PGL-3 was purified from insect cells according to [15]. SF9-ESF cells were infected with baculovirus containing the PGL-3-GFP-6HIS protein under the polyhedrin promoter. Cells were harvested after 3 days of infection by centrifugation at 500 g for 10 min and then resuspended in lysis buffer (25 mM HEPES 7.25, 300 mM KCl, 10 mM imidazole, 1 mM DTT, 1 protease inhibitor). Cells were lysed by passing the cells 2 times through the LM20 microfluidizer at 15,000 psi. The lysate was then centrifuged at 20,000 rpm for 45 min at 15°C. The lysate was loaded in a pre-equilibrated Ni-NTA column with lysis buffer at 3 mL/min. The Ni-NTA column was rinsed with 10 C.V of wash buffer (25 mM HEPES 7.25, 300 mM KCl, 20 mM imidazole, 1 mM DTT, 1) and the protein was eluted in 1.5 mL fractions with elution buffer (25 mM HEPES 7.25, 300 mM KCl, 250 mM imidazole, 1 mM DTT). After elution the GFP tagged was cleaved to produce untagged PGL-3. The cleavage was performed using a TEV protease overnight at 4°C. PGL-3 and PGL-3-GFP proteins were diluted with Dilution buffer (25 mM Tris pH 8.0, 1 mM DTT) to reach 50 mM KCl before loading the protein in an anion exchange HiTrapQ HP 5 mL column. The HiTrap column was previously equilibrated first with HiTrapQ elution buffer (25 mM Tris pH 8.0, 50 mM KCl, 1 mM DTT) and then with HiTrapQ binding buffer (25 mM Tris pH 8.0, 1 M KCl, 1 mM DTT). The column was mounted in a Äkta Pure FPLC system. After the sample was loaded the column was washed with HiTrapQ binding buffer. The sample was finally eluted with a linear gradient from 0% to 55% of HiTrapQ elution buffer (25 mM Tris pH 8.0, 1 M KCl 1 mM DTT) for 25 C.V. Finally a 100% HiTrap elution buffer step was performed for 5 C.V. The pooled fractions were then loaded in a HiLoad 16/60 Superdex 200 size exclusion chromatography column that was previously equilibrated with superdex buffer (25 mM HEPES 7.25, 300 mM KCl, 1 mM DTT). After size exclusion, the final samples were collected.

Unlabeled FUS purified from a baculovirus construct containing N-HIS-MBP-FUS-TEV-SNAP. SF9-ESF cells were harvested after 3 days of infection by centrifugation at 500 g for 10 min. The cell pellet was resuspended using 50 mL of lysis buffer (50 mM Tris pH 7.4, 500 mM KCl, 5% glycerol, 10 mM imidazole, 1 mM PMSF, 1X protease inhibitor) for every 50 mL of cultured cells. The cells were lysed by passing them 2 times through the LM20 microfluidizer at 15,000 psi. The lysate was then centrifuged at 20,000 rpm for 45 min at 15°C. The supernatant was collected and loaded into a Ni-NTA column that was previously equilibrated with lysis buffer. After loading the sample the column was washed for 10 C.V. with Ni-NTA wash buffer (50 mM Tris pH 7.4, 500 mM KCl, 5% glycerol, 20 mM imidazole). The protein was then eluted with Ni-NTA elution buffer (50 mM Tris pH 7.4, 500 mM KCl, 5% glycerol, 300 mM imidazole). The collected fractions where then loaded into a MBPTrap HP column preequilibrated with Ni-NTA elution buffer. The MBP column was washed for 10 C.V with MBP wash buffer (50 mM Tris pH 7.4, 500 mM KCl, 5% glycerol). After washing, the sample was eluted with MBP elution buffer (50 mM Tris pH 7.4, 500 mM KCl, 5% glycerol, 500 mM arginine, 20 mM maltose). The protein was diluted to a concentration of less than 15 μM using MBP elution buffer. 3C and TEV proteases were then added to cleave the MBP and SNAP tags from the FUS construct. The cleavage reactions were incubated overnight at 18°C. Finally the protein was loaded in a SepFast GF-HS-L 26 mm × 600 mm gel filtration to remove the cleaved MBP and SNAP tags and exchange the buffer. The SepFast column was previously equilibrated in storage buffer (50 mM HEPES pH 7.25, 750 mM KCl, 5% glycerol, 1 mM DTT). The sample was concentrated to a final concentration of 15 μL using 30 kDa Amicon centrifuge filters. FUS-GFP was purified as previously described in [14].

