1. Author Response:

    Reviewer #1 (Public Review):

    The gist of this work is that the simple concept of a solubility product determines a threshold for phase separation, thereby enabling buffering even in systems where phase separation is driven by heterotypic interactions. The solubility product or SP is determined by the number of complementary interaction sites and the coordination number i.e., the number of bonds one can make per site.

    The work appears to be motivated by two questions: Are concentrations buffered in systems where heterotypic interactions drive phase separation thereby negating the presence of a rigorously definable saturation concentration? This question was motivated by work from Klosin et al., showing how phase separation can enable buffering of noise in transcription. They relied on the concept of a saturation concentration. In a paper that followed a few months after, Riback et al., showed that the concept of a saturation concentration ceases to exist, as defined for systems where phase separation is driven purely by homotypic interactions. This was taken to imply that the formation of multicomponent condensates via a blend of homotypic and heterotypic interactions causes a loss of buffering capacity afforded by phase separation. The second question motivating the current work is the apparent absence of a theoretical framework for "varying threshold concentrations" in systems governed by heterotypic interactions.

    Using two flavors of simulations, the authors propose that the SP sets an upper limit on the convolution of concentrations that determine phase separation. They show this via simulations where they follow the formation of clusters formed by linear multivalent macromolecules and monitor the emergence of a bimodal distribution of clusters. In 1:1 mixtures of multivalent macromolecules they find that SP sets a threshold beyond which a bimodal distribution of clusters emerges. The authors further find that SP sets an upper limit even in systems that deviate from the 1:1 stoichiometry.

    The authors proceed to show that the SP is influenced by the valence of multivalent macromolecules. They also demonstrate that short rigid linkers can cause an arrest of phase separation through a so-called "dimer trap" reminiscent of the "magic number" postulate put forth by Wingreen and colleagues.

    Is the work significant, novel, and timely? Effectively the authors propose that the driving forces for phase separation can be distilled down to the concept of a solubility product. Given prior knowledge of the valence, coordination number, and affinities can one predict concentration thresholds for phase separation? The authors suggest that this can be gleaned from either network based simulations, which are very inexpensive, or through more elaborate simulations. They further propose that it is the solubility product that sets the threshold.

    It is worth noting that the authors are quantifying what is known in the physical literature as a percolation threshold. The seminal work of Flory and Stockmayer dating back to the 1940s showed how one can calculate a percolation threshold by taking in prior knowledge of valence, coordination numbers, and affinities whilst ignoring cooperativity. These ideas have been refined and advanced in several theoretical contributions by various labs. While none of the papers in the physical literature use the concept of a solubility product, they rely on the concept of a percolation threshold because the transition to large, system-spanning clusters is a continuous one and it is debatable if this is a bona fide phase transition. Rather it is a topological transition.

    Yes, we agree that the novelty and importance of our work rests in the application of the simple and accessible concept of solubility product, which has not been previously considered in relation to LLPS. The relationship of our analysis to the physics underlying phase diagrams is discussed in a new paragraph within the Discussion.

    As for novelty, unfortunately the authors disregard prior work that showed how linker length impacts local vs. global cooperativity in phase transitions that combine phase separation and percolation. Ref. 23 is the work in question and it is mentioned in passing, even though the contributions here are entirely a redux.

    We have eliminated the results on how molecular structural features control LLPS to fully focus our paper on the Ksp concept, as suggested by the Editor. However, in our original manuscript, we described results not just related to linker length, but also steric effects.

    The concept of a solubility product, introduced here to model / understand phase behavior of multivalent macromolecules, is an interesting and potentially appealing simple description. It might make the understanding of phase transitions more accessible, but it has problems: (a) it does not define phase separation; rather it defines percolation transitions; (b) without prior knowledge of the relevant quantities, the solubility product cannot be readily inferred, even from simulations, although one can scan parameter space to arrive at predictions regarding the apparent valence and coordination numbers. (c) the solubility product does not tell us much about properties of condensates, interfaces, or the driving forces for phase transitions that are influenced by the collective effects of interaction domains / motifs and spacers.

