Phase Separation in Mixtures of Prion-Like Low Complexity Domains is Driven by the Interplay of Homotypic and Heterotypic Interactions

Prion-like low-complexity domains (PLCDs) are involved in the formation and regulation of distinct biomolecular condensates that form via coupled associative and segregative phase transitions. We previously deciphered how evolutionarily conserved sequence features drive phase separation of PLCDs through homotypic interactions. However, condensates typically encompass a diverse mixture of proteins with PLCDs. Here, we combine simulations and experiments to study mixtures of PLCDs from two RNA binding proteins namely, hnRNPA1 and FUS. We find that 1:1 mixtures of the A1-LCD and FUS-LCD undergo phase separation more readily than either of the PLCDs on their own. The enhanced driving forces for phase separation of mixtures of A1-LCD and FUS-LCD arise partly from complementary electrostatic interactions between the two proteins. This complex coacervation-like mechanism adds to complementary interactions among aromatic residues. Further, tie line analysis shows that stoichiometric ratios of different components and their sequence-encoded interactions jointly contribute to the driving forces for condensate formation. These results highlight how expression levels might be tuned to regulate the driving forces for condensate formation in vivo. Simulations also show that the organization of PLCDs within condensates deviates from expectations based on random mixture models. Instead, spatial organization within condensates will reflect the relative strengths of homotypic versus heterotypic interactions. We also uncover rules for how interaction strengths and sequence lengths modulate conformational preferences of molecules at interfaces of condensates formed by mixtures of proteins. Overall, our findings emphasize the network-like organization of molecules within multicomponent condensates, and the distinctive, composition-specific conformational features of condensate interfaces.


Introduction
Biomolecular condensates are membraneless bodies that provide spatial and temporal control over cell signaling and cellular responses to various stresses 1,2,3,4,5,6,7 . Functional roles for condensates have been implicated in transcriptional regulation 8,9,10,11,12,13,14 , cytosolic and nuclear stress responses 3,15,16,17,18,19,20,21 , tra cking of cellular components 22,23 , RNA regulation and processing 24,25,26,27,28 , mechanotransduction 29,30,31 , and protein quality control 32,33,34,35,36 . The working hypothesis, based on a growing corpus of data, is that condensates form via spontaneous and driven phase transitions of networks of multivalent biomacromolecules 1,37,38 . The relevant processes involve a coupling of associative and segregative macro-or microphase transitions 39 . These include processes such as phase separation coupled to percolation (PSCP) 39,40,41 and complex coacervation 42,43,44 . Multivalent proteins that scaffold and drive phase transitions encompass different numbers and types of oligomerization and substrate binding domains 41 . Most, although not all protein scaffolds also feature intrinsically disordered regions (IDRs) that drive or modulate phase separation of protein scaffolds through a blend of homotypic and heterotypic interactions 45 . Here, homotypic, and heterotypic interactions refer to intermolecular interactions between the same versus different molecules, respectively. This concept can be extended to distinguish interactions between the same versus different motifs on molecules 46 . driving forces for PSCP that can span several orders of magnitude 54 . Condensates such as stress granules and P bodies, and nuclear bodies such as nucleoli house ~40 or more known proteins with distinct PLCDs (Figure 1B). Within a condensate, the PLCDs can be quite different from one another. The sequence lengths of PLCDs can vary by a factor of four, with most PLCDs being ~150 residues long ( Figure 1C). Therefore, the driving forces for condensate formation in mixtures of PLCDs are of direct relevance for understanding how different PLCDs work together or in opposition to in uence condensate formation, internal organization, and interfacial properties.
Here, we report results from studies of condensate formation in mixtures of PLCDs from two proteins, hnRNPA1 62 and FUS 63 . These two proteins are components of stress granules, paraspeckles, and other condensates 64 . Hereafter, we refer to the two PLCDs as A1-LCD and FUS-LCD, respectively. Mutations within both PLCDs are associated with the formation of aberrant stress granules in the context of Amyotrophic Lateral Sclerosis (ALS) 20,62,65 . The compositional differences between human versions of the two PLCDs are summarized in Figure 1D. These differences translate to a positive net charge per residue (NCPR) for A1-LCD and a negative NCPR for FUS-LCD. In addition to being compositionally different, the two sequences have different lengths, with FUS-LCD being ~1.6 times longer than A1-LCD.
