A scale-invariant log-normal droplet size distribution below the transition concentration for protein phase separation

Tommaso Amico; Samuel Dada; Andrea Lazzari; Antonio Trovato; Michele Vendruscolo; Monika Fuxreiter; Amos Maritan

doi:10.7554/eLife.94214.1

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In this useful study, the authors analyze droplet size distributions of multiple protein condensates and their fit to a scaling ansatz, highlighting that they exhibit features of first- and second-order phase transitions. The experimental evidence is still incomplete as the measurements were apparently done only at one time point, neglecting the possibility that droplet size distribution can evolve with time. The text would benefit from a connection to and contextualization with the well-understood expectations from the coupling of percolation and phase separation in protein condensates - a phenomenon that is increasingly gaining consensus amongst the community and that emphasizes "liquid-gas" criticality.

https://doi.org/10.7554/eLife.94214.1.sa2

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Abstract

Many proteins have been recently shown to undergo a process of phase separation that leads to the formation of biomolecular condensates. Intriguingly, it has been observed that some of these proteins form dense droplets of sizeable dimensions already below the transition concentration, which is the concentration at which phase separation occurs. To understand this phenomenon, which is not readily compatible with classical nucleation theory, we investigated the properties of the droplet size distributions as a function of protein concentration. We found that these distributions can be described by a scale-invariant log-normal function with an average that increases progressively as the concentration approaches the transition concentration from below. These results suggest the existence of a universal behaviour independent of the sequences and structures of the proteins undergoing phase separation, which is typically observed for second-order phase transitions. Based on these observations, we show that it is possible to use the scale invariance to estimate the critical concentration for phase separation.

Introduction

Many proteins have been shown to undergo a phase separation process into a liquid-like condensed state^1-5. This process appears to be of physiological significance since it may lead to the formation of biomolecular condensates^1-7. As a consequence, it is closely controlled by the protein homeostasis system^8,9, and its dysregulation has been associated with a broad range of human diseases^10,11. It is therefore important to understand the fundamental nature of this process^5,12,13 to provide insights for the identification of ways of modulating it through pharmacological interventions.

Several models have been proposed to explain the phenomenon of protein phase separation. A commonly adopted theoretical framework is the Flory-Huggins theory of phase separation^14-16. In its simplest form, this theory describes a first-order phase separation in a system of homopolymers, which can also be adapted to polyampholites¹⁷. The Flory-Huggins theory has been extended to associative polymers, with the aim of modelling the sequence composition of proteins, by Flory and Stockmayer, who described the phase separation in terms of a thirdorder gelation process^18,19. Semenov and Rubinstein modified the Flory-Stockmayer theory, reporting that the gelation process is not a phase transition, as all the derivatives of the free energy are analytical at the gelation point²⁰. It was also argued that another generalization of the Flory-Stockmayer theory²¹ could describe phase transitions in cytoskeletal networks²². More recently, it has been suggested that the protein phase separation process is coupled with percolation²³. In an older study, γ-crystallin was reported to undergo a second-order phase separation process²⁴, consistently with observations in lysozyme solutions²⁵.

To understand better the nature of the transition underlying protein phase separation on the basis of recent experimental observations^26,27, here we study the distribution of the size of the droplets below the value of the concentration at which the transition occurs, which here is referred to as the transition concentration, ρ_t. This question appears as a promising starting point to develop new insights since it has been reported that proteins can form droplets of sizeable dimensions already well below the concentration at which phase separation occurs^26,27. This behaviour is not predicted by classical nucleation theory²⁸, and not readily consistent with the idea that the protein phase separation process can be described as a firstorder phase transition. This is because, in a first-order phase transition, nucleation takes place in a supersaturated system²⁹, while in a subsaturated system particles can still self-assemble, but with a probability that decreases exponentially with the size of the assemblies.

In the present study, we report the observation that the distributions of the sizes of the droplets of the proteins FUS and α-synuclein follow a scale-invariant log-normal distribution. These findings are consistent with a universal behaviour resulting from the presence of an increasingly large correlation length, ξ, as the concentration approaches the transition concentration from below. The correlation length is an emergent characteristic, and it is related to the typical spatial range over which density fluctuations are correlated. When ξ is sufficiently large, one can expect scale invariance and finite-size scaling³⁰ to occur within a range of lengths spanning from the molecular size to ξ. This means that physical observables do not depend explicitly on many microscopic details characterizing spatial scales smaller than ξ, thus leading to a universal behaviour characterized by quantities obtained by coarse graining over scales smaller than ξ.

At a first-order phase transition, ξ is finite, and if it is not large enough, the length range discussed above remains too short to observe scale invariance. However, the vicinity of the spinodal line (where nucleation disappears as the dilute phase becomes unstable) to the coexistence curve (where nucleation appears as the dilute phase becomes metastable) might cause a large increase of ξ as the first-order phase transition is approached by increasing the concentration. If it is not preceded by a first-order phase transition, the spinodal line would correspond to a second-order phase transition, resulting in infinite ξ, and thus scale invariance holds on all length scales larger than the molecular scale.

