Sample preparation

Recombinant protein constructs used in this work were purified and stored as described previously6,7. Constructs6,7,85 are given in Supplementary Table 3, and explicit sequences in Supplementary List 1. To induce phase separation, we mixed protein in high-salt storage buffer (300 mM KCl for PGL-3 constructs, 500 mM KCl for TAF15 and FUS constructs) with storage buffer lacking monovalent salt (‘Dilution Buffer’) to reach the desired final salt concentration. Generally, an aliquot of Dilution Buffer was supplemented to 1 mM with fresh dithiothreitol prior to each day’s experiments. For the ternary and five-solute systems, the dilution buffer was also supplemented with poly(A) RNA (#P9403, Sigma). After induction of phase separation, dilute phase was obtained by centrifugation at 20,800g for 30 min in a tabletop centrifuge (5417R, Eppendorf) pre-equilibrated at the desired temperature. For control measurements, 10-µm silica microspheres were purchased from Whitehouse Scientific, and glycerol–water mixtures were prepared by weight to the desired refractive index. Bead-containing dispersions were prepared by gently dipping a 10-µl pipette tip into a stock of dry beads, transferring the pipette tip to a 40-µl volume of glycerol–water mixture, and pipette mixing to disperse. Aqueous two-phase systems with PEG-35k (Sigma) and Dextran T500 (Pharmocosmos) were prepared as described previously45. BSA was purchased from Sigma and used without further purification.

Quantitative phase imaging and analysis

QPI measurements were performed using a coherence-controlled digital holographic microscope (Q-Phase, Telight (formerly TESCAN)) based on the set-up in ref. 35. Most data were acquired on a Generation-1 (G1) instrument with a tungsten–halogen bulb as light source, although some data were acquired on a Generation-2 (G2) instrument with a 660-nm light-emitting diode as light source. In each case, the holography light source was filtered by a 10-nm bandwidth notch filter centred at 650 nm. All measurements were performed with 40× dry objectives (0.9 numerical aperture (NA), Nikon) except those for SNAP-TAF15(RBD) reported in Fig. 2f and Supplementary Fig. 8, for which 20× dry objectives were used. In all cases, the condenser aperture was set to an NA of 0.30. Immediately following phase separation, ~5 µl of sample was loaded into a temperature-controlled flowcell, sealed with two-component silicone glue Twinsil (Picodent), and allowed to settle under gravity for ~10 min prior to data collection. Flowcells were constructed with a 30 × 24 × 0.17 mm3 PEGylated coverslip and a 75 × 25 × 1 mm3 sapphire slide as bottom and top surfaces, respectively, using parafilm strips as spacers. Proportional-integral-derivative (PID)-controlled Peltier elements affixed to the sapphire slide enabled regulation of flowcell temperature, as previously described48. The sapphire, coverslip and spacers were adhered by heating the assembled flowcell to 50 °C for 2–5 min, then returning to the desired temperature for the first measurement, typically to 20 °C. For each sample, hologram z stacks (∆z = 0.2 µm, first plane typically near the coverglass surface) were acquired for several fields of view. SophiQ software (Telight) was used to construct amplitude and compensated phase images from the raw holograms. Pixels in 40× phase images are 0.157 µm per side for the G1 system and 0.1234 µm per side for the G2. To aid interpretation by persons with red/green colour perception deficiencies, phase images are displayed using the Ametrine colourmap86.

All phase images were analysed in MATLAB using custom code. In each image, the pixel intensity is taken as equal to the local value of the phase profile at the coverslip, Δφ. For each z plane, compensated phase images were first segmented to identify individual droplets. To determine the background phase value, φ0, the image’s pixel intensity histogram was fitted to a Gaussian, and the Gaussian centre taken as φ0. Pixel intensities φ ≥ nsigσφ are considered above threshold, where σφ is the standard deviation extracted from the Gaussian fit. Typically, nsig = 5. A binary mask was generated with this threshold and individual objects were identified using the MATLAB function bwconncomp.m. For each object, a region of interest slightly larger than the object’s bounding box was fitted twice to phase functions of the form given by equation (1). These phase functions incorporate geometric models for the local thickness of the droplet, defined as the vertical distance between the upper and lower droplet surfaces. First, we fit using the local thickness of a sphere,

