A model of the cytoplasm of Mycoplasma genitalium established previously (Feig et al., 2015; Yu et al., 2016) was simulated at a coarse-grained (CG) level with one sphere per macromolecule or complex (Supplementary file 1). CG particle interactions were calibrated against results from atomistic MD simulations of concentrated protein solutions. The parameters involve only two particle-dependent properties, namely size and charge. Droplet-like condensates formed spontaneously within 20 µs (Figure 1A/B) and remained present during 1 ms simulation time. Similar results were obtained with an alternate effective charge model that resulted in better agreement between theory and experiment (see below; Figure 1—figure supplement 1) Two types of condensates were observed: one type contained predominantly tRNAs and positively charged proteins; the other type contained ribosome particles (RP) and positively charged proteins. The RP condensates also attracted the weakly negatively charged GroEL particles at the surface (Figure 1A). The condensates increased in size as the system size was increased from 100 to 300 nm (Figure 1A). This observation is consistent with PS rather than finite-size cluster formation. The presence of multiple droplets in the 300 nm system suggests incomplete convergence, but as the droplets grow in size, further merging becomes kinetically limited due to slowing diffusion. We did not find evidence for growth via Ostwald ripening where particles preferentially evaporate from smaller condensates and redeposit onto larger condensates. Further analysis focused on the condensates observed in the 100 nm system.

Cluster analysis considered interactions between the nucleic acids and positively charged proteins to obtain trajectory-averaged cluster size distributions (Figure 1C). Most tRNA (87%) was part of a condensate. The remaining fraction of tRNA existed as monomers or small clusters, suggesting coexistence of dilute and condensed phases. RP were only found in the RP condensates. Total macromolecular volume fractions inside tRNA and RP condensates were 0.42 and 0.58, respectively, whereas volume fractions for just tRNA and RP inside their respective condensates were 0.07 and 0.26. The volume of the condensates was estimated based on the overlapping van der Waals volumes of spheres inside the largest cluster with an additional probe of 2.2 nm in consistent with our cluster definition. The dilute phase volume was estimated as the remaining accessible volume after subtracting volume of condensates from the total volume. The condensates had significantly higher macromolecular densities than the rest of the simulated system (Figure 1—figure supplement 2). The moderately high volume fractions for tRNA condensates are still within the range of concentrated liquid phases (Dumetz et al., 2008), but the higher volume fractions in the RP condensate tend toward solid- or gel-like phases (Dumetz et al., 2008). Radial distribution functions of tRNA and RP from the center of the corresponding condensates show a relatively smooth decay with a soft boundary for tRNA condensates (Figure 1—figure supplement 3), that are consistent with a more dynamic phase, whereas distinct peaks and a sharper boundary for RP indicate a highly ordered arrangement in the RP condensates. The more ordered structure of the RP condensates may be an example of the kind of structured condensates resulting from a balance between homotypic and heterotypic interactions as described recently (Regy et al., 2020).

We observed separate condensates involving tRNA or RP, presumably due to the large difference in size of RP vs. tRNA that may be explained at least in part by the Asakura-Oosawa depletion model (Asakura and Oosawa, 1958). Both tRNA and RP condensates contained (positively charged) proteins at high concentrations. tRNA and RP interactions with those proteins were favorable as evidenced by a strong peak in the pairwise radial distribution function g(r) at contact distance (Figure 1—figure supplement 4). The charge and size of the proteins attracted to the condensates differed between tRNA and RP condensates (Figure 1—figure supplement 5). In the tRNA condensates, large proteins with radii of 3 nm and above and with charges of 10 and above were preferred. In contrast, the proteins in the RP condensates were smaller, with radii of 3 nm or less, and many proteins had charges below 10. This suggests that differential interactions between different size and charge nucleic acid and protein particles may further explain the formation of separate condensates involving tRNA and RP.

The dynamics inside and outside the condensates was analyzed in terms of translational diffusion coefficients (D_tr) calculated based on mean-squared displacements (Figure 1—figure supplement 6). Diffusion during the last 1 μs of the simulation was compared with diffusion during the first 1 μs when condensates were not yet formed. Molecule-specific values of D_tr are given in Supplementary file 1. As a function of the radius of the macromolecules (Figure 1—figure supplement 7), D_tr values follow a similar trend as observed before in atomistic simulations of the same system. Diffusion outside the condensates resembled diffusion in the dispersed phase. In tRNA condensates, the diffusion of macromolecules is similar to the dispersed phase or is moderately retarded, depending on the molecule, and consistent with reduced diffusion in increased protein concentrations seen in experiment (Muramatsu and Minton, 1988; Zimmerman and Minton, 1993). In RP condensates, diffusion is reduced to a greater extent, but significant dynamics is still maintained for most types of macromolecules as they diffuse around a relatively static RP cluster (Video 1).

A simplified five-component system was constructed to reproduce the RNA condensation observed in the cytoplasmic model. The simplified model consisted of tRNA, ribosome particles (RP), large (POS_L, q = 20, r = 3.5 nm) and small (POS_S, q = 1, r = 2.52 nm) positively charged proteins as well as neutral crowders (CRW, q = 0, r = 2.52 nm). tRNA and RP concentrations were initially set as in the cytoplasmic model while concentrations, sizes, and charges of the other three particle types were adjusted to match the total number of particles, total molecular volume, and total charge of the cytoplasmic system as closely as possible. Subsequently, a series of simulations were run at different concentrations and with different parameters (Supplementary file 2).