However, despite being shorter than FUS-LCD, the driving forces for PSCP are stronger for A1-LCD when compared to FUS-LCD 55 . This is gleaned from comparative measurements of the temperature-and solution-condition-dependent saturation concentrations (c sat ) for phase separation. Here, we focused on understanding how the interplay of homotypic and heterotypic interactions in uences the driving forces for condensate formation in mixtures of A1-LCD and FUS-LCD molecules. For this, we deployed a recently developed coarse-grained model to simulate temperature-dependent phase transitions of mixtures of A1-LCD and FUS-LCD 55 . In the model, each PLCD residue is modeled as a single bead with distinct inter-bead interactions. The simulations were performed using LaSSI 51 , which is a lattice-based Monte Carlo simulation engine.
We nd that heterotypic interactions are the dominant contributors to condensate formation in 1:1 mixtures of A1-LCD and FUS-LCD. This leads to two-component phase diagrams with distinctive shapes and slopes for tie lines. We demonstrate the accuracy of our computational results using in vitro experiments that leverage a novel analytical HPLC-based method for measuring phase diagrams in multi-component mixtures 66 . Further, through additional simulations, we uncover general rules for how the interplay between homotypic and heterotypic interactions in uences the internal organization and interfacial properties of multicomponent condensates.

Results
Heterotypic interactions enhance the driving forces for phase separation of mixtures of A1-LCD and FUS-LCD: We performed a series of simulations for mixtures of A1-LCD and FUS-LCD. The sequences of A1-LCD and FUS-LCD are shown in Fig. 2A. The total protein concentrations were xed in the simulations, and the ratios of FUS-LCD-to-A1-LCD were varied from one set of simulations to another. Treating the mixture as a system with one type of macromolecule, we computed binodals in the plane of total protein concentration along the abscissa and temperature along the ordinate. This approach to depicting phase boundaries for mixtures parallels that of Elbaum-Gar nkle et al., 67 and Wei et al., 68 . Based on the computed binodals (Fig. 2B) we predict that a 1:1 mixture of FUS-LCD and A1-LCD undergoes phase separation at a lower total protein concentration than either FUS-LCD or A1-LCD on its own. This is suggestive of the presence of heterotypic interactions that enhance phase separation.
To better understand the interplay between homotypic and heterotypic interactions, we recast the results as a two-dimensional phase diagram at a xed temperature, where each axis is de ned by the concentration of one protein. In Fig. 2C we show expectations for the dilute arms of two-dimensional phase diagrams for a system of two components that undergo co-phase separation from solution.
At the temperature of interest, we shall denote the saturation concentration of A1-LCD in the absence of FUS-LCD as c sat,A1 . Likewise, at the same temperature, the saturation concentration of FUS-LCD in the absence of A1-LCD is denoted as c sat,FUS . If the dilute arm of the two-dimensional phase diagram is a straight line joining the individual c sat values, then the contributions to phase separation of the mixture of PLCDs are purely additive. In this case, homotypic and heterotypic interactions make equivalent contributions to the driving forces phase separation and there is no cooperativity in the system. As a result, the total protein concentration in the dilute phase of the two-phase system can be written as: c dilute = ac sat,A1 + (1 -a)c sat,FUS , where a is the fraction of A1-LCD molecules in the system; when a = 1, c dilute = c sat,A1 , and when a = 0, c dilute = c sat,FUS . If the computed or measured dilute arm of the two-dimensional phase diagram is concave, then heterotypic interactions enhance the driving forces for phase separation. In this scenario, there is positive cooperativity, whereby phase separation is enhanced by mixing the two components. Conversely, if the dilute arm of the two-dimensional phase diagram is convex, then heterotypic interactions weaken the driving forces for phase separation. This is a manifestation of negative cooperativity, whereby phase separation is weakened in a mixture of the two components. The degree to which the driving forces for phase separation are enhanced or weakened depends on the composition of the mixture i.e., the stoichiometric ratio of the two components.
The results shown in Fig. 2B were recast by xing the simulation temperature and plotting the computed dilute phase concentrations of A1-LCD and FUS-LCD for different stoichiometric ratios (Fig. 2D). We observe a concave shape, indicating an enhancement of phase separation via heterotypic interactions.
FUS-LCD is negatively charged, and A1-LCD is positively charged. Accordingly, we reasoned that complementary electrostatic interactions are likely to enhance the driving forces for co-phase separation.