$$\begin_}\left(x,,_,_\right)\\=\varTheta \left(^-_\right)}^-_\right)}^\right)\sqrt^-_\right)}^-_\right)}^},\end$$

to obtain estimates for the parameters Δn, R, xc, yc, where Θ(x) is the Heaviside function. These estimates were then used to initialize a fit to a regularized version of equation (1),

$$\Delta_}\left(x,y\right)=\frac\Delta\;n_}\left(x,,_,_,_}\right)+_+A\left(_},R\right),$$

using the local thickness of a spherical cap,

$$\begin_}\left(x,,_,_,_}\right)\;=\;\sqrt^-_\right)}^-_\right)}^}\\\qquad\qquad\qquad\qquad\qquad\quad\left(1+\varTheta \left(_}^+_\right)}^+_\right)}^-^\right)\right)\\\qquad\qquad\qquad\qquad\qquad\quad\varTheta \left(^-_\right)}^-_\right)}^\right)\\\qquad\qquad\qquad\qquad\qquad\quad+_}\varTheta \left(^-_}^-_\right)}^-_\right)}^\right).\end$$

The regularization is given by \(A\left(_},R\right)=__}-R\right)}^\varTheta \left(_}-R\right)\) with A0 = 105 and φ0 fixed at the value obtained from the pixel intensity histogram fit. After all z planes were processed, the objects were tracked through z using track.m (https://site.physics.georgetown.edu/matlab/index.html). For each tracked object, the representative fit parameters are taken as those from the fit with the highest adjusted R2 value, which are typically in the plane acquired nearest to the equatorial plane of a given droplet (Extended Data Fig. 1). The particle list was then automatically filtered for fit quality (typically retaining only adjusted R2 values > 0.95) and overlap with dead pixels on the detector. Droplets with irregular wetting or that are not isolated in z (that is, are situated beneath other droplets in solution) are removed manually following inspection of the raw data.

ODT

ODT measurements were performed using a custom-built microscope based on a Mach–Zehnder interferometer that enabled the acquisition of multiple complex optical fields from various incident angles, as previously described87,88. Briefly, a 532-nm solid-state laser beam (Torus, Laser Quantum) was divided into reference and sample beams by a 2 × 2 single-mode fibre-optic coupler. The sample beam illuminated the specimen mounted on a custom-made inverted microscope stage through a tube lens (focal length, 175 mm) and a high-NA objective lens (63×, water immersion, 1.2 NA, Carl Zeiss). A dual-axis Galvano mirror (GVS012/M, Thorlabs) placed in the conjugate plane of the sample was used to scan 150 different incident angles. The scattered light from the sample was collected by a high-NA objective lens (100×, oil immersion, 1.3 NA, Carl Zeiss). This scattered light interfered with the reference beam at the image plane to form spatially modulated holograms, which were recorded using a CMOS camera (FL3-U3-13Y3M-C, FLIR Systems).

To retrieve the complex optical fields, the resulting holograms were processed using a Fourier-transform-based field retrieval algorithm. These fields were then used to reconstruct the three-dimensional (3D) refractive index distribution of the samples via the Fourier diffraction theorem under the first-order Rytov approximation89,90. The refractive index we report for individual droplets is computed as the mean refractive index value of voxels at the droplet centre plane. Additional technical details on the reconstruction procedure are available in previous literature10,91.

Confocal fluorescence microscopy and analysis

Confocal imaging was performed on an inverted Zeiss LSM 880 point-scanning confocal microscope with a 40× water-immersion objective (1.2 NA, C-Apochromat, Zeiss) at room temperature. We note that this highly corrected objective is designed for 3D imaging of aqueous samples and has the highest NA compatible with water immersion. We preferred water as an immersion medium despite the higher NA achievable with oil immersion to reduce imaging artefacts from a refractive-index mismatch between the immersion medium and the sample (primarily the dilute phase with a refractive index close to water). mEGFP was excited with a 488-nm argon laser and emission detected with a 32-channel GaAsP photomultiplier tube (PMT) array set to accept photon wavelengths between 499 and 569 nm. Alexa Fluor 546 (AF546) was excited with a 561-nm diode-pumped solid-state laser and emission detected between 570 and 624 nm with the same spectral PMT array. For both fluorophores, the confocal pinhole diameter was set to 39.4 µm, corresponding to 0.87 and 0.96 Airy units for mEGFP and AF546, respectively. For each field of view, scanning was performed with a lateral pixel size of 0.415 µm and z stacks acquired with a spacing of 0.482 µm. We note that the choice of a comparatively large lateral pixel size was made (1) to reduce the overall laser dosage, (2) to reduce the image acquisition time, and (3) to give a voxel aspect ratio close to 1 with our axial step-size (Extended Data Fig. 2).