In our analysis we mainly use the MetaDisorder predictor by [17] to identify the disordered and low complexity regions giving a disorder tendency score for each amino acid (Figure 1C). MetaDisorder queries other predictors and generates a consensus answer, with an algorithm that tests the strength of each method against several datasets. It thus addresses the issue of training-set dependent model predictions. In particular, MetaDisorder includes also widely used IUpred predictors [18].

Here we detail the procedure for measuring c _out , the saturation protein concentration above which LLPS occurs and c _in , the corresponding protein concentration in the condensed phase.

A master-mix of 95% unlabeled and 5% labeled protein was prepared from high-salt stock solutions (PGL-3: 300 mM KCl, 25 mM HEPES, 1 mM DTT; FUS: 750 mM KCl, 25 mM HEPES, 1 mM DTT, 5% Glycerol). Phase separation was initiated by diluting the stock salt concentration directly before encapsulation in water-in-oil emulsions that were created using Pico-Surf (2% (w/w) in Novec 7500, Sphere Fluidics). Emulsions were loaded on a temperature-controlled stage and the sample chambers were sealed with a two-component silicone (Picodent, Twinsil Speed). After a 30 min waiting time at the desired temperature the emulsion droplets were imaged using a 40X, 0.95 N.A., air objective mounted on an Olympus IX83 microscope stand controlled via CellSens. Confocal Z-stacks were recorded using a Hamamatsu Orca Flash 4.0 connected to a Yokogawa W1 spinning disc unit. Large 3D tile-images were collected to increase the statistics of individual emulsion droplets.

The fluorescence and bright-field images were analyzed using a custom MATLAB code. This allowed us to derive the volume fraction V _frac of the condensed phase in each emulsion droplet by image segmentation. Using volume and mass conservation we can derive a linear relationship volume fraction and total protein concentration c _tot : V _frac = 1/(c _in − c _out ) ⋅ c _tot − c _out /(c _in − c _out ). We then used a set of total protein concentrations to determine both c _out and c _in via linear regression to this equation. Thus, this allows for experimental measurements of temperature- and salt-dependent phase diagrams using small amounts of protein sample.

For the purpose of this study, i.e., analysing the impact of different domains of an IDP on its propensity to phase separate, we use field theoretic approaches rooted in statistical mechanics that are able to incorporate detailed structural information of polyampholytes such as proteins. They are obtained from the partition function for ensembles of coarse-grained polyampholytes, which are represented, via the Hubbard-Stratonovich transformation, as multi-dimensional functional integral (path integral) over all possible states or the polymeric system. Field theoretic simulations (FTS) are numerical methods that consider the full functional integral of the associated partition function. However, the integration of the full integral relies on stochastic sampling via Monte-Carlo methods that have shown considerable convergence problems due to the oscillatory nature of the resulting distribution function [19]. Nevertheless, this approach has recently been used successfully for the tau protein [9]. It holds promise for the development of an appropriate and efficient theory that will allow to characterize a whole class of proteins such as IDPs and proteins that contain intrinsically disordered domains, and can be used to deliver analytical insight into the underlying biophysical principles leading to LLPS. For our analysis we use the Random Phase Approximation (RPA). As is the case for self-consistent field theory, it can be derived from a saddle-point approximation, taking into account the asymptotically dominant contribution (a single Gaussian distribution) of the functional integral of the partition function, reducing the field theory to a mean-field model where a single Gaussian configuration interacts with an average effective field. It is one of the simplest analytical theories that can account for small site-specific fluctuations, e.g., of charge patterns and structural features of the protein.

The RPA approach we use here has been set-up previously by Lin et al. [10,20] for the IDR of Ddx4. For our analysis we use the extended model that includes salt as an additional variable, apart from the protein (here for IDRs of PGL-3 and FUS) (ϕ _aa ), counterions (ϕ _c ), KCl (ϕ _s ) and water. The free energy f = − S + f e l , (1) combines the entropic part S, coming from Flory-Huggins solution theory: − S = ϕ a a N log ϕ a a + ϕ c ⁡ log ϕ c + 2 ϕ s ⁡ log ϕ s + ϕ w ⁡ log ϕ w . (2) where N denotes the number of amino acids and the solvent fraction can be written as ϕ _w = 1 − ϕ _aa − ϕ _c − 2ϕ _s , with the electrostatic part of the free energy f _el . Note that we only have a single volume fraction ϕ _s to represent the salt concentration, which implies that the concentration of both ions conforming the salt is considered equal in all phases. Here we follow [10], but this assumption is not necessarily correct, see the discussion.