In simulations of the five-component model, tRNA and RP condensed separately as in the cytoplasmic model (Figure 2—figure supplement 1). Again, the condensates formed quickly, within 50 μs (Figure 2—figure supplement 1), and cluster size distributions of tRNA and RP resembled the results from the cytoplasmic system (cf. Figure 1 and Figure 2—figure supplement 1). However, in contrast to the cytoplasmic system, we found a small fraction (2% on average) of RP in the dilute phase. As in the cytoplasmic model, tRNA strongly preferred interactions with the larger POS_L particles, whereas RP interacted favorably with both POS_S and POS_L (Figure 2—figure supplement 2). tRNA condensates remained highly dynamic as in the cytoplasmic system. From the last 100 μs of the simulation, we obtained diffusion coefficients D_tr for tRNA of 28.3 ± 0.7 and 59.0 ± 0.5 nm²/µs inside and outside of the condensates, respectively, similar to values of 16.3 ± 0.1 and 55.5 ± 0.8 nm²/µs in the cytoplasmic system. Diffusion coefficients for RP inside and outside of the RP condensates were 0.49 ± 0.01 and 0.80 ± 0.4 nm²/µs, respectively, compared to D_tr = 0.34 ± 0.01 nm²/µs for RP in the cytoplasmic condensates.

RP and POS_L concentrations were varied systematically, while the concentration of POS_S was kept constant and the number of CRW particles was adjusted to maintain a constant total molecular volume (Supplementary file 2). Cluster size distributions were extracted (Figure 2—figure supplement 3) and the fraction of tRNA and RP in the large clusters was determined (Figure 2). Some degree of clustering occurs at all concentrations, but condensation requires that a significant fraction of particles is found in the largest clusters. Based on a criterion that at least half of the particles are found in one or few large clusters, tRNA and RP condensation occurs for [POS_L]>100 μM (Figure 2).

Increasing [RP] reduces the amount of tRNA in the tRNA condensates and effectively raises the critical POS_L concentration above which tRNA forms condensates (Figure 2). This can be understood from competition for POS_L. tRNA only interacts significantly with POS_L (Figure 2—figure supplement 4) and needs POS_L to form condensates, whereas RP interacts with both POS_S and POS_L (Figure 2—figure supplement 5) and therefore draws POS_L from tRNA condensates (Figure 2—figure supplement 6). For [POS_L]>500 μM, the fraction of tRNA particles in the tRNA condensates is relatively constant (Figure 2). However, the number of POS_L particles in the condensates increases as the total [POS_L] increases (Figure 2—figure supplement 6). This results in larger clusters and lower effective [tRNA] in the condensates at the highest values of [POS_L] (Figure 2—figure supplement 7). The effect of increasing [RP] is again a depletion of POS_L in the tRNA condensates, so that [tRNA] in the condensates increases with [RP] for a given value of [POS_L] (Figure 2—figure supplement 7).

In the simulations described so far, the total volume fraction of the system was kept constant by reducing the crowder (CRW) concentration as [POS_L] and [RP] increased. Therefore, the decrease in [tRNA] inside the condensates with increasing [POS_L] could be due to reduced crowder interactions in the condensate environment. To test this further, we reduced [CRW] without changing [POS_L]. Reduced [CRW] also led to reduced [tRNA] in the condensate, but the effect is much smaller than when [CRW] is reduced along with an increase in [POS_L] (Figure 2—figure supplement 8).

In order to construct phase diagrams, simulations of the five-component model phases were carried out at a range of temperatures for selected values of [RP] and [POS_L]. Cluster size distributions were extracted (Figure 3—figure supplements 1–3) and the volume fractions of tRNA in dilute and condensed phases as a function of temperature were determined based on the number of tRNA outside and inside the largest tRNA clusters. The volume of the condensed phase containing the largest tRNA cluster was calculated as described above. The resulting curves (Figure 3) show the typical features of phase diagrams with phase coexistence below critical temperatures T_c of 400–535 K. In the absence of ribosomes, that is, [RP]=0, an increase in [POS_L] lowers T_c and narrows the two-phase regime (Figure 3C). This is consistent with reentrant phase behavior expected for complex coacervation of a binary mixture. However, in the presence of ribosomes, that is, [RP]=55 μM, T_c increased at the same time as the two-phase regime narrowed with increasing [POS_L] (Figure 3D). Moreover, when [POS_L]=180 μM, near the minimum needed for PS, an increase in [RP] slightly decreased T_c (Figure 3A/E), whereas at a higher concentration, that is, [POS_L]=880 μM, T_c increased with increasing [RP] up to a maximum at 55 μM before decreasing (Figure 3B/E). These observations reflect competition between ribosomes and tRNA for interactions with POS_L and more generally highlight the effects of a complex interplay between interactions in non-binary mixtures that are more representative of biological environments than simple binary mixtures.

The results presented so far have focused on multi-component systems that were modeled to reflect the density and distribution of particle sizes and charges in cytoplasmic environments. A key prediction is that PS due to complex coacervation may occur for a wide range of nucleic acids and positively charged proteins simply based on electrostatic complementarity. To test this idea experimentally, we now turn to binary mixtures of globular RNA and positively charged proteins. We focused on the 47-nucleotide J345 Varkud satellite ribozyme RNA, that folds into an approximately globular shape (Bonneau and Legault, 2014) and that was mixed at high concentration with common proteins with positive charges and varying sizes for which we may expect PS: myoglobin (q = +2), trypsin (q = +6), lysozyme (q = +8), lactate dehydrogenase (LDH; q = +4), and alcohol dehydrogenase (ADH; q = +8). Bovine serum albumin (BSA; q = −17, r = 2.58 nm) was added as a control, for which condensate formation is not expected due to its negative charge.