As an application of this observation, we address the question of whether scale invariance holds for droplet size distributions near the coexistence curve. As a practical consequence, we use this observation to propose a procedure to overcome the challenge of estimating the critical concentration, ρ_c. Such challenge arises from the fact that close to the critical concentration, the timescale required for the equilibration of a system grows together with ξ, thus exceeding the timescale amenable to experimental observation. If a second order-phase transition is present, then ρ_c is the concentration where the transition occurs (with ρ_c = ρt), whereas in the case of a first-order phase transition, ρ_c is where the spinodal occurs (with ρ_c > ρt).

Our analysis of experimental data indicates that: (i) scale invariance does indeed hold near the coexistence curve, and (ii) the droplet size distribution is log-normal. Based on the properties of the scale-invariant log-normal distribution of droplet sizes, we investigate a correlation between the moments of the distribution and the distance from ρ_c.

Finally, we note that methods to assess the critical concentration are crucial for understanding the location of proteins in their phase diagram, their proximity to the phase boundary between the native and droplet states, and how pharmacological interventions can modify their phase behaviour. To address this problem, we report how the moments of the distribution can be used as a scale-invariant gauge to estimate the critical concentration. In this way, an accurate estimate of the critical concentration is possible because it is based on measurements carried out away from the critical point, under conditions such that fluctuations are small, and hence experimental errors are smaller than in the proximity of the transition.

Results

Formulation of the scaling ansatz

Empirical evidence indicates that protein self-assembly into liquid-like condensates is characterized by: (i) a phase separation transition at a concentration ρt, (ii) a formation of droplets of sizeable dimensions already below ρt, and (iii) a droplet size distribution that, after an initial transient, does not change with the experimental observation time, although individual droplets can form, grow, shrink and dissolve. An initial analysis of droplet size distributions observed experimentally led us to ask whether the region near the transition could be described in terms of a scaling theory, as commonly done for critical phenomena (29), as summarized above. We also note that this approach is analogous to analyzing the cluster size distribution in a percolation problem³¹.

In our analysis, we called P>(s|ρ) the survival distribution function (SDF = 1-CDF, where CDF is the cumulative distribution function) corresponding to the probability to observe a droplet of size greater than s, when the concentration is ρ. The distance from the critical concentration ρ_c is measured in terms of the dimensionless variable , which also allows to compare data from different experiments, as explained below. P>(s|ρ) in general depends separately on s, ρ and many other parameters characterizing the process, including temperature, with ρ_c and ρt approaching each other as the temperature approaches the critical temperature from below. However, if scale invariance holds in the vicinity of ρ_c, i.e. when is small and the correlation length of the system is large enough, we would expect P>(s|ρ) to depend on ρ and on other details pertaining to the microscopic scales only through the characteristic droplet size, sc. The characteristic size sc is defined, apart from a proportionality constant (see below), as the ratio of the second to the first moment of the droplet size distribution. This leads us to formulate the following scaling ansatz³¹

where, sc depends on ρ, and it is expected to diverge at the critical concentration as^30,31

where α ≥ 0 and φ > 0 are critical exponents, a is a constant and f is the so-called scaling function³⁰.

Thus, the scaling of the SDF is equivalent to saying that, apart from the singular behaviour s^−α, the remaining s and ρ dependence occurs only through the ratio s/s_c. All extra dependencies are encapsulated in s_c through the constant a, ρ_c and, possibly, on the specific form of the scaling function f. A consequence of scaling and singular behavior described by Eq. (2), i.e. the divergence of the characteristic size of the droplets, is that details of any specific system may not affect the value of the critical exponents, which are therefore expected to be universal, i.e. independent of specific details of the system.

To determine the exponents α and φ, we introduce the moments of P>(s|ρ)

where the scaling ansatz Eq. (1) has been used in the last step and ck is given by

which depends on the function f but is independent of .From Eq. (3), we deduce that

If the scaling ansatz of Eq. (1) is correct, by plotting the ratio of moments ⟨s^k+1⟩/⟨s^k⟩ for various values of k as a function of in a log-log scale, we should obtain straight parallel lines with slope φ and intercept -ln(ack+1/ck).

Scaling behaviour of the droplet size distributions of FUS

We investigated the validity the ansatz in Eq. (2) using experimental data on the RNA-binding protein FUS, which are available for both the untagged and the SNAP-tagged protein²⁷ (Table S1). We calculated the survival distribution function, Eq. (2), and its moments, Eq. (3). We then plotted the moments versus the inverse distance from the critical concentration in log-log scale (Figure 1). We used the estimates of the critical concentrations, ρ_c = 5.0 μM for FUS and ρ_c = 5.4 μM for SNAP-tagged FUS, obtained below, in a self-consistency check of the validity of the scaling ansatz. We observed that the moment ratios at different distances from the critical concentration fall onto straight lines, as predicted by the scaling ansatz. In addition, the weighted average slope (see Eqs. (15) and (16) below) of the lines for different moment ratios is 0.95 ± 0.05 for untagged FUS, and 0.95 ± 0.05 for SNAP-tagged FUS, which is in good agreement with φ=1 for the exponent in the scaling ansatz in Eq. (1) (Figure 1A,C).