For the evaluation of f _el the Random Phase Approximation is commonly used [21–24]. It can be expressed in its simplest non-dimensional form as the following integral: f e l = ∫ 0 ∞ k ̃ 2 4 π 2 log 1 + G k ̃ − G k ̃ d k ̃ , (3) with G k = 4 π k 2 1 + k 2 T * 2 ϕ s + ϕ c + ϕ a a N σ i G M i j σ j , (4) where T* is the non-dimensionalised temperature which includes the relative permittivity of the medium. The correlation matrix for the chain can be written as G M i j = e − k 2 | i − j | / 6 , (5) where the lengths have been non-dimensionalized with the characteristic length a of the polymeric link of the associated Gaussian chain, here the domain of interest for protein PGL-3 or FUS. Note that the neutrality condition implies ϕ _c = |σ|ϕ _aa , where σ = ( 1 / N ) ∑ i = 1 N σ i is the average charge density of the protein, and σ _i is the charge of each of its amino acids. Together, this reduces the system to two quantities to be solved for, ϕ _aa and ϕ _s .

To compute the phase diagrams including salt dependence, in particular when solving for the tie lines of the model system, we integrate the system using Gauss-Laguerre quadrature, with the points computed using both our own implementation of the classic method by Golub et al. [25] and the state-of-the-art method by Glaser et al. [26]. In order to find the tie lines between two coexisting points (ϕ _aa , ϕ _s )| _α and (ϕ _aa , ϕ _s )| _β , we solve the following system, which is equivalent to the equality of the electrochemical potentials and the common tangent construction: ∂ ϕ a a f ϕ a a , ϕ s α = ∂ ϕ a a f ϕ a a , ϕ s β , (6a) ∂ ϕ s f ϕ a a , ϕ s α = ∂ ϕ s f ϕ a a , ϕ s β , (6b) f ϕ a a , ϕ s α − ∂ ϕ a a f ϕ a a , ϕ s α ϕ a a α (6c) − ∂ ϕ s f ϕ a a , ϕ s α ϕ s α = f ϕ a a , ϕ s β − ∂ ϕ a a f ϕ a a , ϕ s β ϕ a a β − ∂ ϕ s f ϕ a a , ϕ s β ϕ s β , where the non-linear system of equations is solved by using the trust-region-dogleg algorithm as implemented in Octave.

Note that this is a system of three equations with four unknowns, the volume fraction of salt and protein in the dilute and in the condensed phases. Because of the way in which the protein concentrations are measured experimentally, we know the salt concentration in the dilute phase, which allows the system of equations to be determined when comparing with the experiments.

In order to draw the complete phase diagram, we can take one of the unknowns as a parameter and solve for the others. In order to solve effectively the system above it is imperative to develop a continuation strategy. Naively, one could vary the unknown taken as a parameter and solve for the others using the prior solution as a guess. This strategy is known as natural parameter continuation [27], but the system above presents turning points for the parameter, which requires the development of a pseudo-arclength continuation algorithm. The latter implies the definition of a new variable, the pseudo arclength, which is defined as the arclength of the solution curve in the four-dimensional space spanned by the four unknowns in the system above. In practical terms, using this method implies adding an additional equation to the previous system that imposes a fixed increase of the arclength from the prior solution. Thus, we obtain a system of four solutions with four unknowns that is similarly solved using the trust-region-dogleg algorithm.

We now express T* and ϕ _aa in terms of experimentally accessible variables, concentrations (c) and temperatures (T). In the case of ϕ _aa , we know that it corresponds to the total volume occupied by amino acids over the volume of the total lattice. We take the volume of a lattice site to be that of a single water molecule, which corresponds to 1/55.5 M. We would have then for PGL-3 in the presence of KCl (note that PGL-3 consists of 693 amino acids): ϕ a a = 693 55.5 M PGL - 3 and ϕ s = 1 55.5 M KCl . (7)

There are two implicit assumptions in this calculation. One, it is assumed that each amino acid takes the same volume as a water molecule, which is clearly not true, and two, that the volume of a protein scales linearly with the number of amino acids. The latter can be argued to be false, since different configurations of the protein will give different volumes. Therefore, there is not a clear and easy way of relating the volume fraction and the protein concentration.