    Recent papers have drawn attention to the potential importance of buffering as a biological function of biomolecular condensation, and also the failure of buffering in heterotypic LLPS. We felt that the Ksp would help “rescue” the idea of buffering, as Reviewer 1 has so aptly put it below. We have refocused the paper to emphasize this. Of course, we describe this for a series of ideal systems with known valency and affinities. However, theoretical systems are always “ideal” and the deviations from ideality are what make experiments so vital. We have added a paragraph in the Discussion that relates our work to the physics of phase transitions, providing 2 citations, [13, 21], to support taking the percolation threshold as a proxy for the phase boundary. We also point out at the end of the Discussion, how the Ksp concept might be validated experimentally and might be useful in categorizing the effective valency of molecules comprising a cellular condensate.

    Finally, as for the absence of a theoretical explanation for the apparent loss of buffering in systems with heterotypic interactions, the authors would do well to see the work of Choi et al., published in PLoS Comput. Biol. in 2019. Figure 12 in that work clearly establishes that the concentrations of A and B species in the coexisting dilute phase are set by the slopes of tie lines - the lines of constant chemical potential. These slopes are set by the relative strengths of homotypic vs. heterotypic interactions, and to zeroth order, that is the physical explanation.

    We apologize for missing this very relevant work and have now cited it several times in the paper. However, as Reviewer 1, states, Figure 12 treats the potential competition between homotypic and heterotypic interactions within a system. We did not address this in our paper. Rather, for our purposes, homotypic interactions are a special case that still fits within the solubility product framework. We do now address the relationship of tie-lines in phase diagrams to the Ksp in the Discussion paragraph mentioned above.

    Reviewer #2 (Public Review):

    This paper asks whether systems composed of more than one component (heterotypic) that undergo liquid-liquid phase separation will follow the same rules as ionic solutions. The question is motivated by (i) the behavior of homotypic solutions, where after phase separation, monomer concentrations remain fixed despite addition of new components, which is not true for heterotypic systems and (ii) the known behavior of multivalent ionic salts. This idea has not previously been tested. They show quite clearly through simulations that the solubility product, Ksp, can be used as a quantitative metric to delineate phase transition behavior in heterotypic systems. This is a valuable contribution to the understanding of phase separation in these systems, and could be impactful in analyzing experimental observables, at least in vitro, to determine the valency of interacting systems. It provides a relatively straightforward conceptual basis for observed partitioning of components into dilute and dense phases. The result seems robust and likely to be reproducible experimentally and through alternative simulation studies, particularly given its established history in quantifying the related phenomena in ionic salts.

    A weakness is the rather qualitative comparison to experiment, which is justified by the authors based on the unknown valency of the experimental system. There is also no quantitative comparison between simulation types (spatial vs non-spatial). However, the simulations do seem sufficiently detailed to test and validate the Ksp concept.

    Strengths:

    • The paper is very focused, and uses multiple simulation 'experiments' to test the role of the Ksp in delineating the phase transition, showing good agreement for multiple systems, with both matched and distinct stoichiometries between the components. They see typical behavior at the phase transition point, where they observe the largest variability or fluctuations in the formation of the dense phase. Thus the results strongly support the conclusion that the Ksp delineates phase transitions in these 2-3 component systems.

    • A comparison is made to a recent experimental result with three components, showing qualitative agreement with an observed lack of buffering, which was unexpected at the time due to the behavior observed for homotypic systems. Here this result is now rationalized via the Ksp, which does plateau despite the monomer concentrations changing.

    • Spatial simulations probe the role of structure and flexibility in impacting phase separation, finding general agreement with previously published experimental and modeling work. These observations about flexibility and matched valency are also relatively intuitive.

    Weaknesses

    • There is no quantitative comparison between the two simulation approaches (spatial and non-spatial), which should be straightforward. By using the same composition and KD in both types of simulations and directly comparing outcomes, it would help explain when and why the spatial simulations differ from the non-spatial ones-see subsequent comments below:

    • A related methodological point: On Line 97 it states that NFSim does not allow intramolecular bonds to form, but this is not true. On one hand, they can be written out explicitly. E.g. A(a1!1, a2).B(b1!1, b2)->A(a1!1, a2!2).B(b1!1, b2!2), would form a second bond between an AB complex that already had one bond. While quite tedious, these could be enumerated, allowing for the zippering effect they see spatially, although the rates would not be bimolecular. This would still leave out intra-complex bonds between proteins without a direct link. However, based on the NFsim website, by default it does in fact allow these types of intra-complex bonds to be formed (http://michaelsneddon.net/nfsim/pages/support/support.html) see "Reactant Connectivity Enforcement". So it is not clear to me which option was used in this paper. According to what is written in the methods, no intra-complex bonds are formed, but this is not the default in NFsim and is indeed allowable.