These interactions are likely to contribute in addition to heterotypic aromatic sticker interactions. To test this hypothesis, we performed simulations for mixtures of FUS-LCD and a variant of A1-LCD denoted as A1-LCD + 12D 54 . In this variant, twelve Asp residues were substituted across the sequence, replacing extant spacers, giving rise to a sequence with a net negative charge. Although the aromatic sticker interactions remain unchanged, the electrostatic interactions should be weakened. We reasoned that this would generate a dilute arm with a more convex shape, and this is precisely what we observe (Fig. 2E).
Results from in vitro measurements are in accord with computational predictions Next, we measured co-phase separation in aqueous mixtures of A1-LCD and FUS-LCD molecules (see Materials and Methods, Supplemental Material and Figure S1). Diffraction-limited uorescence microscopy shows that the PLCDs co-localize into the same condensates upon phase separation (Fig. 3A). This is true for all concentration ratios studied. We then used our recently described analytical high-performance liquid chromatography (HPLC) method 66 to determine dilute and dense phase concentrations of both species in mixtures with different mass concentration ratios of the two PLCDs.
As in the simulations, a 1:1 mixture of A1-LCD and FUS-LCD undergoes phase separation at a lower concentration than either protein on its own (Fig. 3B). Recasting the results as a two-dimensional phase diagram shows the same patterns as the simulations, where the dilute arm of the phase boundary has a concave shape (Fig. 3C). These results suggest that complementary electrostatic interactions contribute to enhance co-phase separation of FUS-LCD and A1-LCD. In contrast, and in agreement with computational predictions, the dilute arm of the measured phase diagram for mixtures of the FUS-LCD and A1-LCD + 12D system has a more convex shape (Fig. 3D), suggestive of negative cooperativity due to electrostatic repulsions.
Notably, although mixtures of FUS-LCD and A1-LCD + 12D have a weakened driving force for phase separation relative to FUS-LCD and A1-LCD, the two proteins still co-localize into a single dense phase as gleaned from diffraction-limited microscopy (Fig. 3E). We measured how the total dilute phase protein concentration varied as a function of salt concentration and the ratio of FUS-LCD-to-A1-LCD (Fig. 3F). Increasing the salt concentration from 150 mM NaCl to 300 mM NaCl resulted in decreased c sat values for either protein on its own, but an increased c dilute value for a 1:1 mixture. We rationalize this as follows: In solutions with only one type of PLCD, all proteins have the same sign and magnitude of charge.
Accordingly, the intermolecular electrostatic interactions will be repulsive. Increasing the salt concentration screens repulsive interactions, thereby lowering the intrinsic c sat values. In contrast, attractive electrostatic interactions in the 1:1 mixture will be weakened due to charge screening, and this leads to an increase in c dilute for the mixture as salt concentration is increased. These results suggest an interplay between homo-and heterotypic aromatic interactions with intermolecular electrostatic interactions in mixtures of FUS-and A1-LCD.
Tie lines and their slopes help uncover the relative contributions of homotypic versus heterotypic interactions to phase separation in mixtures of PLCDs In a system that forms precisely two coexisting phases, the generalized tie simplex 39 for a given composition will be a tie line. The tie line is a straight line that connects points of equal chemical potentials and osmotic pressures on the dilute and dense arms of phase boundaries 69 . The signs and magnitudes of the slopes of tie lines provide quantitative insights regarding the interplay between homotypic versus heterotypic interactions 69 .
In a system with macromolecules A and B that undergo co-phase separation from a solvent, we can map the full phase diagram on a plane with the concentration of A being the variable along the abscissa, and the concentration of B being the variable along the ordinate (Fig. 4). In these titrations, we x the solution conditions including the temperature. The slope of the tie line will be unity if heterotypic interactions are the dominant drivers of phase separation or if the homotypic interactions are equivalent to each other while also being on par with the heterotypic interactions. In the A-B mixture, a tie line with a slope that is less than unity will imply that homotypic interactions of component A are the main drivers of phase separation. Conversely, if tie lines have slopes that are greater than unity, then the homotypic interactions among B-molecules are the stronger drivers of phase separation.
If the total concentration of the mixture is c tot , we can extract tie lines by joining three points corresponding to c dilute , c tot , and c dense . Here, c dilute and c dense are the macromolecular concentrations in the coexisting dilute and dense phases, respectively. A single line connects the three points if and only if phase separation gives rise to precisely two coexisting phases. To test that this is the case, we plotted two sets of lines for each mixture viz., one that joins c dilute and c tot and another that joins c tot and c dense .