Imaging via confocal microscopy of dye-labeled RNA (Figure 4 and Figure 4—figure supplements 1–6) shows well-defined fluorescent clusters for mixtures of RNA with trypsin, ADH, lysozyme, and LDH, but not for RNA with myoglobin or BSA. The background fluorescence varies significantly with protein. It is especially high for the mixtures with LDH, suggesting that only a fraction of RNA is participating in the condensates and a larger fraction of RNA remained in the dilute phase.

Individual condensates are relatively small, and many appear to have sizes near or below the diffraction limit of the microscope. For RNA-trypsin mixtures, we clearly observe single droplet-shaped condensates of varying sizes that follow roughly an exponential distribution (Figure 4—figure supplement 7). We note that the concentration of Cy3-labeled RNA is only 8 µM, corresponding to 1 in 56 RNA at 0.45 mM total RNA concentration. Therefore, the fluorescent images in Figure 4 are biased toward clusters that contain at least 50 RNA molecules, whereas smaller clusters are imaged incompletely. RNA-LDH condensates appear similar but we did not attempt a quantitative size analysis due to the high background fluorescence of the RNA-LDH sample. Diffusing droplets in the RNA-trypsin mixture merge over the course of 1 min when they come into proximity (Figure 4M and Videos 2 and 3), indicative of liquid behavior inside the condensates.

For other proteins (lysozyme and ADH), we found more complex condensate morphologies (Figure 4), where smaller condensates associate to form larger, irregular-shaped condensates without merging as seen for RNA-trypsin condensates. This suggests that the condensates with these proteins are less liquid-like, although the exact nature of the condensates not involving trypsin is unclear.

To further study the particle size distributions, we carried out dynamic light scattering (DLS) analysis on RNA/lysozyme and RNA/trypsin samples (Figure 5, Figure 5—figure supplements 1–2, and Table 1). The light scattering correlation functions indicate a polydisperse sample that is dominated by very long correlation times up to 1 s (Figure 5). Those long correlation times theoretically correspond to macroscopic-size particles (Stetefeld et al., 2016), but since no such particles were readily visible in the sample, we may conclude that a significant fraction of condensates exhibited very slow diffusion due to surface adsorption. From the correlation function at shorter times, multi-exponential fits suggest particles in two size regimes for RNA-trypsin and in three regimes for RNA-lysozyme. In both cases, the data indicate the presence of 10 nm-scale particles that are consistent with oligomer-size clusters of RNA and protein molecules. Such small clusters between RNA and/or proteins are expected to be present in the dilute phase due to transient associations (Nawrocki et al., 2017; Yildirim et al., 2018; Barhoum and Yethiraj, 2010; Kowalczyk et al., 2011). In both, RNA-trypsin and RNA-lysozyme sample, the DLS analysis suggests the presence of µm-size particles (somewhat smaller for trypsin than for lysozyme). In addition, the DLS data indicate the presence of particles at the light microscopy diffraction limit, around 300 nm, for the RNA-lysozyme system but not for RNA-trypsin mixtures. In fact, the DLS results are qualitatively consistent with the microscopy images and provide additional insights into the particle size distributions at and below the light diffraction limit. However, an exact quantitative interpretation of the DLS results is challenging due to the polydispersity and dynamic nature of our samples and for that reason we also did not attempt to quantify what fraction of particles would be expected in the different size regimes.

To map out a phase diagram, we prepared RNA-trypsin mixtures at various experimentally feasible RNA and protein concentrations. PS required a minimum protein concentration, for example with [RNA]=100 µM, PS was found with [trypsin]=150 µM but not with [trypsin]=50 µM (Figure 6—figure supplement 1). At the same time, PS was lost when RNA concentrations were too high. The resulting phase diagram based on confocal microscopy imaging is shown in Figure 6 in comparison with results from theory that are discussed below.

Förster resonance energy transfer (FRET) experiments also showed a significant increase in FRET efficiencies from 50 to 150 µM (Figure 7A). The comparison between the microscopy and FRET results furthermore establishes that RNA condensates at this RNA concentration can be recognized by FRET efficiencies above 0.26, whereas lower values may indicate a disperse phase. The gradual increase in FRET efficiencies from 0.24 to 0.26 upon increase of trypsin concentrations from 0 to 50 µM is interpreted to result from increasing non-condensate cluster formation (see cluster size distributions in Figure 2—figure supplement 3 at [RP]=0 with increasing protein concentration). However, as in the confocal microscopy experiments, the low concentration of fluorescence-labeled RNA limits the detection of very small clusters where only one or zero of the RNA would be labeled. The FRET results are compared with theoretical predictions (Figure 7B) as detailed below.

We applied circular dichroism (CD) and nuclear magnetic resonance (NMR) spectroscopy with the goal of examining whether the proteins and RNA retain their folded states upon condensate formation. The CD spectra in Figure 4—figure supplement 8 show that there is no substantial change in the shape of the spectrum of trypsin in the presence of the RNA from 225 to 250 nm, which would be expected if the protein had unfolded, as a random coil spectrum has essentially no ellipticity in this wavelength range and the spectrum. The key feature of the RNA spectrum, the broad peak at 250–290 nm, is also retained in the mixture. In fact, the spectrum of the trypsin-RNA mixture appears to be simply a linear combination of the spectra of each of the components measured separately.