All confocal fluorescence images were analysed in MATLAB using custom code. Partition coefficients of fluorescently labelled species into condensates are estimated on the basis of the fluorescence intensity along a line-scan through the droplet centre. The analysis pipeline begins with determining the location of each condensate and an appropriate line-scan orientation angle. To determine lateral positions of condensates in each field of view, a z plane was selected slightly above the coverglass such that even small droplets appeared bright. Following convolution with a 2D Gaussian (σx = σy = 0.5 pixels) to suppress shot noise, a threshold of Ithresh = max(I(x,y))/2 was applied to obtain a binary mask. The lateral positions and approximate sizes of objects were determined from the mask with bwconncomp.m. Only the largest ~120 objects for each condition were analysed further. For each object, partition coefficients were calculated using the z plane for which the mean intensity in a 5-pixel-radius disk concentric with the object was greatest. Line-scans were 51 pixels long, concentric with the object and averaged over a width of 3 pixels. Suitable line-scan orientations were determined in a semiautomated manner by superimposing reference lines rotated through 15° increments on each object and manually selecting an orientation that best avoided neighbouring objects. Objects for which no suitable line-scan orientation could be found were discarded.

Line-scans for each droplet were automatically subdivided into three domains, corresponding to pixels in the dilute phase, the condensed phase or the exclusion zone. The positions of the left and right dilute/condensed interfaces are estimated as those at which the intensity profile reaches its half-maximal value above detector background Ibkgd (see below). To reduce artefacts stemming from the finite point-spread function of the microscope, pixels within an exclusion zone, defined as the greater of 1 pixel or lEZ = 1.22λ/(2NA) on either size of the half-maximum, were excluded from the analysis (Extended Data Fig. 2). We estimate the decay length scale of the axial point-spread function for light emitted at the peak fluorescence wavelength λem as92

$$_=\frac_}}_}-\sqrt_}^-}\right)}^}},$$

where nimmersion = 1.333 is the refractive index of the immersion medium (water). Using λem = 510 nm for mEGFP (www.fpbase.org (ref. 93)) and NA = 1.2, this gives a theoretical axial PSF resolution of 596 nm. The remaining profile pixels outside the droplet were averaged to give Idil, while the profile pixels inside are averaged to give Icond. The partition coefficient for each object was calculated according to

where Ibkgd is the average of all pixels in a background image acquired immediately following the fluorescence z stack. Background images were acquired with the light source blocked to measure the contribution of detector noise to the signal.

Bulk refractometry

The data in Fig. 1k were acquired at λ = 653.3 nm and 21 °C with a DSR-L multiwavelength refractometer (Schmidt + Haensch) using a 200-µl sample volume. All other bulk refractive index measurements were acquired at λ = 589.3 nm with a J457 refractometer (Rudolph Research Analytical). The refractive index of glycerol/water mixtures was adjusted from λ = 589.3 to 650 nm using empirical dispersion relations for distilled water, nwater(λ) (ref. 94), and glycerol, nglycerol(λ) (ref. 95) according to

$$_}\left(\lambda \right)=_}_}\left(\lambda \right)+\left(1-_}\right)_}\left(\lambda \right).$$

The glycerol weight fraction in the mixture was calculated from the refractive index measurement of the mixture at 589.3 nm as

$$_}=_}\left(\lambda \right)-_}\left(\lambda \right)\right)/\left(_}\left(\lambda \right)-_}\left(\lambda \right)\right)\right|}_.$$

Bead porosity models

The two models used to account for the porosity p of the silica microspheres (Fig. 1j and Supplementary Fig. 6) are a weighted linear sum, \(\Delta n=p_}+\left(1-p\right)_}\) (simple model), and a more detailed model,

$$\Delta n=_}\right)+2\left(_}\right)}_}\right)+\left(_}\right)}\right)}^-_}$$

based on the Lorentz–Lorenz relation96, wherein

$$}=\frac_}\right)-f\left(_}\right)}_}\right)-f\left(_water\,mixture}}\right)}$$

with \(f\left(n\right)\equiv \left(^-1\right)/\left(^+2\right)\). For comparison, reference values for fused silica were taken from ref. 97.