Determination of the critical exponents for FUS of the scaling invariance.
**(A,C)**Determination of the exponent φ for FUS (A) and SNAP-tagged FUS (C). The ratios of the average moments of the droplet sizes (<s^k+1>/<s^k>, at k=0.5, 1, 1.5, 2, Eq. (4)) are represented at various distances from the critical concentration . The exponent φ for each value of k was determined by error weighted linear regressions. The exponent and its error were determined as mean and standard deviation of the three independent measurements (Eqs. (15) and (16)). Error bars are shown in inset for graphical clarity. (**B,D)** Determination of the exponent α for FUS (B) and SNAP-tagged FUS (D). The mean of the droplet size distributions is plotted at various distances from the critical concentration . The value of the exponent α was determined by error weighted linear regression (Eq. (6)), using 𝜑 =1, where the errors were standard deviations of the three independent measurements (Eq. 16). Error bars, which were obtained as the standard deviation of the three independent measurements are shown in inset for graphical clarity. Error weighted linear regressions are performed in both cases excluding the data point at the lowest concentration ρ = 0.125 μM. The fit corresponding to the scaling ansatz, compatible with φ=1 and α=0, is represented by a dashed gray line with a slope of 1.

Having determined the exponent φ, we can also determine the exponent α using Eq. (3)

The exponent α is then calculated from

where m is the slope of the linear fit of the log-log plot of <s> (the first moment, k=1) vs . We then plotted the mean droplet size, <s>, versus the distance from the critical concentration on a natural logarithm scale, which could be fitted using a line with a slope m= 0.99 ± 0.05 for untagged FUS, and 0.93 ± 0.07 for SNAP-tagged FUS (Figure 1B,D), which is consistent with m=1. Using the value of φ = 1, determined based on Eq. (4), we obtain α = 0.

Taken together, these data support the validity of the scaling ansatz of Eq. (1).

The droplet size distribution of FUS is log-normal

The above analysis suggests that the droplet size distribution may follow a log-normal distribution. The scaling ansatz of Eq. (1) for the SDF is equivalent to the following scaling for the probability density distribution

where f, the scaling function in Eq. (1), and F are related as follows

The log-normal droplet size distribution P(s|ρ) is

with

being the characteristic droplet size distribution as defined above. Consequently, the size survival distribution function is

The values of s₀ and 0 can be determined as the average and the variance, of ln (s/u) obtained at each concentration:

where u is an arbitrary (and irrelevant) constant with the same units as s. In the following, when not stated, it is implicitly assumed that u=1 in the same units as s. The droplet sizes follow a log-normal distribution only if the survival distribution functions or, equivalently, the size distribution functions, multiplied by s, collapse when plotted versus Eq. (9) or Eq. (10) with the values of s0 and σ of each droplet size distributions obtained at different concentrations (Eq. (11)). We determined s₀ and σ values for each distribution (Figure 2A,B) and plotted the properly rescaled size distribution functions versus [ln(s/s0)]/0 (Figure 2C). We observed that the size distribution functions collapsed for both FUS and SNAP-tagged FUS (Figure 2C). Furthermore, the collapsed curve overlapped with the analytic log-normal distribution we computed with s0=1, σ=1, the normal distribution in the rescaled variables.

Log-normal behavior of FUS and SNAP-tagged FUS size distributions below the critical concentration.
(**A, B**)Variation of the size distribution with protein concentration: lns₀ (A) and σ (B). lns₀ and σ (inset) were computed for FUS (blue) and SNAP-tagged FUS (red) using Eq. (11). Error bars (in inset for graphical clarity) are estimated from three independent measurements. While droplet sizes increase with concentration, the width of distribution does not change considerably. **(C)** The collapse of the droplet size distribution functions is consistent with a log-normal behaviour. The droplet size distribution functions for both untagged FUS (blue) and SNAP-tagged FUS (red) are plotted after rescaling the sizes by the lns₀ and σ values, the first and second moment of the logarithm of the droplet size distribution, which are a function of the concentration. The rescaled curves for both the untagged and the tagged protein collapse to the normal distribution (gray dashed), as expected when the non-rescaled droplet sizes follow a log-normal distribution.

The observed collapse supports the observation that droplet size distributions from different experiments follows a log-normal behaviour.

Independence of the variance of the distribution from the concentration of FUS

The lognormal behaviour described above is consistent with the scale invariance underlying Eq. (1) if the variance of the log-normal distribution, σ, is independent of ρ or, equivalently, of ³². Indeed, comparing Eq. (1) with Eqs. (7), (9a,b) and (10) we obtain that the scaling invariance holds with

with α=0 and φ = 1 (Figure 1). Furthermore, the k-th moment of the log-normal distribution is

which, compared with the scaling prediction Eq. (3), is also consistent with α = 0, appropriate for the log-normal, and

which is independent of if σ is independent of , see Eq. (3).

This prediction is verified in Figure 2B, where the σ² values are shown to be nearly uniform at different concentrations, with the exception of the data at the lowest concentration values, i.e. those furthest away from the critical concentration. These results are consistent with the scaling ansatz in the vicinity of .