NMR spectroscopic analysis of RNA-trypsin and RNA-lysozyme samples at PS-inducing concentrations focused on the structure of the RNA. We observed the characteristic ¹H spectrum of a solution containing only J345 RNA that matches previously matched spectra for the same structure (Bonneau and Legault, 2014; Figure 4—figure supplement 9). In the presence of proteins, the characteristic peaks were retained at the same positions, although with greatly attenuated intensities (Figure 4—figure supplement 9). This was interpreted to mean that only a fraction of RNA remained sufficiently dynamic to achieve rotational averaging via molecular tumbling. From comparing the signal-to-noise ratios, we estimate that about 80% of the RNA is not visible in the RNA-lysozyme sample and 90% is invisible in the RNA-trypsin sample. Since the majority of RNA is expected to be found in the condensates, this suggests that rotational diffusion of individual RNA molecules in the condensates is retarded significantly because the condensates themselves are too large (>100 nm) to tumble on time scales allowing NMR signals to be observed (<100 ns). Moreover, if one assumes that only RNA in the dilute phases remains visible in NMR spectroscopy, the experiments provide an estimate of the fraction of RNA in the dilute vs. condensed phases, that is 20:80 in the presence of lysozyme and 10:90 in the presence of the trypsin for the concentrations studied here. Unfortunately, that also implies that there is no information about the structure of RNA inside the condensates from these experiments.

To compare with the experimental findings, we carried out CG simulations again with the model described above but for binary mixtures of spherical particles equivalent in size and charge to the experimentally studied systems, that is, J345 RNA (q=-46, r = 1.47 nm), myoglobin (q = +2, r = 1.64 nm), trypsin (q = +6, r = 1.81 nm), lysozyme (q = +8, r = 1.54 nm), lactate dehydrogenase (LDH; q = +4, r = 2.68 nm), alcohol dehydrogenase (ADH; q = +8, r = 2.79 nm), and bovine serum albumin (BSA; q = −17, r = 2.58 nm). We also tested a spherical particle equivalent to cytochrome C (q = +11, r = 1.45 nm) which was not studied experimentally because of heme absorption. We observed the formation of condensates at sufficiently high-salt concentrations. With κ = 0.7 (about 20 mM salt), condensates formed with lysozyme, trypsin, LDH, and ADH, but not with cytochrome C, myoglobin, or BSA (Figure 8). Very similar results were also found with an alternative effective charge model (according to Equation 6) as shown in Figure 8—figure supplement 1.

The simulation results qualitatively match the experimental results in terms of which proteins promote PS. Moreover, the fraction of RNA in the dilute phase is higher with lysozyme than with trypsin (32% vs. 26–27% using Equation 5 or Equation 6 from averages over the last 100 µs) in qualitative agreement with the estimates from the NMR experiments. We note that an overall larger fraction of RNA is expected in the dilute phase in the simulations due to an excess concentration of RNA (0.439 mM) compared to the protein concentration (0.350 mM), whereas concentrations of RNA and protein were equal in the NMR experiments (0.150 mM). However, the scale of the simulations is too small to directly compare the condensate sizes with the experimental size distributions.

To generate more extensive phase diagrams, a theoretical model was developed based on the CG simulations. Briefly, the model approximates the chemical potential for either RNA or proteins in condensed and dilute phases based on a decomposition into enthalpy and entropy: µ=∆h-T∆s. The enthalpy is determined from convoluting the coarse-grained interaction potential U(r) (Equation 3) with radial distribution functions g^(r) of RNA-RNA, RNA-protein, and protein-protein interactions in the condensed and dilute phases extracted from CG simulations and scaled by particle densities ρ:(1)Δh=2πρ∫g^(r)U(r)r2dr

The entropy was estimated from the ratio of particle densities ρ between the entire system and either the dilute or condensed phase:(2)∆s=Rlogρtotalρphase

Solutions with respect to the concentrations of protein and RNA in dilute and condensed phases were determined numerically under the conditions that µ_condensed = µ_dilute for either RNA, protein, or both, and that molecular volume packing fractions did not exceed maximum packing densities. Total free energies were then calculated, taking also into account mixing entropy contributions between RNA and protein particles. PS was predicted based on the solution with the lowest free energy.

The theoretical approach is essentially a variation of Voorn-Overbeek theory (Overbeek and Voorn, 1957) for spherical particles. While this theory has seen numerous applications, especially to polyelectrolyte fluids (Priftis et al., 2014; Spruijt et al., 2010), the specific model described here emphasizes an interaction potential that is parameterized based on atomistic simulations of biological macromolecules and that was further tuned to match experimental data. Therefore, the theory is expected to make predictions that are more relevant for globular biological macromolecules than previous studies.

In developing the theory, we found that using the alternative effective charge model according to Equation 6 (Figure 1—figure supplement 8) results in better agreement between theory and experiment and therefore we used this model here. We also use a slightly different Debye-Hückel screening term, that is, κ = 1.17, which gave better agreement between theory and experiment.

Application of the theory predicts that PS should occur for a wide range of protein radii and charges as long as proteins are large enough and carry sufficiently positive charge (Figure 9). More specifically, radius/charge combination corresponding to lysozyme, trypsin, LDH, and ADH are predicted to lead to PS as in the experiments and CG simulations. The radius and charge corresponding to myoglobin is just outside the PS region (Figure 9) again consistent with the lack of PS in the experiment and simulations. The theory also predicts PS for cytochrome C, for which PS was not seen in the simulations.

The theory reproduces an expected temperature dependence of PS with protein-dependent critical maximal temperatures (Figure 8—figure supplement 2). The electrostatic nature of PS also suggests that changes in salt concentrations would affect the findings and the results are indeed sensitive to the value of κ. However, the theoretical treatment is too limited due to the mean-field nature of the Debye-Hückel formalism to make meaningful predictions of salt effects. More specifically, the model is only valid for low ionic strengths and ignores entropic consequences of ion partitioning between condensed and dilute phases that are an important contribution to PS in complex coacervates (Vis et al., 2015).