Calculation of dn/dc, \(\bar}}\) and polymer volume fraction

The refractive index increment and partial specific volume were estimated for each protein construct using the calculator tool within SEDFIT32 and the protein sequences listed in the Supplementary Information. The partial specific volume of 0.5773 ml g−1 for poly(A) RNA was estimated using consensus volumes per base from ref. 98 and assuming a typical chain length of 500 bases. The partial specific volumes of 0.8321 and 0.6374 ml g−1 for PEG-35k and Dextran-500k, respectively, were taken from ref. 45. The polymer volume fraction in the dense phase (Figs. 4g and 5g) for polymer i is given by \(_^=_^\bar_\).

Estimation of capillary length

We estimate the capillary length for a droplet-forming system as \(_=^\), where γ and Δρ are the interfacial tension and mass density differences between the two phases, respectively, and g is the acceleration due to gravity. The range of γ values we explored was informed by the values measured in ref. 99.

Calculations for ageing systems

The volumes of individual ageing condensates (Fig. 3c,f) were calculated assuming the shape of a spherical caps as \(V=\uppi}R^[1-\frac\left(2+\frac_}}\right)_}}\right)}^]\). The number of molecules in each condensate (Fig. 3d,g) was calculated as \(N\left(t\right)=c\left(t\right)V\left(t\right)\). Given c(t), the relative volume change expected if \(N\left(t\right)=N\left(0\right)\) is given by \(V\left(t\right)/V\left(0\right)=c\left(0\right)/c\left(t\right)\) (Fig. 3f, black line).

UV–vis spectroscopy

Absorption spectra of dilute-phase and reference samples were collected on an NP-80 spectrophotometer (IMPLEN). All spectra were acquired at room temperature over λ ∈ [200 nm, 900 nm]. Depending on signal strength, spectra were acquired with a pathlength of 0.07 mm, 0.67 mm or 10 mm. All absorption spectra were converted to absorbance units (AU) appropriate to a 10-mm pathlength prior to analysis. For each raw spectra \(\widetilde\left(\lambda \right)\), a linear fit on λ ∈ [550 nm, 750 nm] was used to determine a baseline correction, SBL(λ). Corrected spectra are given by \(S\left(\lambda \right)=\widetilde\left(\lambda \right)-_}\left(\lambda \right)\). At least three replicate spectra were acquired for each condition and averaged following baseline correction to give the final representative spectra. The uncertainty in the spectra at each wavelength was estimated as the standard deviation of the corrected replicates.

For FUS/RNA ternary mixtures, dilute-phase spectra were demixed (Fig. 4b) into a weighted sum of three contributions

$$_}\left(\lambda \right)=^}_}\left(\lambda \right)+^}_}\left(\lambda \right)+^}^}_}_}\left(\lambda \right),$$

where pI and rI are the protein and RNA concentrations in the dilute phase. Sp and Sr are reference spectra for protein and RNA, respectively. The final term captures the effect of protein–RNA interactions on the absorbance of a mixture, which could physically stem from binding-induced changes in extinction coefficients. A reference spectrum for the interaction was calculated from the spectrum of a protein–RNA mixture of known composition (p,r) in the one-phase regime according to \(_}=S\left(p,r\right)-p_}-r_}\). The parameter αint captures the approximately linear increase of Sint with p (Extended Data Fig. 4). The same value of αint was used to demix all dilute-phase spectra. The dilute-phase concentrations pI,rI were therefore the only free parameters for the demixing.