If the slopes of the two lines are identical or nearly identical, then the system forms two coexisting phases upon phase separation. The tie lines computed in this manner are shown in Fig. 5A and 5B for simulations of different ratios of A1-LCD-to-FUS-LCD. The slopes of the two sets of tie lines are essentially equivalent to one another (Fig. 5C). We repeated this analysis for measured phase diagrams of the corresponding system in vitro and nd similar values for the slopes of the tie lines ( Fig. 5D-F).
In computations and measurements, the 1:1 mixtures of A1-LCD and FUS-LCD have tie lines with slopes that are essentially one. Therefore, phase separation involves changes in concentrations from the dilute to the dense phase that are similar for both sets of molecules. This suggests that heterotypic interactions are the dominant drivers of phase separation in 1:1 mixtures. Conversely, in a mixture with a 3:1 ratio of A1-LCD-to-FUS-LCD, the slope of the tie line is signi cantly greater than one. The implication is that homotypic interactions among A1-LCD molecules play a dominant role in driving phase separation along this tie line. Likewise, in a mixture with a 1:3 ratio of A1-LCD-to-FUS-LCD molecules, the slope of the tie line is signi cantly less than one, highlighting the importance of homotypic interactions among FUS-LCD molecules along this tie line.
Next, we asked if the tie lines for the mixture of FUS-LCD and A1-LCD + 12D would differ from those of FUS-LCD and A1-LCD. FUS-LCD has a lower intrinsic c sat than A1-LCD + 12D. First, we plotted the low concentration arms obtained from in vitro measurements (Fig. 6A). Following Qian et al. 69 , the agreement between the two sets of tie lines in Fig. 5F suggests that we can infer the slope of the whole tie line based on the slope of the line that connects c dilute to c tot . Unlike the mixture of A1-LCD and FUS-LCD, we nd that for the FUS-LCD and A1-LCD + 12D mixture, the slope of the tie line for the 1:1 mixture is signi cantly less than one (Fig. 6B). This suggests that along this tie line, the homotypic interactions among FUS-LCD interactions are major drivers of phase separation. This effect is further enhanced for the mixture with a 3:1 ratio of FUS-LCD-to-A1-LCD + 12D. Finally, for a mixture with a 1:3 ratio of FUS-LCD-to-A1-LCD + 12D, the slope of the tie line is closer to unity, indicating that along this tie line, the interplay among heterotypic interactions, homotypic FUS-LCD interactions, and homotypic A1-LCD + 12D interactions, are relatively well-balanced (Fig. 6B). Taken together, we nd that when there are repulsive interactions between molecules in the mixture, the co-phase separation is driven by molecules that have stronger intrinsic driving forces for phase separation.
How are condensate interfaces in uenced by the balance of homotypic versus heterotypic interactions?
Next, we followed recent approaches 55 and computed radial density distributions, and logistic ts to these distributions 70 for a 1:1 mixture of A1-LCD and FUS-LCD (Fig. 7A). We nd that the concentration of A1-LCD is greater than that of FUS-LCD in the dense phase and less than that of FUS-LCD in the dilute phase. This is because the intrinsic c sat of A1-LCD is less than that of FUS-LCD. The thickness of the interfacial region predicted by the logistic t to the radial density distribution for FUS-LCD is larger than that of A1-LCD. This is because FUS-LCD is ~ 1.6 times longer than A1-LCD.
We analyzed the ensemble averaged radius of gyration, R g , of each species, normalized by , where N is the protein length (Fig. 7B). In the dense phase, each species has a similar value of R g / , because the dense phase is an equivalently better solvent for both species when compared to the dilute phase, where the solvent is relatively poor for both species 55 . In the dilute phase, R g / is slightly larger for FUS-LCD than for A1-LCD, and this is in accordance with the weaker homotypic interactions among FUS-LCD molecules. At the interface, both species show the predicted chain expansion 55 .
Next, we probed the internal organization of A1-LCD and FUS-LCD molecules with respect to one another.
To quantify this, we deployed a crosslinking parameter L i−j to determine the relative likelihood that protein species i interacts with species j, given a xed total number of proteins from each species in the condensate. Details of how L i−j are computed are described in the Supplemental Material.