Our analysis does not exclude the possibility that σ might depend on specific experimental conditions, even though the data that we analysed are suggestive of at most a weak dependence. We note that such dependence, even if present, does not invalidate the scaling, as long as the exponents do not depend on the experimental conditions.

Estimation of the critical concentration of FUS using the scale invariance

The fact that the scaling ansatz is satisfied for different set of experiments opens a possibility to estimate the critical concentration. The scaling predicts (⟨S_k⟩)^-1/k vs ρ to be a straight line with a slope depending on k. It is important to note that this is the consequence only of the scaling ansatz and not of the log-normal distribution. The line should intersect the ρ-axis at the critical concentration providing an estimate of ρ_c.

We illustrate this process using the FUS data by plotting (⟨S_k⟩)^-1/k vs ρ. As expected, for different values of k we obtained a straight line fit of the points near ρ_c (Figure 3). Due to experimental uncertainties, the various lines, one for each value of k, lead to a slightly different estimate of ρ_c. The average of the estimated ρ_c values is 5.0 ± 0.2 μM for FUS and 5.4 ± 0.4

Estimation of the critical concentration of FUS using the scale invariance.
(**A,B)** Critical concentration of FUS (A) and SNAP-tagged FUS (B). The scaling model predicts that the function of the moments plotted versus the concentration ρ becomes a straight line near the critical concentration ρ_c and intersects the ρ-axis at ρ_c, independently of the value of k. Error weighted linear regressions are performed in both cases excluding the data point at the lowest concentration ρ = 0.125 μM. The resulting estimate of the critical concentration is shown in green along with the corresponding standard deviation, estimated from three independent measurements.

μM for SNAP-tagged FUS (Figure 3). The different ρ_c. values predicted based on the scaling ansatz using different k values enable the estimation of error of the predicted critical concentration (Figure 3). Both estimates are higher than the values of the transition concentration ρt originally reported from the plateau of the absorbance of a spin-down assay (ρt = 2 μM of untagged and ρt = 3 μM of tagged FUS²⁷), as expected, since ρt is always smaller than ρ_c. We also note that our estimates are compatible with other ones recently reported³³.

We then used both the values of ρ_c to probe the collapse of the size distribution functions (Figure 4). We observed that both in cases of untagged and tagged FUS, the SDFs collapsed with our estimated value of ρ_c (Figure 4). These results indicate that the scaling model can be used to estimate the critical concentration based on the distribution of droplet sizes.

Collapse of the droplet size distributions of FUS as predicted by the scale invariance.
If the scaling ansatz of Eq. (2) holds, the standard deviation 0 of the log-normal distribution should not depend on the distance from the critical concentration, and a collapse should be achieved by rescaling the size with the distance from the critical concentration. **(A,C)** Droplet size distributions derived from the experimental data of untagged FUS at 0.125, 0.25, 0.5, 1.0 and 2.0 μM concentrations (A) and SNAP-tagged FUS at 0.125, 0.25, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 μM concentrations (C), and their standard error of the mean from three independent measurements (26). (**B,D)** Collapse of the droplet size distributions rescaled by the estimated critical concentration. The error bars show the standard error of the mean from the three independent measurements (26).

Estimation of the critical concentration of α-synuclein using the scale invariance

As heterogeneous clusters of α-synuclein were recently reported below the critical concentration³⁴, we aimed at characterising whether such clusters also follow scale invariant size distribution. Using A90C α-synuclein labelled with Alexa Fluor 647, we monitored droplet formation as previously described³⁵. We measured droplet sizes at 5% PEG concentration 10 minutes after detection of liquid-like condensates using increasing concentrations of α-synuclein (20, 40, 50, 60, 75, 80 and 100 μM) (Table S2). Using k values of 0.25, 0.75, 1.25, 1.75, we determined the critical exponents in the scaling ansatz (Eqs. (4) and (6)), obtaining φ = 1.3 ± 0.2 and α = 0.9 ± 0.2. As control, we also determined the critical exponents using the scaling ansatz in a different way, using Eq. (19), obtaining φ = 1.1 ± 0.2 and α = 0.8 ± 0.2, corroborating the validity of the scale invariant model. We then estimated the critical concentration using the two methods obtaining ρ_c = 137 ± 10 μM, using Eq. (18) (Figure 5C) and ρ_c = 125 ± 7 μM, using Eq. (19) (Figure 5D).

Discussion

Although growing experimental evidence indicates the presence of protein condensates both in vitro and in vivo^1-5,36, the mechanisms responsible for the formation of these condensates are not fully understood. There is still the need of a general framework capable to fully account for all the experimental observations of protein phase separation. Here, to help build this framework, we aimed at describing phase separation as a universal phenomenon using a scaling ansatz.

With this aim, we analyzed a series of droplet size distributions of FUS and α-synuclein. We found that the droplet size distributions follow a log-normal behaviour and can be described by a scale-invariant model. Scale invariance means that the description of the behaviour of a system remains the same regardless of the scale of observation. It has been seen for example in self-similar systems like fractals, which repeat patterns at different length scales³⁷. Scale invariance is vital in physics, biology, and economics, helping understand complex systems by identifying consistent patterns and fundamental properties³⁸. The critical exponents of the scaling model are consistent for different systems. The compatibility of the droplet distributions with a scale-invariant model underscores its universality, whereby the properties of a broad range of systems remain unaffected by specific dynamical details. We also note the generality of the log-normal model for nucleation and grain size growth ranging from crystal seeding^39,40 to the mass size distributions in organism growth of various sea organisms³².