The non-dimensional temperature T* is related to T by T * = 4 π ε 0 ε r k B a e 2 T , (8) where ɛ ₀ is the vacuum permittivity, ɛ _r is the relative permittivity of the medium, k _B is Boltzmann’s constant, a the link length and e is the charge of the electron. The two unknowns in Eq. 8 are a and ϵ _r . We take a to be the Cα-Cα virtual bond length of 3.8 Å and fit ɛ _r , due to the lack of a complete theory to derive its value, we only know that it should be between 2 (typical value for hydrocarbon crystals) and 80 ( ε r , H 2 O = 78 ) .

The impossibility of obtaining accurate theoretical estimates for the relation between volume fraction and the concentration of the protein and for the permittivity of the solution make it necessary to proceed with a fitting procedure. There exist two parameters, β ₁ and β ₂ that allow us to scale the volume fraction and the non-dimensional temperature to fit the data of a T-c phase diagram, i.e., T = β ₁ T* and c = β ₂ ϕ _aa . Once the phase diagram is computed, we obtain a functional relation between T* and ϕ _aa , ϕ _aa = f(T*) (note that f has different branches, a dilute and a condensed branch). If we know both parameters we can rescale the previous relation to obtain c = β ₂ f(T/β ₁), a predictor for the concentration. We can find both parameters by minimizing the following function: χ 2 = ∑ i = 1 N c exp , i − c th , i σ 2 , (9) where c _exp are the experimental values of the concentration, σ is the standard deviation of the experimental points. The sum runs over all experimental points including all branches and all values of the salt concentration. The parameter β ₁ is allowed to vary with the salt concentration on account of the strong dependence of the permittivity on salt concentration.

Note that we do not assume any particular functional dependencence of ɛ _r on salt, we simply fit a value of β ₁ at each salt concentration. On the other hand, β ₂ is considered independent of salt concentration, which is the most parsimonious choice. Finally, note that the lack of a good theory for the dependencence of ɛ _r on salt implies that a single value of β ₁ has to be selected for the computation of the salt-concentration phase diagrams, which limits the agreement of the model with the experimental data.

Equation 9 is minimized using a sequential quadratic programming algorithm as implemented in Octave. The parameter χ ² will provide then a measure of the goodness of fit for each case. In Table 1 we give a summary of all fitting parameters for all cases we have considered.

We first test the predictions of this model with experimental results for a range of different bulk concentrations of recombinant PGL-3 in an in vitro phase-separation assay. The experiments exhibit the generic behaviour that higher PGL-3 concentrations are necessary at increasing temperatures in order to initiate LLPS at physiological salt (150 mM KCl) concentration. This is quantified at each temperature with a corresponding protein saturation concentration c _out (Figures 2B, C).

To investigate the role of intrinsically disordered regions (IDRs) of PGL-3, we determine (using MetaDisorder MD2) four small disordered regions (IDRs) and one large IDR (IDR-5: aa 515-693) near the C-terminus, which we denote by PGL-3-C1, see Figure 2A. It is the longest linear sequence predicted to lack a secondary structure. Indeed, this region has previously been shown to include a set of six C-terminal RGG repeats (aa 633-695), which bind RNA and promote droplet formation [15]. For the derivation of the binodal, specifically to find the RPA free energy due to the electrostatic interactions (f _el ), we use the longest IDR PGL-3-C1.

For the quantitative comparison, we need to relate the PGL-3 volume fraction ϕ _aa and the non-dimensional temperature T* in terms of the experimentally accessible PGL-3 concentration (c) and temperature (T), respectively.

To relate ϕ _aa with c we could assume, as it is sometimes done, that we have a lattice with each lattice site having the volume of a water molecule and that an amino acid occupies exactly one site. This provides a conversion factor (Eq. 7), but it is at best a crude estimation. Similarly, the non-dimensional temperature depends on the relative permittivity ɛ _r (see Eq. 8). A complete theory to derive the value of ɛ _r is lacking, thus leaving us with little choice but to fit both conversion factors β ₁ and β ₂ for T = β ₁ T* and c = β ₂ ϕ _aa . Note that ɛ _r is expected to have a strong dependence on salt concentration, and hence we will fit β ₁ independently for each temperature phase diagram at each salinity. (See Materials and Methods for the details).