Using the theory, we constructed concentration-dependent phase diagrams that can be compared with experiment. Figure 6 shows the prediction of the two-phase region for RNA-trypsin in good agreement with the experimental data. Figure 9—figure supplements 1–6 show the phase diagrams for all proteins studied here over a wider range of concentrations. All phase diagrams exhibit reentrant behavior with minimal and maximal protein and RNA concentrations as expected for complex coacervates. It should be noted, though, that the full range of concentrations cannot be realized in practice for all systems due to limited solubilities.

Predictions from the theory also allowed a quantitative interpretation of the FRET experiments. Using the predicted fraction of RNA in the condensates for the RNA-trypsin mixtures at different RNA and protein concentrations (Figure 9—figure supplement 7), FRET efficiencies were estimated (Figure 7B). The theoretical predictions qualitatively reproduce the experimental data with an onset of increased FRET efficiencies due to condensation. Moreover, the gradual increase in FRET efficiencies after condensates form is predicted from a growing number of RNA in the condensed phase as protein concentration increases.

We report phase separation of RNA and proteins based on a universal principle of charge complementarity that does not require polymers or multi-valency via specific interactions. The results are supported by coarse-grained simulations, theory, and experimental validation via microscopy, FRET, and NMR spectroscopy as well as DLS experiments. Condensate formation depends on concentration, size, and charge of the proteins but appears to be possible for typical RNA and common proteins. Simulation results, furthermore, suggest that such phase separation may occur in heterogenous cellular environment, not just between tRNA and cellular proteins but also, in separate condensates, between ribosomes and proteins. Further computational and experimental studies are needed to gain more detailed insights into the exact molecular nature of the condensates described here.

The larger implication of the work presented here is that charge-driven phase separation appears to be a broad phenomenon in biology, particularly because intrinsically disordered proteins and disordered RNA are not required. As a result, cellular cytoplasms could be phase-separated extensively. The observation that tRNA could condense and co-locate near ribosomes suggests a mechanism in which the rate of protein translation is increased because the diffusional wait time for the correct tRNA arriving at the ribosome is decreased. It remains to be explored through in vivo experiments how widely charge-driven phase separation may present itself in cellular environments and what additional factors may modulate it.

CG simulations were run using a modified version of a previously introduced colloid-type spherical model (Mani et al., 2014). In this model, pair interactions consist of a short-range 10–5 Lennard-Jones potential and a long-range Debye Hückel potential according to:(3)Urij=4εσijrij10-σijrij5+Aij+A0κσijrije-rijκσijwhere r_ij is the inter-particle distance, σ_ij is the distance between particles at which the potential is zero, ε is the strength of short-range attraction, A_ij+A₀ describes attractive or repulsive long-range interactions, and κσ_ij is the Debye-Hückel screening length. Only A_ij and σ_ij vary between different particles according to charge and size.

The model was initially parameterized from previously published all-atom simulations of homogeneous mixtures of chicken villin headpiece (‘villin’) (Nawrocki et al., 2017) and subsequently validated with heterogeneous mixtures of protein G, villin, and ubiquitin (Nawrocki et al., 2019a) as summarized in Table 2.

A common value of ε = 4.0 kJ/mol was used for all particles in the short-range 10–5 Lennard-Jones potential. Particle size was taken into account by first determining the radii r_i of spheres with equivalent volumes to the atomistic molecular volumes of a given macromolecule or complex (see Supplementary file 1 for molecules in the cytoplasmic model system). Lennard-Jones parameters σ_i were obtained from the radii r_i according to:(4)σi=2-16∙ri

Pairwise parameters σ_ij were calculated as σ_ij = σ_i + σ_j.

In the long-range Debye-Hückel type potential, a common value of A₀ = 3.0 kJ/mol was used to reflect effective repulsion between charge-neutral, but still polar molecules due to solvation effects. Net charges led to additional repulsive or attractive contributions.

The nominal net charge of a given molecule was converted to effective charges to account for counterion condensation around highly charged macromolecules (Manning, 1978). We distinguish here effectively bound ions that lead to an effectively reduced charge vs. ions that remain mobile in solution and give rise to Debye screening as described below. Generally, the effective charge remains close to nominal charges for small charges, but for highly charged molecules, in particular negatively charged nucleic acids and nucleic acid complexes such as the ribosome, the effective charge is reduced significantly (Diehl and Levin, 2004; Wishnia and Boussert, 1977; Trylska et al., 2004). Charge neutralization is more pronounced with divalent ions such as Mg²⁺ vs. monovalent ions such as Na⁺ or K⁺ (Diehl and Levin, 2004; Templeton and Elber, 2018; Yoo and Aksimentiev, 2012). But the amount of Mg²⁺ ions in biological systems is limited and typically not high enough to neutralize the charge of all the nucleic acids so that additional charge neutralization by monovalent ions remains a significant factor (Akanuma et al., 2014).