These observations may appear to be in contrast with the Flory-Huggins theory, which characterizes droplet formation as a consequence of nucleation processes within a metastable state of a supersaturated system. In such cases, however, when the correlation length exceeds the cellular dimensions, the remnants of what should have been a first-order phase transition in an infinite system may effectively exhibit indistinguishable characteristics from a secondorder phase transition. We also note that a power law distribution, but above the transition temperature, has recently been reported for nucleoli⁴¹.

Quite generally, the existence of scale invariance within the droplet size distribution, at least under the conditions investigated here, imposes stringent constraints on theoretical models that aim to elucidate protein phase separation. The elegance of a scaling analysis lies in its ability to uncover the fundamental aspects of universal phenomena, transcending models confined solely to specific systems for which they were originally designed.

As a practical consequence of the scaling model, we found that the moments of the droplet size distribution correlate with the critical concentration. This observation suggests that the analysis of the moments of the droplet size distribution can provide a quantitative gauge of the proximity to the phase boundary irrespective of the actual mechanism of phase transition.

Conclusions

We have addressed the problem of understanding the nature of the transition that leads to protein phase separation. By analysing the functional form of the droplet size distribution, we have reported a scale-invariant log-normal behaviour, which suggests that protein phase separation exhibits features typical of both second-order and first-order phase transitions.

As a prediction derived from this functional form of the droplet size distribution, we have presented a method of estimating the critical concentration for protein phase separation. This distribution can be determined in the test tube, cell cultures or human tissues. The droplet dimensions could be assessed in 1D (length), 2D (area) or 3D (volume).

We note that second-order phase transitions usually result from critical phenomena, and that these processes tend to be highly sensitive to environmental conditions. The modulation of the formation and dimensions of protein droplets therefore could be more tunable than what expected from a standard first-order phase transition. This feature would appear to be favourable for the control of the formation of biomolecular condensates by the protein homeostasis system.

Methods

Expression and purification of α-synuclein

Human wild-type α-synuclein and A90C cysteine variants were purified from Escherichia coli BL21 (DE3)-gold (Agilent Technologies) expressing plasmid pT7-7 encoding for α-synuclein as previously described^35,42,43. Following purification in 50 mM trisaminomethane-hydrochloride (Tris-HCL) at pH 7.4, α-synuclein was concentrated using Amicon Ultra-15 centrifugal filter units (Merck Millipore). The protein was subsequently labelled with a 1.5-fold molar excess of C5 maleimide-linked Alexa Fluor 647 (Invitrogen Life Technologies) overnight at 4 °C with constant mixing. The excess dye was removed using an Amicon Ultra-15 centrifugal filter unit and used immediately for phase separation experiments.

Determination of the droplet distribution of α-synuclein

The experimental conditions were determined from a previously described phase boundary³⁵. To induce liquid droplet/condensate formation, wild-type α-synuclein was mixed with an A90C variant labelled with Alexa 647 at a 100:1 molar ratio in 50 mM Tris-HCL and 5% polyethylene glycol 10,000 (PEG) (Thermo Fisher Scientific). The final mixture was pipetted onto a 35-mm glass-bottom dish (P35G-1.5-20-C; MatTek Life Sciences) and immediately imaged on Leica Stellaris Will inverted stage scanning confocal microscope using a 40×/1.3 HC PL Apo CS oil objective (Leica Microsystems) at room temperature. The excitation wavelength was 633 nm for all experiments. For liquid droplet size characterisation images were captured 10 minutes post-liquid droplet formation. All images were processed and analysed in ImageJ (NIH). Images were analysed by applying a threshold function in ImageJ that excluded the background of the image and identified the liquid droplets as having a circularity of 0.8-1.

Determination of the droplet distributions of FUS

We used previously published data on FUS droplets²⁷, where the fluorescence intensity of the FUS droplets was reported to be proportional to their diameters²⁷. The abundance of untagged FUS droplets formed at 0.125, 0.25, 0.5 and 1.0, 2.0 μM concentrations were measured by nanoparticle tracking analysis for 51 different droplet sizes with three repetitions for each measure²⁷. Similarly, the abundance of SNAP-tagged FUS droplets at 0.125, 0.25, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 μM concentrations were measured for 64 different sizes with 3 repetitions for each measure²⁷. For each data point, we determined the mean droplet abundance and the standard error of the mean from the three experimental replicates. The survival distribution function was computed as P_>x = N_>x/N_tot, where N_>x is the number of droplets above size x, and Ntot is the number of droplets. The values of x were chosen according to the original data²⁷.

