell in the range of −4 to −3 μm⁻¹. This is about an order of magnitude smaller than a previous estimate based solely on the relative sizes of the hydrophobic and hydrophilic blobs of CPEB4¹¹, indicating that the blobs are highly deformable and soft.

In contrast to CPEB4, the DLS data for FUS suggest that ${\mathrm{ln}}\left(P\left(R\right)/P({R}^{* })\right)$ of large clusters at high protein concentration is dominated by parabolic and cubic terms in the cluster radius, while ${\mathrm{ln}}\left(P\left(R\right)/P({R}^{* })\right)$ for the small clusters measured by NTA are linear, as CPEB4. This indicates that the larger clusters have non-vanishing interfacial tension (${A}_{2}$) and\or core energy (${A}_{3}$), while the smaller ones are dominated by their curvature tendency (${A}_{1}$). This suggests that the large clusters, particularly those close to LLPS (occurring at about 3 μM⁴), are fluctuations and rather unstable, characteristic of a system below the CAC, and consistent with the experiments and theory for the concentration dependence of ${R}^{* }$ in the previous section.

We found, based on our fits to FUS DLS data (Fig. 3B and Table S3), that ${A}_{1}$ decreases from 0.76 k_BT (1/3 of the value for CPEB4) at 0.25 μM to vanishing values at 0.7 μM. This is reasonable since one cannot extract from the distribution of such large clusters any tendency to bend on significantly smaller scales. The core and surface tension energies, which are hard to numerically distinguish due to the limited number of available data points (especially for the data sets at 2 and 3 μM), increase from vanishing values at 0.25 μM to 1.2 k_BT at 3 μM. The tension and core energies dominate the distribution of large clusters as they scale as ${R}^{2}$ and ${R}^{3}$, respectively (${R}^{* }$ is 49 nm at 0.25 μM and 381 nm at 3 μM). The fits to NTA measurements of FUS (Fig. 4 and Table S4), which are most sensitive to the radii of small clusters, indicate that the core and tension energies are approximately zero. The small FUS clusters are dominated by curvature tendency (2 k_BT at 0.25 μM decreasing to 0.8 k_BT at 2 μM), similar to the CPEB4 clusters.

Fig. 4: Normalized FUS nanoparticle tracking analysis (NTA) number distribution.

Experimental data taken from ref. ⁴. Open circles—experimental data, dashed lines—theoretical fit to Eq. 16. The protein concentration is indicated in the legends. The transition to LLPS occurs at a concentration of ~3 µM. The result of the fitting is summarized in Table S4. A linear-linear plot is presented in Fig. S2C. Measurement was done at 298 °K, 20 mM Tris, pH 7.4, and 100 mM KCl.

Based on these results, we estimate the FUS clusters’ interfacial tension is in the 10⁻³–10⁻² μN/m range (Tables S3, S4, for large and small clusters, respectively), indicating a tight packing of the ‘amphiphiles’ configuration on the clusters’ surface. These tension values are an order of magnitude smaller than those of CPEB4 (~10⁻¹ μN/m, Table S2) and are the likely reason that allows the cluster size to increase as proteins are added to the system, since increasing the surface area involves only a negligible free energy cost. Based on the bending rigidity of FUS clusters estimated in the previous section (1.3 k_BT) and the fits to the NTA data (Table S4) that represent the curvature tendency of the clusters more accurately than DLS, we estimate the spontaneous curvature, ${J}_{s}$, of the FUS proteins to be between −1.7 μm⁻¹ to −2.3 μm⁻¹, similar in magnitude to that of CPEB4.

Finally, we calculate the solubility energy, ${\epsilon }_{B}$, which is the energy difference between a protein molecule in the core of the cluster relative to the aqueous solution, based on the fitted parameter ${A}_{3}\,$ (Tables S3, S4) and the monomer volume fraction, ${\phi }_{m}$. Kar et al. estimated that 0.15% of the proteins are in clusters at 0.25 μM⁴. The fit of our model to the DLS and NTA data showed that at this protein concentration ${A}_{3}\ll {k}_{B}T$, meaning that ${\epsilon }_{B}\cong k_{B} T \; {\mathrm{ln}}{\phi }_{m}$ (Eq. 17C). The protein volume fraction, ${\phi }_{m}$, is estimated to be 2.7 ∙ 10⁻⁵, and the solubility energy we estimate is thus −10.5 k_BT. To compare, the CPEB4 solubility energy was estimated as −7.5 k_BT¹¹. We note that this is an energy per protein molecule and thus coarse-grains over all the amino acids; converting it to an energy per amino acid gives only a fraction of k_BT.