    The reviewer misinterpreted this admittedly unclear statement: “The binding rules only allow inter-molecular binding; internal bond formation within the molecular clusters is not permitted, as NFsim cannot account for proximity of binding sites within clusters.” We did not intend this to imply that NFSim does not support intramolecular binding; rather we meant that our choice was to only allow intermolecular bond formation. We made this choice because, being non-spatial, NFSIM cannot account for spatial proximity or steric effects. We have clarified this in the revised ms as follows: “We chose binding rules to only allow inter-molecular binding; we felt this was appropriate because NFsim cannot account for spatial proximity of binding sites or steric crowding within clusters.”

    • The spatial simulations do not show the bimodal distribution under the fixed concentrations (Fig S9). This is a significant difference from the non-spatial result. They attribute this to a 'dimer trap', but given they see the dense phase in the clamped monomer simulations, this cannot be the only explanation. What about kinetic effects, due to the differences in initial concentrations of monomers in the two simulation approaches? The rate constants are not listed anywhere. They only seem to see large clusters at fixed concentrations for the mismatched sizes (Fig S12B), where the Ksp behavior does not hold. Can they increase monomer flexibility more and start to see bimodal at fixed concentration, or change the rates and see a bimodal distribution?

    In general, there is a limited ability of a small number of molecules in the FTC simulations to form a clear bimodal distribution, whether spatial or non-spatial. This is directly demonstrated in Figure 1C, where the non-spatial simulations become increasingly bimodal as the number of molecules increases, keeping concentration constant. Because of the greater computational cost of SpringSaLaD calculations, we kept the FTC simulations in Figure 7 to 200 molecules. However, the histograms that are averaged over 50 runs obscure the clear separation that is apparent when examining molecule size distribution in individual trajectories for the FTC case. We now include these in the supporting figures as Figure 1- figure supplement 3 (NFsim) and Figure 7- figure supplement 2 (SpringSaLaD). Above Ksp, we see a consistent group of small oligomers (which is reinforced in the averaged histograms) and individual large clusters (which are smeared out in the average histograms). As Reviewer 2 noted, we were also able to convincingly demonstrate bimodality at and above Ksp with the CMC simulations, which are allowed to continue until they stochastically nucleate large clusters and take off.

    All the FTC simulations are run to steady state, so only the Kds should matter, not the rate constants, which were actually available in the input files in the Git repository; we have now included the SpringSaLaD rate constants in the manuscript as well.

    • Related-I am surprised that the sterically hindered monomers would not form large clusters at fixed concentration, as it looks like it is impossible for them to 'zipper' up their binding sites and become trapped in dimers. Is the distribution at fixed concentrations bimodal? The data is not shown.

    We have removed the additional spatial simulation Results for structures other than the one in Figure 7 as requested by the Editor. We hope to thoroughly explore the molecular-structural determinants of Ksp and LLPS in a subsequent paper.

    Reviewer #3 (Public Review):

    In this work, Chattaraj and colleagues utilize simulation models to study collective behaviors of molecules with multiple binding sites (multivalency). When the concentrations are low, the molecules do not bind to each other frequently, and they are called free. On the other hand, if the concentrations increase, they start to bind and eventually form a wide network of molecules connected by molecular binding. This transition can be considered as a model for liquid-liquid phase separation. Their major claim is that the solubility product, a simple product of the concentrations of the free molecules, can be used as a proxy to the phase separation threshold (known as the saturation concentration). They observed in various simulation conditions that as the total concentration of molecules increases, the solubility product first increases but eventually converges to a certain value, and the value is consistent over different simulation conditions. The value is the upper limit of the solubility product, after which the molecules start to form a molecular network.

    After establishing the model, they tested systems with different valences. Higher valency leads to reduction of the threshold (and phase separation occurs at lower concentrations). The theory was also valid for systems with non-equal valences (e.g. pentavalent A + trivalent B). They applied their models to a three-component system, and found that the results qualitatively explain the published experimental patterns. Lastly, using off-lattice coarse-grained simulations, they show that the linker flexibility and the spacing of binding sites are important determinants of the threshold, which confirms the findings from other computational and experimental works.