Here, we propose the following two expressions to obtain effective charges:(5)qeff,1=signq∙20∙logq20+1(6)qeff,2=signq∙0.6q∙logq2+1

Both empirical formulae give effective charges close to nominal charges for molecules with small charges and highly reduced charges for molecules with large formal charges (Figure 1—figure supplement 8). For a DNA molecule with a nominal charge of −45, atomistic MD simulations suggest effective charges of −10 to −20 under the assumption that ions within 1 nm from the solute surface are effectively bound (Templeton and Elber, 2018; Yoo and Aksimentiev, 2012); at the other end, effective charges between −100 and −800 are estimated for ribosomal particle with a nominal charge of about −4000 based on colloid models (Diehl and Levin, 2004) or electrostatic potential calculations (Trylska et al., 2004), assuming a mixture of divalent and monovalent ions is involved in neutralization. Equations 5 and 6 are both consistent with these estimates. Equation 5 was used initially and screens smaller charges less and larger charges more strongly compared to Equation 6 which was adopted after adjusting the theory to better match experimental results. Neither expression considers ionic concentration as counterion condensation does not depend strongly on concentration (Yoo and Aksimentiev, 2012). Moreover, negatively and positively charged solutes are treated in the same manner even though the binding strength of biological anions (Cl^-) and cations (K⁺, Na⁺, Mg²⁺) to oppositely charged macromolecules may be asymmetric. However, since highly positively charged macromolecules are uncommon, this assumption may not have significant consequences for the systems studied here.

The effective charges calculated either via Equation 5 or Equation 6 were then converted to A_i values:(7)Ai=signqi34qi,eff

Pairwise values were determined as A_ij=A_i*A_j and the factor ¾ was determined by parameterization against the atomistic MD simulations.

The Debye screening length in Equation 3 is κσ_ij, that is, it depends on particle size as in the original model by Mani et al., 2014 in order to better model screening interactions between particles of very different sizes with screened charges that are mostly near the surface. This complicates interpretation of κ in terms of specific salt concentrations. However, as an illustration one may consider a typical smaller protein or RNA with σ_ii of 3 nm where κ = 0.5, 1.0, and 1.5 would correspond to monovalent ion concentrations of 40, 10, and 5 mM, respectively. Note, that these ion concentrations reflect excess ion concentrations after subtracting condensed counterions as those are accounted for in the effective charges according to Equation 5 or 6. Therefore, total ion concentrations in experiment corresponding to a given value of κ in our model should be significantly higher by factors of 2–10 depending on the charges of the considered macromolecules.

MD simulations of the CG model were run up to 1 ms using OpenMM (Eastman et al., 2017) on GPU machines. The interaction potential from Equation 3 was implemented as a custom non-bonded interaction potential via OpenMM’s Python interface. A Langevin thermostat was applied with a temperature of 298 K unless noted otherwise and a friction coefficient of 1 ps⁻¹. As a result, the simulations described here reflect stochastic dynamics of our CG model. A value of κ = 1.5 was used to describe salt screening unless noted otherwise. The timestep for the simulations was set to 1 ps. Frames were saved every 1 ns for simulations of the 100 nm cytoplasm model, every 10 ns for the concentrated protein simulations used for parameterization, and every 100 ns for all other systems. The pairwise potential in Equation 3 was evaluated with a cutoff 49.5 nm. A switching function was applied to be effective at 49 nm. In total, about 270 ms of combined simulation time was run for all systems described here. The total computational cost for these simulations was around 350 GPU days based on timing on a single NVIDIA GeForce GTX 1080 Ti GPU card.

For validation, CG simulations of the systems with the same concentrations as in the atomistic simulations were performed for 100 μs. The CG simulations compared favorably with the atomistic simulations based on pairwise radial distribution functions and cluster size distribution (Figure 1—figure supplement 11).

We constructed a coarse-grained model of Mycoplasma genitalium cytoplasm based on our previously established atomistic model (Yu et al., 2016; Feig et al., 2015). All the macromolecules and complexes were converted to single spherical particles where the particle center initially coincided with the center of mass of the molecules in the atomistic model. Sphere radii were determined as described above based on equivalent volumes, and effective charges were determined from nominal charges according to Equation 5 or Equation 6. A list of all particles with their size, charge, effective charge and concentration is given in Supplementary file 1. The initial system is a cubic box with a size of 100 nm. Additional systems were generated with 200 and 300 nm box sizes by replicating the initial system accordingly. MD simulations were run up to 1 ms as described above.

A representative model of the cytoplasmic system consisted of five components, with an effective charge and volume fraction matching the values in the cytoplasmic system. The components consist of tRNA, ribosome particles (RP), positively charged proteins with small (POS_S) and large (POS_L) sizes and charges and neutral crowders (CRW) (Supplementary file 2). tRNA and RP have the same size and charge as in the full cytoplasmic system. The RP concentration includes RP, that is, complete ribosomes, in the cytoplasmic model as well as additional numbers of ribosomal fragments RR23, R50P RR16 and R30P (Supplementary file 1). The tRNA concentration was adjusted to include all particles with a nominal charge between −100 and −25, except for GroEL, which has a very large size and was not found as part of the tRNA condensates in the cytoplasmic simulations. Concentrations of the positively charged proteins were adjusted to keep the total effective charge of the system close to the cytoplasmic model. The system components were then varied to achieve different concentrations of RP and positively charged particles (Supplementary file 2). Simulations of the five-component system were performed as described above over 1 ms using only effective charges calculated via Equation 5.

Two-component RNA-protein systems were simulated with the same CG model as described above for 1 ms to make predictions for experimentally testable systems. Effective charges were calculated either via Equation 5 or Equation 6. RNA particles were modeled after the 47-nucleotide J345 Varkud satellite ribozyme RNA, that folds into an approximately globular shape (Bonneau and Legault, 2014) with r_RNA = 1.47 nm and q_RNA = −46. Proteins were considered with the following charges and radii: myoglobin (+2, 1.64 nm), trypsin (+6, 1.81 nm), lysozyme (+8, 1.54 nm), cytochrome C (+11, 1.45 nm), lactate dehydrogenase (+4, 2.68 nm), alcohol dehydrogenase (+8, 2.79 nm), and bovine serum albumin (−17, 2.58 nm).