The experimental values were obtained using a method that calculates c _in and c _out by measuring the volume of the condensed phase in an enclosed compartment with fluorescence microscopy. This method relies on the linear relationship between total protein concentration and condensed phase volume fraction. A visual example of this linear relationship can be observed in Figure 2B for PGL-3 at different temperatures. Figure 2B shows that an increase in temperature is accompanied by an increase in the proetin concentration required for phase separation (c _out ). Here, we used water-in-oil emulsions to encapsulate the protein solutions immediately after triggering phase separation. The corresponding experimental values of c _out are presented in the table in Figure 2C.

The theoretical phase diagram in Figure 2D reaches its maximum near 160°C. At this point the slope of the phase diagram is clearly discontinuous, thus creating a pointy feature. In the zoom on the temperature range of the experimental data (Figure 2D, right panel), we observe a good agreement between theory and experimental data, particularly for lower temperatures. The value of the concentration for the theoretical condensed branch at 10°C does not fall strictly within the 95% confidence interval, but it is nevertheless close. Considering that we only have two parameters at our disposal, we can consider this agreement of experiments and theory very good. The goodness of fit is quantified by the comparatively low value of the χ ² parameter (see Table 1 in Methods and Materials).

Our theoretical model allows for a variable salt concentration. As a consequence, the protein-poor and protein-rich phases may have different equilibrium salt concentrations, which implies in turn that the tie lines connecting both equilibria have a non-zero slope. We note that this is responsible for the characteristic pointy shape of the temperature phase diagram in Figure 2D, which is a path along the complete 3D phase diagram (shown in the Supplementary Figure S1). Experimentally, the value of the saturation concentration c _out is found by extrapolation at a value where the salt concentration in the supernatant phase is fixed. This implies a constant salt concentration in the protein-poor branch but a varying salt concentration in the protein-rich branch. This is the underlying cause of the non-smoothness of the binodal curve at the point where both branches meet.

Starting with the protein-poor branch we measure how salt affects c _out . Again we obtain a generic behaviour where higher salt concentration requires higher protein concentration for phase separation of PGL-3 to occur (Figure 3A). Compared to the dependence we saw for temperature, however, our experimental results suggest that salt has a stronger influence on c _out . Specifically, in the range from 100 to 220 mM KCl at 20°C, c _out changes approximately 30-fold from 1.12 to 34 μM (Figure 3B). For the quantitative comparison against experimental results for the salt-concentration phase diagram we use the now already determined scaling parameters to derive the phase diagram from our theoretical model. For that purpose we use the scaling parameters fitted at a constant salt concentration of 150 mM KCl, since a model that captures how permittivity changes with salt concentration is not available.

The resulting theoretical curve at 20°C (Figure 3C) has a maximum near 250 mM KCl, a salinity above which we do not expect LLPS.

The agreement of the overlaid experimental points with the theoretical dilute branch (Figure 3C, right panel) is very good, with the discrepancy with the confidence interval stemming from the fact that the value of the permittivity varies strongly with salt concentration, and we are considering it to be constant. At the lower temperature of 10°C we observe a similar good agreement between theory and experiments in the dilute branch (Figure 3D) and on the condensed branch, again considering we only have two parameters at our disposal.

Note that our theoretical model provides a prediction for the slope of the tie lines and the salt content of the protein-rich branch. As an example, for 10°C and a salt concentration of 150 mM in the protein-poor branch, we obtain a salt concentration of 187 mM in the protein-rich branch. This corresponds to an increase of salt in the condensed phase of 25% and a slope of the tie line equal to 51.9 (M KCl/M PGL-3).

We expect to test these results in future experiments.

We have shown, in the case of PGL-3, how the prediction of its temperature vs. protein concentration or salt vs. protein concentration phase diagrams can be achieved using only minimal experimental input data. However, the actual predicting power of our approach is revealed for structurally more complex proteins such as FUS. In contrast to PGL-3, which has only one long, continuous IDR, the application of the same machine learning tools yields that FUS has two long IDRs. Note that the structure of FUS is well known [28], but we proceed according to the output of the disorder prediction software, since our stated goal is to apply our method to proteins with unknown structure. Nevertheless, it is known that FUS contains a prion-like low complexity domain [29] (LC) (aa 1-214), which overlaps with the predicted IDR near the N terminus (aa 1-285, FUS-N1) (Figure 4A) and is mostly devoid of charged amino acids.

We predict an additional IDR near the C terminus (aa 367-526, FUS-C1) (Figure 4A). Within the latter IDR it is localized a Zn-finger domain, which is reported to have a role in RNA binding and sequence recognition [30]. The role of these domains in LLPS of FUS is still not known and in particular the role of the LC domain is still debated, even if it is known to be necessary for phase separation, see the discussion below.