Values of L i−j that are close to one indicate that species i interacts with species j in a manner that is proportional to the number of i and j proteins in the condensate i.e., the proteins are randomly mixed. In contrast, values of L i−j that are greater than or less than one point to a non-random organization of different molecules within the condensate. If L i−j > 1 then species i is more likely to interact with species j than would be expected from random mixing. Conversely, if L i−j < 1, then species i is less likely to interact with species j than would be expected for a random mixture.
We calculated L i−j for simulations of FUS-LCD and A1-LCD at 1:1 ratios and a series of temperatures below the critical phase separation temperature and found the following trends: In general, FUS-LCD molecules are more likely to interact with A1-LCD molecules than with other FUS-LCD molecules. In contrast, A1-LCD molecules show relatively equal preferences for interacting with FUS-LCD or A1-LCD molecules. As the temperature increases, both FUS-LCD and A1-LCD molecules show an increased preference for interacting with A1-LCD molecules. This is because the critical temperature for FUS-LCD is lower than the critical temperature of A1-LCD 55 . The results outlined in Fig. 7 set up predictions for First, we set all homotypic and heterotypic interactions to be equivalent. In this scenario, we nd that L i−j equals one for all species combinations and at all temperatures, suggesting A and B are randomly mixed within the condensate (Fig. 8A). However, if heterotypic interactions are stronger than homotypic interactions, then we observe an increase in A-B contacts, indicating that chains within the condensate are organized to maximize heterotypic interactions (Fig. 8B). Setting the homotypic interactions to be stronger than heterotypic interactions give rise to internally demixed condensates with an A-rich region forming an interface with a B-rich region (Fig. 8C). Similar ndings were recently reported by Welles et al., 71 . Finally, adding asymmetry to the energetics by keeping A-B and B-B interactions equivalent, but weakening A-A interactions leads to a strong preference for A molecules to interact with B molecules over other A molecules, whereas B molecules show indifference at low temperatures, but a preference for other B molecules at high temperatures (Fig. 8D). These observations are akin to those made for mixtures of A1-LCD and FUS-LCD molecules (Fig. 7C).
The lengths of macromolecules will in uence the material properties 72 and interfacial features of condensates 55 . Therefore, we asked how sequence length affects internal and interfacial features of condensates using simulations of homopolymers of various lengths. We performed simulations with equivalent mass concentrations for mixtures of homopolymers of lengths 150 (H150) and 300 (H300) where all homotypic and heterotypic interactions were set to be equal (Fig. 9A). Calculated phase diagrams show that H300 has a lower c sat than H150 (Supplemental Material, Figure S2A). In addition, analysis of L i−j for the mixture shows that there is a strong preference for interacting with H300 above T 56 in reduced units. Beyond this temperature, H150 no longer phase separates on its own ( Figure S2A). Radial density distributions show that H300 has a stronger preference for the dense phase and a weaker preference for the dilute phase when compared to H150. Logistic ts of the radial densities, used to determine interfacial regions corresponding to each homopolymer, show that the interfacial region is wider for H300 than for H150. We also analyzed the normalized radius of gyration and found that H300 has a greater degree of chain expansion in the interface when compared to H150. This is true even though it is more compact than H150 in the dense and dilute phases. Along the radial coordinate from the center of the condensate, the location within the interface that corresponds to the peak of chain expansion of H300 is shifted closer to the dilute phase when compared to the corresponding peak for H150.
To assess the extent to which our ndings re ect the stronger intrinsic driving forces for phase separation of longer homopolymers, we titrated the strengths of homotypic and heterotypic interactions of H300 and H150 so that the c sat values of H300 and H150 were equivalent ( Fig. 9B and Supplemental Material, Figure S2B). In mixtures of equivalent mass concentrations, the mixture of H150 and H300 molecules forms apparent core-shell structures that allow both H150 and H300 to maximize their interactions with H150 molecules, as shown by an analysis of L i−j (Supplemental Material, Figure S2B). In turn, the width of the interfacial region de ned by H300 shrinks when compared to the results shown in Fig. 9A. The scaled R g values for H150 and H300 are similar in the dilute and dense phases. However, the locations and heights of the peaks of chain expansion show the same pattern as in Fig. 9A.
Finally, we asked if interfacial features vary when we extend to three-component systems. In simulations that use equal mass concentration of homopolymers of lengths 200, 300, and 400, with all interaction strengths being equal, we nd that interfacial organizational features follow rules that were uncovered for binary mixtures. This is evident in comparisons of results shown in Fig. 9C and Supplemental Material, Figure S2C to those shown in Fig. 9A. Longer polymers make up more of the outer regions of the interface. They also show a greater level of expansion at the interface, and the peak of chain expansion shifts toward the dilute phase relative to that of the shorter polymers.