Analysis of the CG simulations was performed for the simulation time between 500 µs to 1 ms unless stated otherwise using in-house code in conjunction with the MMTSB Tool Set (Feig et al., 2004).

An analytical model was constructed to reproduce the phase behavior seen in the simulations and allow a wider range of parameters to be explored. The analysis focuses on a two-component system consisting of a mixture of negatively charged particles R, equivalent to the RNA in the simulations, and particles P, equivalent to proteins, typically with a positive charge. The particles have charges q_R, q_P and radii r_R, r_P. We consider a system of volume V in which R and P particles are present in total concentrations of c_R and c_P. However, we do not include any finite-size effects and therefore the following analysis is scale-independent.

We assume that a phase-separated state is formed with a high-density condensate of volume V_c and a low-density dilute phase of volume V_d = V-V_c, that is, there is no change in the total system volume upon phase separation. The concentrations of R and P particles in the dilute and condensed phases are denoted as c_R,d, c_P,d, c_R,c, and c_P,c. From the concentrations, number densities ρR,d, ρP,d, ρR,c, and ρP,c for R and P particles in the dilute and condensed phases are obtained according to ρ=cmM∙NA1027nm3.

Mass conservation requires that:V-VcρR,d+VcρR,c=VρRand(11)V-VcρP,d+VcρP,c=VρPleaving V_c and ρ_R,c, and ρ_P,c as independent variables to be determined for a given system in case of phase separation.

In general, the following scenarios are possible:

Which of these possible scenarios is assumed, depends on the total free energy of the system.

In order to determine the total free energy of the system, we begin by estimating the chemical potential for a particle either in the dilute (d) and condensed (c) phase from enthalpies and entropies according to:μR,d=ΔhR,d−TΔsR,dµP,d=∆hP,d-T∆sP,dµR,c=∆hR,c-T∆sR,c(12)µP,c=∆hP,c-T∆sP,c

In the following, only terms for the dilute phase are given. The terms for the condensed phase are obtained in an equivalent manner.

The enthalpy terms are decomposed into interactions of R-R, P-P, and R-P pairs:(13)∆hR,d=∆hd, RR+∆hd, RP(14)∆hP,d=∆hd, PR+∆hd, PP

Each pairwise interaction energy is estimated from the coarse-grained interaction potential by assuming a spherically symmetric distribution of particles but modulated as a function of distance according to radial distribution function extracted from simulations for each pair. This amounts to convoluting the pairwise interaction potential U (see Equation 3) with scaled volume- and density-normalized radial distribution functions g^ as follows:∆hd, RR=12ρR,d∫Vg^RR,drURRrd3r(15) =2πρR,d∫0rmaxg^RR,drURRrr2dr(16)∆hd,PP=2πρP,d∫0rmaxg^PP,drUPPrr2dr(17)∆hd, RP=2πρP,d∫0rmaxg^RP,drURPrr2dr(18)∆hd, PR=2πρR,d∫0rmaxg^PR,drUPRrr2drwhere the factor 1/2 corrects for double-counted self-interactions.

Different radial distribution functions were used for dilute and condensed environments (Figure 2—figure supplement 10). The g(r) functions extracted from the simulations were truncated at 20 nm and set to a constant value of 1 for larger radii to remove finite-size artifacts. Although the g(r) functions were determined from simulations with specific sizes r_R,MD, r_P,MD of the R and P particles, other particle sizes could be considered by scaling the radial dependence of the g(r) functions according to the ratios r_R/r_R,MD, r_P/r_P,MD, and (r_R+r_P)/(r_R,MD+r_P,MD) for R-R, P-P, and R-P interactions. The upper integration limit r_max was set to 100 nm for all interactions. At that radius and above, U(r) is negligible for the range of radii and charges considered here. With the fixed integration limit, the integrals in Equations 15 to 18 vary only with the charges and radii of particles R and P, and, thus, they are independent of particle concentrations. Then, the enthalpy contributions can be written as:(19)∆hd, RR=ρR,dxd,RR(20)∆hd,PP=ρP,dxd,PP(21)∆hd, RP=ρP,dxd,RP(22)∆hd, PR=ρR,dxd,PRwhere the x values represent the integrals in Equations 15 to 18 multiplied by 2π.

The entropy term was calculated based on the change of concentration in either dilute or condensed phases relative to the concentration in a fully disperse, non-separated system, which is the total system concentration, that is, for the dilute phase:(23)∆sR, d=RlogcRcR,d=RlogρRρR,d(24)∆sP, d=RlogcPcP,d=RlogρPρP,dwhere R is the universal gas constant. In estimating the entropy for the condensed phase, the finite volumes of the R and P particles were subtracted from the condensed phase volume V_c:(25)∆sR, c=RlogρRρR,c∙1-ρR,cVR+ρP,cVP(26)∆sP, c=RlogρPρP,c∙1-ρR,cVR+ρP,cVPwith the molecular volumes calculated from the radii of the spherical R and P particles:VR=4π3rR3and(27)VP=4π3rP3

Coexistence of the dilute and condensed phases assumes equilibrium, that is:(28)µR,d=µR,c(29)µP,d=µP,c

In scenario (3), both, Equations 28 and 29, have to be satisfied simultaneously. For scenario (4), only Equation 28 needs to be satisfied under the condition that ρ_P,d = 0; and for scenario (5), only Equation 29 has to be satisfied with ρ_R,d = 0.