Initially, the comparison of the theoretical and experimental results is performed using FUS-N1, which gives the smallest overall error (see Table 1 in Methods and Materials), and later we also explore the consequences of using the other IDRs we have identified. We show that we can not only predict if FUS undergoes LLPS by itself under physiological conditions, but we can now test whether our approach can determine which domain is responsible for LLPS of FUS.

In a similar fashion as for PGL-3, we first investigate experimentally the temperature-protein concentration phase diagram in order to fit the parameters of the model. In contrast to PGL-3, we observe for FUS that the influence of temperature on the saturation concentration c _out is comparable to the change we see for salt, with a 3.4-fold increase from 0.36 to 1.21 μM between 10°C and 25°C (Figures 4B, C) and a salt dependence in a range of 100–200 mM KCl with a 2.5-fold change from 0.5 to 1.26 μM (Figures 4D, E). Considering the FUS-N1 region for the model (IDR-N1, highlighted in Figure 5A), the theoretical results show good agreement with the experimental results (Figure 5B). The theoretical phase diagram has a maximum near 150°C and has an even more striking pointy feature. If we zoom in to the region with a biologically meaningful temperature (Figure 5A, right panel), we can see how well the overlaid experimental points fall upon the theoretical dilute branch. In this case, the experimental point at 10°C also falls directly upon the condensed branch, thus giving an even better fit that PGL-3, which is manifest in the value of the χ ² parameter (see Table 1 in Methods and Materials).

As in the PGL-3 case, the experimentally fitted parameter values obtained for the temperature-concentration phase diagram are also used to generate the salt-concentration phase diagram for comparison to the experimental data. The overlay of experimental points and the theory captures the trend of the change of concentration with the salinity in the dilute branch (Figure 5C). Again the discrepancy in the values of the concentration stems from the fact that we use the value of the permittivity at 150 mM KCl but we know that its value depends strongly on salt concentration. In the condensed branch we obtain a very good agreement between theory an experiments (Figure 5D), probably on account of the smaller variation of the salinity. For 10°C we can compute the slope of the tie line for a salt concentration of 150 mM in the dilute branch, giving 25.6 (M KCl/M FUS), and a predicted salt concentration of 366 mM, and thus an increase of 144% in the condensed branch, which we also expect to be tested experimentally in the future.

We also investigated the FUS-LC domain and found that the theoretical curve cannot be fitted to allow for comparison with the experimental results, since the scaling factor involves a permittivity smaller than 1 as shown in Table 1 in Methods and Materials. Thus, interestingly, the LC domain should not play a dominant role in LLPS of FUS, according to our approach. Other than FUS-N1, a further IDR should be considered according to the results from MetaDisorder, FUS-C1 (highlighted in Figure 5E). In principle one could compare the fits and reason in terms of the goodness of fit (i.e., the χ ² parameter given in the Methods and Materials section) to decide which region is most likely responsible. But a side-by-side comparison of the overlay of the experimental data points for FUS-N1 and FUS-C1 shows a much more clear-cut result, which is that the experimental results cannot be fitted to the FUS-C1 curve (See Figure 5F). This is a consequence of salt partitioning and our parsimonious fitting approach.

Salt partitioning and the requisite that the dilute branch has a constant salt concentration give a very constrained form of the phase diagram (Figure 5F, left panel). This is in contrast with constant-salt models, which give a binodal curve that can be conveniently scaled to fit most experimental data. In particular, salt partitioning imposes a maximum ratio of the protein concentration in the condensed branch with the concentration in the dilute branch. This maximum ratio is marked explicitly with an arrow in Figure 5F, left panel. The arrow has a length of log c i n / c out , which is the log of the maximum of the ratio c _in /c _out . If the experimental c _in /c _out is greater than this maximum value, the theoretical curve cannot be scaled to fit the experiments. This means that either the condensed or the dilute branch can be fitted, but not both. In Figure 5F we observe that the dilute branch is well captured by the theoretical model, but there is a large discrepancy with the condensed branch, which is even more manifest in a linear concentration scale in the salt-protein phase diagram (Figure 5F, right panel).

This surprising result is in clear contrast with models where the salinity is considered constant in both phases (as shown in Supplementary Figure S1) and thus clearly shows the predictive power of our model, discarding regions for being responsible for LLPS while giving a testable prediction for the salinity of the condensed phase.