Discussion
Condensates are characterized by distinctive macromolecular compositions 73,74,75,76,77 . It is thought that macromolecular compositions, delineated into categories of molecules known as scaffolds, clients, regulators, crowders, and ligands, contribute to the functions of condensates 78, 79, 80, 81, 82, 83 . Here, we explored how the interplay between homotypic versus heterotypic interactions of PLCDs in uence the driving forces for phase separation. The FUS-LCD and A1-LCD are intrinsically disordered scaffolds with at least cursorily similar compositional biases. Our results set the stage for understanding how control over compositions in complex mixtures might in uence condensate formation of bodies such as 40S hnRNP particles 84 or stress granules. There is also growing interest in understanding how the interplay of homotypic and heterotypic interactions impact the dynamically controlled 85 or purely thermodynamically in uenced miscibility of condensates formed in multicomponent systems 71,86,87,88 .
Key ndings that emerge from our investigations are as follows: In 1:1 binary mixtures of PLCDs, heterotypic interactions can enhance the driving forces for phase separation. This enhancement comes, in part, from complementary electrostatic interactions, thus suggesting a role for complex coacervation 43,44 even in systems where fewer than 10% of the residues are charged. This highlights additional roles for spacer residues, which determine the solubility and the extent of coupling between associative and segregative transitions for individual PLCDs, while they determine the electrostatic complementarity of mixtures of PLCDs. Within condensates, the concentration of macromolecules will be above the overlap threshold 55 . As a result, the interplay between homotypic and heterotypic interactions will also contribute to the internal organization of molecules within condensates 89 . Clearly, physicochemical control of cellular biomolecular condensates, which typically contain dozens of distinct macromolecular species, will involve a complex and dynamic interplay of different types of interactions between macromolecules.
Our ndings are conceptually reminiscent of the results of Espinosa et al., who used ultra-coarse-grained patchy colloid models for simulations of complex mixtures of associative macromolecules 90 .
If phase separation gives rise to precisely two coexisting phases, then a 1-simplex 39 or tie line will connect the two points that de ne the coexisting phases. Each phase is de ned by concentrations of macromolecules and solution components that enable equalization of osmotic pressures and chemical potentials across the phases. The slopes of tie lines tell us about the relative contributions of homotypic and heterotypic interactions to phase separation for speci c stoichiometric ratios of the macromolecules that drive phase behavior. Using a tie line analysis, we nd that FUS-LCD and A1-LCD engage in strong heterotypic interactions when present in equal mass concentration ratios. However, when present in unequal ratios, homotypic interactions involving the predominant species become the primary driving force for phase separation. This behavior changes for FUS-LCD and A1-LCD + 12D mixtures. For this system, phase separation in mixtures of equal mass concentration ratios is driven mainly by homotypic interactions among FUS-LCD molecules. Tilting the balance of interactions toward heterotypic interactions can be achieved by increasing the ratio of A1-LCD + 12D to FUS-LCD. These results highlight the central importance of expression levels and stoichiometric ratios in mixtures of macromolecules 91 .
We nd that molecules within condensates are organized to maximize the most favorable interactions, be they homotypic or heterotypic. Accordingly, segregative transitions i.e., phase separation transitions drive the formation of condensates wherein associative interactions, i.e., physical crosslinking among favorably interacting molecules, are maximized 39 . Further, molecules that engage in stronger interactions are more likely to be in the core of the condensate as opposed to in the interface. These ndings highlight the crucial importance of the coupling of segregative and associative transitions as joint drivers of condensate formation and determinants of condensate internal structures.
In agreement with prior work 55 , we nd that polymers at the condensate interface are likely to adopt expanded conformations. We probed the effects of polymer lengths and found that the degree of expansion increases with polymer length. The strengths of homotypic versus heterotypic interactions and polymer length contribute to preferential localization of molecules at the interface, the degree of chain expansion at the interface, and the thickness of the interface. All polymers show an increase in their R g values near the condensate interface, and the precise location of the maximal expansion increase is controlled by the species-speci c interface. In general, interfaces de ned by longer polymers are thicker than those formed by shorter polymers. Interfacial thickness can be altered by modulating the strengths of homotypic and heterotypic interactions, allowing for ner control over spatial localization of each polymer species. The physical properties of interfaces 92 are likely to contribute to capillary forces of condensates 93 , interactions between condensates and emulsi ers in vivo 94 or in vitro 95 , and chemical reactions that are likely controlled at interfaces 96, 97 .