Determination of the critical exponents for FUS

We computed the k-th moment of each droplet size, determined the average for each value of k and computed the ratios of the subsequent moments (Eq. (4)). We tested several k values from 0.25 to 3 in 0.25 increments, determined the moment ratios in each case. We selected four k values (k=0.5, k=1.0, k=1.5 and k=2.0) where the moment ratios at each concentration provided the best linear fit for both FUS and SNAP-tagged FUS. Then we generated a log-log plot of the moments and , the distance from the critical concentration (Figure 1). For each value of k, the points were estimated by taking the average of the computation done on the 3 independent measurements. The errors of the data points are estimated as the standard error of the mean. At each k value, we performed a linear regression between the points at each concentration²⁷ determining the slope, resulting in four 𝜑_k values, with an associated error from the weighted linear regression. Given the independent observations 𝜑_kwith variances 𝜎_k, the value of the exponent was obtained as the weighted average

The error on the exponent 𝜑 was computed as

The exponent α was determined based on the log-log plot of the mean of the droplet size distribution at various distances from the critical concentration . (Figure 1). We performed a linear regression weighted by the inverse of the variance of the data points corresponding, as previously, to the computed mean of the three independent experiments for each concentration (Eq. (6)) using 𝜑 =1. The error of the exponent α was computed using Eq. (16), where the variance referred to the k=0 moments.

At each concentration, from the distribution of ln(s) we determined s0 as <ln(s)> =ln(s0), and σ as the standard deviation of ln(s). At each concentration, we plotted the droplet size distribution function versus (Eq. (10)) using the s0 and σ values determined from the corresponding droplet size distribution at the given concentration. The log-normal behaviour is demonstrated by the resulting collapse (Figure 2). Furthermore, the collapsed curves overlapped with a theoretical log-normal curve computed with s0=1, σ = 1, which is the normal distribution in the rescaled log variables.

Determination of the critical concentration for FUS

The k-th moment of the log-normal distribution (Eq. (13)) and the scaling ansatz with α = 0 (Eq. (3)) gives

where , a is a proportionality constant independent of k, ln s₀ is the mean and σ is the standard deviation of the logarithm of the droplet probability size distribution. Therefore

plotted versus the concentration intersects the x-axis at the value of the critical concentration.

Determination of the critical exponents for α-synuclein

For α-synuclein, the critical exponents were determined in two different ways, the first using the same method as for FUS (Eq. (18)), and the second using

Where we constrained 𝒳 ρ to 1. Analogously to what happens in Figure 3, the lines plotted in Figure 5D intercept the x axis on the same point, which corresponds to the critical concentration.

Determination of the critical concentration of α-synuclein

As for the critical exponents, we determined the critical concentration of of α-synuclein using the scaling ansatz in two different ways, either using Eq. (18) or using Eq. (19). We plotted the moments of the droplet size (Eq. (3)) versus the concentration, using a range of values of k to cover majority of the data. We performed a linear regression weighted by the inverse variance, determining the intercept on the x-axis. For each value of k, we calculated the regression errors of both intercept and slopes. We estimated the critical concentration from each fit as the intercept calculated as , where q and m are the y-axis intercept and the slope retrieved by the fit, respectively, where the subscript i indicates each independent experiment. The estimated ρ_c was obtained as the mean of the different independent values ρ_c,i. The error on the estimate of ρ_c is obtained as the standard error of the mean of the ρ_c,i.

Acknowledgements

We acknowledge insightful discussions with Drs. Serena Carra, Jonathan Vinet, Alex Buell and Soumik Ray. This work was supported by AIRC IG 26229 (M.F.).

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Article and author information

Author information

Tommaso Amico
Department of Physics and Astronomy, University of Padova, Italy
Samuel Dada
Centre for Misfolding Diseases, Department of Chemistry, University of Cambridge, UK
Andrea Lazzari
Department of Physics and Astronomy, University of Padova, Italy
Antonio Trovato
Department of Physics and Astronomy, University of Padova, Italy
Michele Vendruscolo
Centre for Misfolding Diseases, Department of Chemistry, University of Cambridge, UK
- Correspondence: mv245@cam.ac.uk, monika.fuxreiter@unipd.it, amos.maritan@unipd.it
Monika Fuxreiter
Department of Physics and Astronomy, University of Padova, Italy, Department of Biomedical Sciences, University of Padova, Italy
- Correspondence: mv245@cam.ac.uk, monika.fuxreiter@unipd.it, amos.maritan@unipd.it
Amos Maritan
Department of Physics and Astronomy, University of Padova, Italy
- Correspondence: mv245@cam.ac.uk, monika.fuxreiter@unipd.it, amos.maritan@unipd.it

Version history

Preprint posted: October 2, 2023
Sent for peer review: November 12, 2023
Reviewed Preprint version 1: February 7, 2024
Reviewed Preprint version 2: August 6, 2024
Version of Record published: November 18, 2024

Copyright

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

Not revised: This Reviewed Preprint includes the authors’ original preprint (without revision), an eLife assessment, and public reviews.