Solutions in terms of ρ_R,d, ρ_P,d, ρ_R,c, ρ_P,c, and V_c were determined numerically by scanning V_c and solving for the densities in the dilute phase (the densities in the condensed phase follow from Equation 11).

Equation 28 combined with Equations 11, 12, 13, 19, 21, 23, and 25 gives the following:(30)0=µR,d-µR,c =∆hR, d-T∆sR,d-∆hR, c+T∆sR,c =ρR,dxd,RR+ρP,dxd,RP-ρR,cxc,RR-ρP,cxc,RP+TRlogρR,dρR,c∙1-ρR,cVR+ρP,cVP =ρR,dxd,RR+V-VcVcxc,RR+ρP,dxd,RP+V-VcVcxc,RP-VVcρRxc,RR+ρPxc,RP +TRlogVcρR,d-VRρR,dVρR-V-VcρR,d-VPρR,dVρP-V-VcρP,dVρR-V-VcρR,d(31) =fRρR,d,ρP,d,Vc

An analogous function fPρR,d,ρP,d,Vc is obtained from Equation 29. There is no analytical solution, but fRρR,d,ρP,d,Vc=0 and fPρR,d,ρP,d,Vc=0 can be solved via the Newton-Raphson method given V_c and either ρ_P,d or ρ_R,d.

For scenario (4), fRρR,d,ρP,d,Vc=0 was solved for different values of V_c and ρ_P,d = 0; for scenario (5), fPρR,d,ρP,d,Vc=0 was solved for values of V_c and ρ_R,d = 0. For scenario (3), ρ_R,d was scanned as well and the value of ρ_P,d was determined for given values of V_c and ρ_R,d by first solving fRρR,d,ρP,d,Vc=0. The resulting value of ρ_P,d was then used with V_c to solve fPρR,d,ρP,d,Vc=0 for a refined value of ρ_R,d.

Mathematically possible solutions include cases where the volume fractions in the cluster exceed what is physically realistic inside the condensed state. In order to exclude such solutions, it was required that the combined macromolecular volume in the condensed phase is less than 30% of the total volume of the condensed phase, that is,:(32)ρR,cVR+ρP,cVP<0.3

We note that most final solutions were found at the 30% vol fraction limit, since the theory did not directly account for volume exclusion between individual molecules and found a gain in energy at higher particle densities. However, similar results were obtained with maximal macromolecular volume fractions according to Equation 32 in a range of 20–40%. The value of 30% was ultimately arrived at by optimal agreement between theory and experiment for the concentration-dependent phase separation between RNA and trypsin shown in Figure 6.

The total system energy is calculated according to:(33)∆G=µR,d∙V-VcρR,d+µR,c∙VcρR,c+µP,d∙V-VcρP,d+µP,c∙VcρP,c-TSmixwhere S_mix is the overall mixing entropy according to the ratio of particles R and P in the dilute and condensed phases according to:(34)Smix=Smix,d+Smix,c(35)Smix,d=RV-VcρR,dlogρR,dρR,d+ρP,d+ρP,dlogρP,dρR,d+ρP,d(36)Smix,c=RVcρR,clogρR,cρR,c+ρP,c+ρP,clogρP,cρR,c+ρP,c

For the five scenarios described above, total free energies were then calculated as follows:

(1) Disperse:(37)∆G1=µR,disperse∙VρR+µP,disperse∙VρP-TRVρRlogρRρR+ρP+ρPlogρPρR+ρPwhere µRP,disperse were calculated according to Equations 12 to 18 using RDFs from the disperse phase extracted from our molecular dynamics simulations before condensates started to form.

(2) Condensed:(38)∆G2=µR,c∙VρR+µP,c∙VρP-TRVcρRlogρRρR+ρP+ρPlogρPρR+ρP

(3) R and P in phase coexistence:(39)∆G3=µR,c∙VρR+µP,c∙VρP -TRV-VcρR,dlogρR,dρR,d+ρP,d+ρP,dlogρP,dρR,d+ρP,d -TRVcρR,clogρR,cρR,c+ρP,c+ρP,clogρP,cρR,c+ρP,csince µ_R,c = µ_R,d and µ_P,c = µ_P,d.

(4) R in phase coexistence, ρ_P,d = 0:(40)∆G4=µR,c∙VρR+µP,c∙VρP-TRVcρR,clogρR,cρR,c+ρP,c+ρP,clogρP,cρR,c+ρP,c

(5) P in phase coexistence, ρ_R,d = 0:(41)∆G5=µR,c∙VρR+µP,c∙VρP-TRVcρR,clogρR,cρR,c+ρP,c+ρP,clogρP,cρR,c+ρP,c

The scenario with the overall lowest free energy was then considered to be the predicted state.

A program implementing this model is available at http://github.com/feiglab/phasesep; copy archived at swh:1:rev:24890516a822b917b76a2730ced19839acbaec3d

The J345 RNA sequence was synthesized and deprotected by Dharmacon (Horizon Discovery Group), both with and without Cy3 or Cy5 on the 3’ end. The 47-base sequence is GCAGCAGGGAACUCACGCUUGCGUAGAGGCUAAGUGCUUCGGCACAGCACAAGCCCGCUGCG.

All measurements were made using the buffer used by Bonneau and Legault for structure determination of this sequence, 10 mM sodium cacodylate (pH 6.5), 50 mM NaCl,. 05% sodium azide, 5 mM MgCl₂. Equine liver trypsin, equine alcohol dehydrogenase, bovine lactic dehydrogenase, equine myoglobin, hen egg lysozyme, and bovine serum albumin were obtained from Sigma-Aldrich and used without further modification.