Refractive index cross-term calculation

The relevant theory and the calculation are presented in detail in Supplementary Note 14. Briefly, the calculation began with the construction of the excess absorption spectra of a ‘homogeneous test mixture’ of FUS and RNA with cFUS = cRNA = 1 mg ml−1 using the interaction term and normalized interaction spectra determined by UV–vis spectroscopy. This excess absorption spectrum was converted to excess extinction and transformed from wavelength space to frequency space. The frequency-dependent excess extinction was then fitted to a model for the dissipation of two driven linearly damped linear oscillators (two-oscillator model) to give an analytic approximation to the spectrum over all frequencies. The real component of the excess optical response function was calculated from a Kramers–Kronig relation via symbolic integration of a normalized version of the fitted two-oscillator model spectrum in MATLAB using int.m with the ‘PrincipalValue’ flag set to true. After rescaling back to the original units and converting back to wavelength space, the value of the refractive cross-term coefficient at the 650-nm wavelength used for QPI measurements was calculated from the excess refractive index as \(_}\left(650\,\text\right)\equiv \frac^n}_}_}\left(650\,\text\right)=\frac_}\left(650\,\text\right)}_}_}}\).

Composition calculation for ternary mixtures with a quadratic refractive index model

A derivation of the ATRI theory for a quadratic refractive index model is presented in Supplementary Note 13. To calculate the composition of FUS/RNA condensates using a quadratic refractive index model in Extended Data Fig. 6, the dense-phase composition was calculated by solving the system of equations given by \(_^}=_}_^}+_}\) together with \(_^}=\frac_^,\max }-r_^}}}__^}}\), where \(_}=\frac}_-_^}}}_-_^}}\) and \(_}=}_-_}}_\) are the slope and y-intercept of the tie-line, respectively, and \(_^,\max }\equiv \Delta n+_^}\fracn}}_}\)\(+_^}\fracn}}_}+_^}_^}\frac^n}_\mathrm_}\), \(r\equiv \fracn}}_}/\fracn}}_}\) and \(}_\equiv \frac^n}_\mathrm_}/\fracn}}_}\). As the second equation for \(_^}\) is non-linear, the system was solved numerically in MATLAB. For each of the tie-lines in Fig. 4, \(_^}\) was calculated first by solving \(\frac_^,\max }-r_^}}}__^}}=_}_^}+_}\) using the non-linear solver fsolve.m. Subsequently, \(_^}\) was determined through the tie-line equation.

The magnitude of the systematic error incurred when assessing FUS/RNA condensate composition in terms of a linear refractive index model relative to a quadratic one was calculated as \(\delta _^}=\left|\frac_^}-_^}}_^}}\right|\), where the superscripts ‘quad’ and ‘lin’ specify the refractive index model used to obtain the concentration value.

Dilute-phase RNA concentration for a five-solute system

Here, the RNA concentration in the dilute phase was determined from UV–vis absorption spectra SI(λ) of the dilute phase at 260 nm after accounting for contributions from protein solutes, the AF546 dye on the TAF15 construct, a UV-active contaminant population in the TAF15 stock, and interactions between the RNA and the RNA-binding proteins present according to \(_}^\mathrm=\frac^\mathrm\left(260\,\text\right)-_^_\left(260\,\text\right)_^\mathrm}_}\left(260\,\text\right)+_^_}_^\mathrm}\), where εi is the extinction coefficient of species i, \(_^}\) is the dilute-phase concentration of species i, and i indexes the non-RNA components considered according to 1 = FL-FUS-mEGFP, 2 = MBP-FUS-mEGFP, 3 = shortFUS-mEGFP, 4 = SNAP-TAF15(RBD) (with and without AF546 dye), 5 = AF546 dye, 6 = effective UV-active contaminant in the SNAP-TAF15(RBD) stock. The central estimate of \(_}^}\) and its uncertainty were computed as the mean and standard deviation of the population of estimates obtained through Monte Carlo simulations100 of the preceding equation using experimentally constrained Gaussian distributions for the measured quantities SI and \(_^}\) (Extended Data Fig. 8).