Reviewing Editor
Bin Zhang
Massachusetts Institute of Technology, Cambridge, United States of America
Senior Editor
Qiang Cui
Boston University, Boston, United States of America

Reviewer #1 (Public Review):

The authors analyse droplet size distributions of multiple protein condensates and fit to a scaling ansatz to highlight that they exhibit features of first-order and second-order phase transitions. While the experimental evidence is solid, the text lacks connection and contextualization to the well-understood expectations from the coupling of percolation and phase separation in protein condensates - a phenomenon that is increasingly gaining consensus amongst the community. The evidence supports the percolation and phase separation model rather than being close to a true critical point in the liquid-gas phase space. Overall, the work is useful to the community.

Strengths:
The experimental analysis of distinct protein condensates is very well done and the reported exponents/scaling framework provides a clear framework to help the community deconvolve signatures of percolation in condensates.

Weaknesses:
The principal concern this reviewer has is that the reviewers adopt a framing in this paper to present a discovery of second-order features and connections to criticality - however, they ignore/miss the connections to percolation (a well-understood second-order transition that is expected to play a major role in protein condensates). I believe this needs to be addressed and the paper suitably revised to help connect with these expectations.

- Protein condensates have been increasingly understood to be described as fluids whose assembly is driven by a connection of density (phase separation, first-order) and connectivity (percolation, second-order) transitions. This has been long known in the polymer community (Flory, Stockmayer, Tanaka, Rubinstein, Semenov, and others) and recently repopularized in the condensate community (by Pappu and Mittag, in particular, amongst others). The authors make no connections to any of these frameworks - which actually seem to be the essence of what they are describing.

- Percolation theory, which has been around for more than half a century, has clear-cut scaling laws that have essentially similar forms to the ansatz adopted by the authors, and the commonalities/differences are not discussed by the authors - this is essential since this provides a physical basis for their ansatz rather than an arbitrary mathematical formulation. In particular, percolation models connect size distribution exponents to factors like dimensionality, valence, etc. and if these connections can be made with this data, that would be very powerful.

- The connections between spinodal decomposition and second-order phase transitions are very confusing. Spindal decomposition happens when the barriers for first-order phase transitions are zero and systems can phase separate without crossing nucleation barriers. Further, the "criticality" discussed in the paper is confusing since it more likely refers to a percolation threshold and much less likely to a "critical temperature" (Tc -where spinodal and binodals become identical). I would recommend reframing this argument.

It's unlikely, in this reviewer's opinion, that the authors are actually discussing a "first-order" liquid-gas critical point - because saturation concentrations of these proteins can be much higher with temperature and the critical point would thus likely be at much higher concentrations (and ofc temperature). Further, the scaling exponents don't fall into that class naturally. However, if the authors disagree, I would appreciate clear quantitative reasons (including through the scaling exponents in that universality class) and be happy to be convinced to change my mind. As provided, the data does not support this model.

https://doi.org/10.7554/eLife.94214.1.sa1

Reviewer #2 (Public Review):

This is a potentially interesting study addressing a possible scale-invariant log-normal characteristic of droplet size distribution in the phase separation behavior of biomolecular condensates. Some of the data presented are valuable and intriguing. However, as it stands, the validity and utility of this study are uncertain because there are serious deficiencies in the execution and presentation of the authors' results. Many of these shortcomings are fundamental, including a lack of clarity in the basic conceptual framework of the study, insufficient justification of the experimental setup, less-than-conclusive experimental evidence, and inadequate discussion of implications of the authors' findings to future experimental and theoretical studies of biomolecular condensates. Accordingly, this reviewer considers that the manuscript should undergo a major revision to address the following. In particular, the discussion should be significantly expanded by including references mentioned below as well as other references pertinent to the issues raised.

1. The theoretical analysis in this study is based on experimental data on condensed droplet size distributions for FUS and α-synuclein. The size data for FUS droplet is indirect as it relies on the assumption that FUS droplet diameter is proportional to fluorescence intensity of labeled FUS (page 10 of manuscript), with fluorescence data adopted from a previously published work by another group (Kar et al. & Pappu, ref.27). Because fluorescence of a droplet is expected to be dependent upon the condensed-phase concentration of FUS, this proportional relationship, even if it holds, must also be modulated by FUS concentration in the droplet. Moreover, why should fluorescence be proportional to diameter but not the cross-sectional area or volume of the FUS droplet, which would be more intuitive? These issues should be clarified. A new measure by microscopy is used to determine the size distribution of condensed α-synuclein; but no microscopy image is shown. It is of critical importance that such raw data (for example microscopy images) be presented for the completeness and reproducibility of the experiment because the entire study relies on the soundness of these experimental measurements.

2. Despite the authors' claim of a universal scaling relationship, the log-log scatter plots in Figure 1 (page 15 of the manuscript) exhibit significant deviations from linearity at low protein concentrations (ρ→0). Given this fact, is universal scaling really valid? Discussion of this behavior is conspicuously absent (except the statement that these data points are excluded in the fit). In any case, the possible origins of these deviations should be thoroughly discussed so that the regime of universal scaling can be properly delineated.

3. Droplet size distribution most likely depends on the time duration after the preparation of the sample. For α-synuclein, "liquid droplet size characterisation images were captured 10 minutes post-liquid droplet formation" (page 9 of the manuscript). Why 10 minutes? Have the authors tried imaging at different time points and, if so, do the distributions at different time points remain essentially the same? If they are different, what is the criterion for focusing only on a particular time point? Information related to these questions should be provided.