To conclude, the normalized size distribution of the FUS clusters changes strongly with increasing protein concentration, shifting the distribution from being dominated by the curvature tendency at low concentrations (small clusters) to being dominated by core and\or tension energies at high concentrations (large clusters). CPEB4 clusters are almost unchanged even as the protein concentration increases by an order of magnitude, and are always dominated by their curvature tendencies. The behavior of FUS is similar to that of a microemulsion below the CAC, while that of CPEB4 is similar to a swollen micelle (microemulsion) above the CAC.

Discussion

The analysis of the measurements of the most common cluster size ${R}^{* }$ as a function of protein concentration as well as the fits to the cluster size distributions indicate that FUS and CPEB4 represent two different paradigms for cluster formation in the one-phase regime: below and above the CAC, respectively. This qualitative difference is due to the difference in surface properties of these clusters, which we inferred from various experiments in this work based on the theoretical model we presented here.

The protein fraction in clusters is mostly determined by the solubility energy (Eq. 2), while the cluster size distribution is determined by the surface (shell) energy (Eq. 1). The ${R}^{* }$ size scaling obtained from FUS NTA, CPEB4 DLS, and mass photometry of Ddx4n1, and α-synuclein measurement (Fig. 2) showed that the bending energy is 10–16 k_BT per cluster. These measurements (in contrast to the FUS DLS) are appropriate for relatively small clusters. The bending rigidity modulus estimated here is of the order of 1 k_BT for all these types of clusters, which is comparable to bending rigidities measured for other condensates³¹. The size distribution analysis around ${R}^{* }$ for FUS and CPEB4 clusters showed that all the other terms (${A}_{1}$, ${A}_{2}$ and ${A}_{3}$, Eq. 17) are order 1–2 k_BT far from LLPS (Tables S2–S4). This means that the surface energy of small clusters is dominated by their bending rigidities. The ultra-low tensions we report here (Tables S2–S7) are in agreement with measurements on condensates^31,34,35.

We note that the scaling of ${R}^{* }$ presented in Eqs. 11–15 holds only if the energy per cluster is constant or weakly changing. The large FUS clusters measured using DLS (Table S3), particularly close to LLPS, do not meet this condition as the tension and\or core energies significantly change with cluster size. Therefore, we estimated the bending energy using NTA measurement of FUS clusters (Fig. 2), which excludes the very large clusters that do not fill this requirement. Our estimation of bending rigidity modulus is also valid for large clusters since the bending energy is independent of the size.

The formation energy (sum of all the energies of the model given by Eq. 5) of small, FUS clusters (far from LLPS) is 17 k_BT, while CPEB4 clusters are less costly and are roughly 13 k_BT. This energy difference of ~4 k_BT per cluster is mostly due to the bending energy ($8\pi \bar{\kappa }$) and accounts for the relative fraction of proteins in clusters, with a ratio of ~90 (~1% for FUS and ~90% for CPEB4), of the same order as the relative Boltzmann factors of ${e}^{4}\approx 55$. In contrast to the total formation energy (including also the surface), FUS clusters’ solubility energy per protein molecule, ${\epsilon }_{B}$, is 3 k_BT lower compared to CPEB4.

The higher bending energy of FUS clusters (16 k_BT compared to 11 k_BT in CPEB4) indicates a tighter packing and more aligned ordering of the proteins in the ‘amphiphilic configuration’ (Fig. 1), reducing the interfacial tension from the exposure of hydrophobic amino acids in the core to the water. The size distribution analysis around ${R}^{* }$ (Figs. 3, 4) obtained by DLS and NTA measurements supports this prediction: These showed that FUS clusters’ surface tension (Tables S3, S4) is an order of magnitude lower than that of CPEB4 (Table S2). The ultra-low tension (~100 nN/m, 5 orders of magnitude smaller than the water-oil tension of ~10 mN/m) and relatively high bending energy of FUS clusters are consistent with a system below the CAC: clusters are rare due to their high formation energy, but grow with little free energy penalty, meaning they exist as fluctuations. In contrast, CPEB4 clusters have a lower bending energy and higher interfacial tensions, so these favor the formation of more clusters over the growth of existing ones. This is consistent with relatively stable clusters, as in a system above its CAC.

Further, our analysis suggests that the energy of a molecule in a cluster relative to the aqueous solution is less than k_BT per molecule (clusters contain 10²–10³ molecules). This is what might be expected for a non-folded, disordered protein, as typical amino acid level interactions are only several k_BT³⁶. Of course, the stickers discussed in ref. ⁴ most probably coarse-grained over several amino acids, resulting in an effectively large interaction. We emphasize that the goal of our model is not to determine the interactions at the amino acid level, but rather to account for the mesoscale properties of the disordered protein as a whole in a coarse-grained model, and we find those energies to be of the order of k_BT for the entire cluster.