Overall, our ndings highlight the complex and designable properties 98, 99, 100 of biomolecular condensates formed by mixtures of PLCDs. These ndings set the foundations for dissecting the phase behaviors and properties of condensates formed by n-nary mixtures of multivalent macromolecules. Our work highlights the promise of using simulations for modeling and designing phase behaviors in multicomponent systems 60, 87, 100, 101, 102 .

Methods
Identifying condensate-associated prion-like lowcomplexity domains (PLCDs) To identify the number of proteins with PLCDs that have been located within different cellular condensates, we used DrLLPS 103 , a database of condensate-associated proteins. We culled all proteins associated with condensates with at least 40 known components. We then used the algorithm PLAAC 104 to identify proteins in the curated database that contain a PLCD. Within PLAAC, we speci ed a 50/50 weighting of background probabilities between H. sapiens and S. cerevisiae. This resulted in 89 distinct condensate-associated proteins with PLCDs. The sequences of these PLCDs were used for analyses that led to the results in Fig. 1A-C.

Monte Carlo simulations using LaSSI:
The coarse-grained model uses one lattice bead per amino acid residue and treats vacant sites as components of the solvent. In this way, solvent is afforded space in the system, but we do not parameterize any explicit interactions between solvent occupied sites and any other sites. Monte Carlo moves are accepted or rejected based on the Metropolis-Hastings criterion such that the probability of accepting a move is the min[1,exp(-∆E/k B T)] where ∆E is the change in total system energy of the attempted move and k B T is the thermal energy, or simulation temperature. The energetic model used is the same as that described by Farag et al., 55 . Here, we perform simulations of multi-component systems.
We modi ed the mean-eld electrostatic model, which was originally based on a single-component system. Instead of using a single NCPR value to determine the effect of electrostatics on a pairwise beadbead interaction, we use the average NCPR of the chains to which the beads belong. The full set of Monte Carlo moves used in each type of simulation performed in this work are shown in Table S1 of the Supplemental Material. All the move sets and frequencies of moves are as described previously 55 .
We performed multi-chain LaSSI simulations at various temperatures. The simulation temperatures are referenced in terms of units where we set k B = 1. Simulations involving FUS-LCD and A1-LCD used a 120⋅120⋅120 cubic lattice with periodic boundary conditions. Those involving homopolymers used a 150⋅150⋅150 cubic lattice with periodic boundary conditions. To speed up condensate formation, the simulations were initialized in a smaller 35⋅35⋅35 cubic lattice, which allows for signi cantly faster equilibration processes. The number of chains in each simulation was chosen to keep the total volume fraction of beads as close as possible to 0.016. Details of the sequences used in this study are shown in Supplemental MaterialTable S2.
A total of 3⋅10 10 MC steps was deployed for multi-chain simulations at each of the simulation temperatures. Simulations typically equilibrated after about 2⋅10 9 steps, as determined by a plateauing of the total system energy. To be conservative, all simulation results were analyzed after the halfway point of 1.5⋅10 10 steps. Multi-chain simulations were performed with ve replicates, with each simulation initiated by a distinct random seed.

Details of constructs used for in vitro measurements
We used the WT low-complexity domain (LCD) (residues 186-320) of human hnRNPA1 (UniProt: P09651; Isoform A1-A), in which the M9 nuclear localization signal had been mutated, substituting the PY motif with GS (referred to as A1-LCD); we also used a variant in which we further substituted several glycine and serine residues with aspartate residues increasing the aspartate content by 12 residues (A1-LCD + 12D ); and third, we used the WT LCD (residues 1-214) of FUS (UniProt: P35637). The gene sequences were synthesized as previously described 54 , including a gene sequence coding for an N-terminal TEV cleavage site followed by the protein coding sequence of interest. The protein sequences are shown in Supplemental MaterialTable S2. The three proteins were expressed and puri ed as previously described 53,54,66 , and the puri ed proteins were stored in 6 M GdmHCl (pH 5.5), 20 mM MES at 4°C until they were buffer exchanged into phase separation buffer.     Schematic depicting how to interpret tie lines for a system of two components that undergo co-phase separation from the solvent to form precisely two coexisting phases.