For \(i\in \left\\right\}\), the \(_^}\) of the corresponding population was determined from MS. For the AF546 dye, we set \(_^}=_}_^}\), where fAF is the fraction of dye-bound SNAP-TAF15(RBD) molecules in the stock solution. This expression is valid in the limit that covalent attachment of the AF546 dye does not significantly perturb the partitioning of SNAP-TAF15(RBD) relative to the construct without dye. fAF was determined from the ratio of the dye concentration \(_}^}\) measured by UV–vis in a homogeneous reference sample formed by dilution of the SNAP-TAF15(RBD) stock solution to the protein concentration \(_}^}\) in the SNAP-TAF15(RBD) population measured by MS of the same reference sample. We define the effective concentration of UV-active contaminants in a stock solution as the difference between the protein concentration inferred from absorbance at 280 nm in a reference sample to that measured by MS. For the SNAP-TAF15(RBD), the equation used is \(_-}}^}=\frac_-_}_}^}-_546}_546}^}}_-}}}\). We set \(_-}}=_}\) for convenience and note that the exact choice is ultimately irrelevant because \(_-}}^}\) appears in the expression for \(_}^}\) only as the product \(_-}}_-}}^}\), which is uniquely determined. Within the uncertainty of the measurements, this difference was zero for the reference sample from the FUS stock solution but non-zero for the reference sample from the SNAP-TAF15(RBD) stock solution (Supplementary Fig. 13). For the effective contaminants from the SNAP-TAF15(RBD) stock, the concentration in the dilute phase was estimated as \(_^}=}_\), where \(}_\) is the average concentration in the demixed system. This expression is valid in the limit where the partition coefficient of the effective contaminant species is close to 1.

MS

Proteins were enzymatically digested and analysed by LC–MS/MS on an RSLC system UltiMate 3000 series coupled with a Q Exactive HF mass spectrometer (Thermo Fisher Scientific). Molar abundances of quantotypic peptide were calculated as described previously70. Each proteotypic peptide was quantified independently. To calculate the molar abundance of proteins, the molar abundances of the corresponding peptides were averaged. Further details can be found in Supplementary Note 15.

Statistics and reproducibility

Individual droplets were excluded from quantitative analysis when they did not meet the physical assumptions of the analysis. As described above and in Supplementary Note 17, droplets were excluded from QPI analysis if the shape inferred from the phase image was not well described by a spherical cap, for instance due to irregular wetting of the coverslip or the presence of an additional object in solution directly above the droplet of interest. Similarly, droplets were excluded from ODT analysis if their tomographic reconstructions were strongly impacted by artefacts due to their position near the edge of the field of view. Droplets with volumes smaller than 0.1194 µm3 (equivalent to fewer than 100 voxels) were also excluded from ODT analysis to avoid misclassification of small segmented regions of background noise as droplets. As samples G and H in Supplementary Fig. 8 were both prepared in the presence of a small number of polystyrene microspheres with radii near 2 µm, we additionally restricted the ODT analysis of these samples to objects with volumes smaller than 15 µm3 to ensure that the few microspheres present in our tomograms were excluded. Droplets were excluded from final fluorescence analysis when they were too small relative to the axial point-spread function for the fluorescence intensity estimated from interior pixels to reach the level observed in larger droplets (see also Extended Data Fig. 2). Otherwise, data were not excluded from the analyses.

To assess the statistical significance of a correlation coefficient robs computed from a set of Nobs paired experimental measurements, we determined one-sided P values from Monte Carlo simulations100. Briefly, we used rand.m in MATLAB to randomly draw Nobsx and y values from a uniform distribution. From this set of simulated data, we computed a correlation coefficient rsim between the randomly paired x and y values. We repeated this procedure Niter = 100,000 times to generate a probability density distribution \(\rho \left(_}|_}\right)\) of rsim values given the sample size Nobs. We then calculated the probability of encountering by chance a correlation at least as large as the correlation found experimentally. We compute this as the fraction of the simulations for which rsim ≥ robs. In the limit that Niter → ∞, this fraction is equivalent to

$$P=\frac__}}^\rho \left(_}|_}\right)}_}}_^\rho \left(_}|_}\right)}_}}$$

that is, a one-sided P value. To assess the uncertainty incurred by using a finite number of simulations, we generated a P-value distribution by repeating the p-value calculation procedure 50 times. The P value we report is the mean from this P-value distribution. For cases where the standard deviation of the distribution is larger than 10% of the mean, we report the standard deviation as well.

No statistical method was used to predetermine sample size. The experiments were not randomized. The investigators were not blinded to allocation during experiments and outcome assessment.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.