We note that the spontaneous curvature of both cluster types considered here is negative. That is, the proteins’ preferred packing is opposite to the actual packing in the cluster shell. With the vanishingly low interfacial tension, this energy cost per cluster suppresses the formation of larger clusters in the distribution. It is tempting to speculate that the relative packing of the hydrophobic and hydrophilic blobs of certain intrinsically disordered proteins is the major factor determining the ability to form clusters that precede LLPS. While the proteins in the shell have an amphiphilic conformation, they must have a negative spontaneous curvature, which limits clusters larger than a certain size.

We speculate that FUS and CPEB4 may follow distinct pathways to LLPS, characterized by different internal structures within the micron-scale LLPS domains (Fig. 5). FUS clusters grow in size with total protein concentration (Table S3). At the saturation concentration of 3 μM, there is a coexistence of micron-size (LLPS) domains, clusters, and a very large fraction of monomers (Fig. 8 of ref. ⁴). The LLPS domains grow via cluster coalescence and become larger as a function of time. This is consistent with the theoretical fits presented here to the DLS data (Table S3): These large clusters, which have costly core and/or surface energies, tend to undergo fusion of their cores to reduce those energies (Fig. 5A). LLPS micron-sized domains consist of fused clusters, which essentially contain only the cluster core. This pathway to LLPS via cluster coalescence is also supported by recent work on a FUS-MBP system showing that FUS-MBP LLPS occurs through the coalescence of FUS clusters¹².

Fig. 5: Liquid-liquid phase separation (LLPS) pathways.

A Coalescence LLPS: The merger of clusters into a micron-scale LLPS domain occurs via fusion. LLPS occurs at a critical cluster radius, where the interfacial tension energy exceeds the cluster’s translational entropy. In addition, the monomers in solution must change from a hydrophilic to a hydrophobic dominant conformation. B Aggregation LLPS: The formation of a large cluster is mediated by an attractive interaction between the intrinsically disordered proteins of the proteins at the amphiphilic configuration. Here, the transition occurs at a critical cluster concentration and depends strongly on the length and interaction energy of the intrinsically disordered proteins.

It is also possible, in principle, that in some systems, the pathway to LLPS is dominated by a drastic change in the dominant conformation of the protein monomers in solution from hydrophilic to hydrophobic without any significant intermediate clusters. Both pathways have the same final configuration—a large macroscopic LLPS domain with most proteins in hydrophobic configurations. A more comprehensive determination of the pathway to LLPS for a broader range of temperatures and concentrations would shed further light on the dominance of cluster coalescence versus monomer associations in FUS and related systems.

In contrast to FUS, CPEB4 clusters grow very slowly with increasing protein concentration, with negligible cluster tension energy even close to LLPS. This system is also significantly more concentrated, with the CPEB4 concentration in the cluster regime (that precedes LLPS) being two orders of magnitude larger than that of FUS and the other proteins in the FET family (100 μM for CPEB4 versus 3 μM for FUS). In addition, EPR experiments showed that the fractions of proteins distributed among the core, shell, and dispersed monomers do not change abruptly at LLPS^5,11, suggesting that the clusters remain intact during the transition. We, therefore, propose that for the paradigm of CPEB4 (above their CAC), LLPS will occur via aggregation (in contrast with fusion) of existing clusters at a critical concentration ${C}_{{sat}}$ (Fig. 5B). A similar process was proposed to occur when the concentration of CPEB4 was held constant, but the temperature changed¹¹. However, this hypothesis remains to be experimentally validated through more in-depth structural analysis of macroscopic CPEB4 condensates.

It is also possible that the protein configuration at the core of the clusters we consider here is not the ‘true’ free energy minimum and that the cluster core ‘age’, as seen in macroscopic FUS LLPS domains³⁷. Some evidence for this came from Cabau and colleagues²⁷ who reported a progressive increase in CPEB4 cluster size from 55 to 90 nm over 15 h, accompanied by a rise in the polydispersity index. They proposed that these multimers evolve into mesoscopic condensates, comparable to those observed by optical microscopy but significantly smaller. However, due to the limitations in physically characterizing these structures, the authors classified them as distinct species. We believe that the identity and nature of these evolving species remain an open question and should be investigated in greater detail before applying our theoretical framework to them.

To conclude, we have presented a theoretical model to describe the size distribution of protein clusters that precede LLPS and used it to analyze new and existing experimental data in vitro. We propose that a similar approach could be used to model the behavior of condensates in vivo as well as other condensate geometries, such as recently observed fibrils in certain FUS clusters²⁸ and Ebola-induced condensates^38,39. Direct measurement and modeling of the protein at the amino acid level, which requires other techniques, are certainly of interest to verify our mesoscale approach.

The main novelty of our proposed theory lies in the extraction of the ultra-low tension of the clusters from their size distribution, as well as the cluster bending rigidity and spontaneous curvature, all of which are attributed to the amphiphilic conformation of the protein at the interface of the cluster core and the solvent. These properties should be considered when analyzing and modeling membrane-less organelles’ equilibrium shapes and kinetics, such as fusion, fission, and shaping processes.