4. At least two well-known mechanisms can lead to the time-dependent distribution of liquid droplet sizes: (i) coalescence of droplets in spatial proximity to form a larger droplet, and (ii) Ostwald ripening, i.e., formation of larger droplets concomitant with the dissolution of smaller droplets without fusion of droplets. The implications of these mechanisms on the authors' droplet size distributions should be addressed. Indeed, maintaining a size distribution against these mechanisms in vivo often requires active suppression [Bressloff, Phys Rev E 101, 042804 (2020)] with possible involvement of chemical reactions [Kirschbaum & Zwicker, J R Soc Interface 18, 20210255 (2021)]. These considerations are central to the basic rationale of this study and therefore should be carefully tackled.

5. If coalescence and/or Ostwald ripening do occur, given sufficient time after sample preparation, the condensed phase may become a single large "droplet" or a single liquid layer. Does this occur in the authors' experiments?

6. It is unclear whether the authors aim to address the kinetic phenomenon of liquid droplet formation and evolution or equilibrium properties. The two types of phenomena appear to be conflated in the authors' narrative. Clarification is needed. If this work aims to address time-independent (or infinite-time) equilibrium properties, how are they expected to be related to droplet size distribution, which most likely is time-dependent?

7. The relationship between the potentially time-dependent droplet size distribution and equilibrium properties of ρt and ρc (transition and critical concentrations, respectively) should be better spelled out. An added illustrative figure will be helpful.

8. The authors comment that their findings appear to be inconsistent with Flory-Huggins theory because Flory-Huggins "characterizes droplet formation as a consequence of nucleation ..." (page 8 of the manuscript). Here, three issues need detailed clarification: (i) In what way does Flory-Huggins mandate nucleation? (ii) Why are the findings of apparent scale invariance inconsistent with nucleation? (iii) If liquid droplet formations do not arise from nucleation, what physical mechanism(s) is (are) envisioned by the authors to be underpinning the formation of condensed liquid droplets in protein phase separation?

9. Are any of the authors' findings related to finite-system effects of phase separation [see, e.g., Nilsson & Irbäck, Phys Rev E 101, 022413 (2020)]?

10. Since the authors are using their observation of an apparent scale-invariant droplet size distribution to evaluate phase separation theory, it is important to clarify whether their findings provide any constraint on the shape of coexistence curves (phase diagrams).

11. More specifically, do the authors' findings suggest that the phase diagrams predicted by Flory-Huggins are invalid? Or, are they suggesting that even if the phase diagrams predicted by Flory-Huggins are empirically correct (if verified by experimental testing), they are underpinned by a free energy function different from that of Flory-Huggins? It is important to answer this question to clarify the implications of the authors' findings on equilibrium phase behaviors and the falsifiability of the implications.

12. How about the implications of the authors' findings on other theories of protein phase separation that are based on interactions that are different from the short spatial range interactions treated by Flory-Huggins? For instance, it has been observed that whereas the Flory-Huggins-predicted phase diagrams always convex upward, phase diagrams for charged intrinsically disordered proteins with long spatial range Coulomb interactions exhibit a region that concave upward [Das et al., Phys Chem Chem Phys 20, 28558-28574 (2018)]. Can information be provided by the authors' findings regarding apparent scale-invariant droplet size distribution on the underlying interaction driving the protein molecules toward phase separation?

13. Table S1 (page 4) and Table S2 (page 7) are mentioned in the text but these tables are not in the submitted files.

14. The two systems studied (FUS and α-synuclein) have a single intrinsically disordered protein (IDP) component. It is not clear if the authors expect their claimed scaling relation to be applicable to systems with multiple IDP components and if so, why.

https://doi.org/10.7554/eLife.94214.1.sa0

Significance of findings

Strength of evidence

Abstract

Introduction

Results

Formulation of the scaling ansatz

Scaling behaviour of the droplet size distributions of FUS

Determination of the critical exponents for FUS of the scaling invariance.

The droplet size distribution of FUS is log-normal

Log-normal behavior of FUS and SNAP-tagged FUS size distributions below the critical concentration.

Independence of the variance of the distribution from the concentration of FUS

Estimation of the critical concentration of FUS using the scale invariance

Estimation of the critical concentration of FUS using the scale invariance.

Collapse of the droplet size distributions of FUS as predicted by the scale invariance.

Estimation of the critical concentration of α-synuclein using the scale invariance

Estimation of the critical concentration of α-synuclein using the scale invariance.

Discussion

Conclusions

Methods

Expression and purification of α-synuclein

Determination of the droplet distribution of α-synuclein

Determination of the droplet distributions of FUS

Determination of the critical exponents for FUS

Determination of the critical concentration for FUS

Determination of the critical exponents for α-synuclein

Determination of the critical concentration of α-synuclein

Acknowledgements

References

Article and author information

Author information

Tommaso Amico

Samuel Dada

Andrea Lazzari

Antonio Trovato

Michele Vendruscolo

Monika Fuxreiter

Amos Maritan

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