    The authors successfully defend their claim by using different types of simulations, and their methods to crosscheck the physical validity of their models may be useful for other simulation works. For example, the authors checked if increasing the number of molecules and reducing the system size give the same results for equal concentrations. Also, they employed two different methods (so-called FTC and CMC in the manuscript) to determine the threshold concentrations. However, the conclusions are not easily transferable to real biopolymer systems, since it is hard to determine the valences (and binding affinities) of biopolymers such as intrinsically disordered proteins.

    Our work was motivated by recent work highlighting the importance of buffering as a biological function of biomolecular condensation, but also the failure of buffering in heterotypic LLPS. We realized that Ksp offers a more general framework than buffering that encompasses complex multicomponent (heterotypic) systems. But our original manuscript was not sufficiently focused on this primary motivation and has been revised accordingly. Of course, we used simulations on ideal systems to establish this idea. We suggest at the end of the discussion that the Ksp concept may potentially be used to derive effective parameters for experimental systems.

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  2. Reviewer #3 (Public Review):

    In this work, Chattaraj and colleagues utilize simulation models to study collective behaviors of molecules with multiple binding sites (multivalency). When the concentrations are low, the molecules do not bind to each other frequently, and they are called free. On the other hand, if the concentrations increase, they start to bind and eventually form a wide network of molecules connected by molecular binding. This transition can be considered as a model for liquid-liquid phase separation. Their major claim is that the solubility product, a simple product of the concentrations of the free molecules, can be used as a proxy to the phase separation threshold (known as the saturation concentration). They observed in various simulation conditions that as the total concentration of molecules increases, the solubility product first increases but eventually converges to a certain value, and the value is consistent over different simulation conditions. The value is the upper limit of the solubility product, after which the molecules start to form a molecular network.

    After establishing the model, they tested systems with different valences. Higher valency leads to reduction of the threshold (and phase separation occurs at lower concentrations). The theory was also valid for systems with non-equal valences (e.g. pentavalent A + trivalent B). They applied their models to a three-component system, and found that the results qualitatively explain the published experimental patterns. Lastly, using off-lattice coarse-grained simulations, they show that the linker flexibility and the spacing of binding sites are important determinants of the threshold, which confirms the findings from other computational and experimental works.

    The authors successfully defend their claim by using different types of simulations, and their methods to crosscheck the physical validity of their models may be useful for other simulation works. For example, the authors checked if increasing the number of molecules and reducing the system size give the same results for equal concentrations. Also, they employed two different methods (so-called FTC and CMC in the manuscript) to determine the threshold concentrations. However, the conclusions are not easily transferable to real biopolymer systems, since it is hard to determine the valences (and binding affinities) of biopolymers such as intrinsically disordered proteins.

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  3. Reviewer #2 (Public Review):

    This paper asks whether systems composed of more than one component (heterotypic) that undergo liquid-liquid phase separation will follow the same rules as ionic solutions. The question is motivated by (i) the behavior of homotypic solutions, where after phase separation, monomer concentrations remain fixed despite addition of new components, which is not true for heterotypic systems and (ii) the known behavior of multivalent ionic salts. This idea has not previously been tested. They show quite clearly through simulations that the solubility product, Ksp, can be used as a quantitative metric to delineate phase transition behavior in heterotypic systems. This is a valuable contribution to the understanding of phase separation in these systems, and could be impactful in analyzing experimental observables, at least in vitro, to determine the valency of interacting systems. It provides a relatively straightforward conceptual basis for observed partitioning of components into dilute and dense phases. The result seems robust and likely to be reproducible experimentally and through alternative simulation studies, particularly given its established history in quantifying the related phenomena in ionic salts.

    A weakness is the rather qualitative comparison to experiment, which is justified by the authors based on the unknown valency of the experimental system. There is also no quantitative comparison between simulation types (spatial vs non-spatial). However, the simulations do seem sufficiently detailed to test and validate the Ksp concept.

    Strengths:

    • The paper is very focused, and uses multiple simulation 'experiments' to test the role of the Ksp in delineating the phase transition, showing good agreement for multiple systems, with both matched and distinct stoichiometries between the components. They see typical behavior at the phase transition point, where they observe the largest variability or fluctuations in the formation of the dense phase. Thus the results strongly support the conclusion that the Ksp delineates phase transitions in these 2-3 component systems.

    • A comparison is made to a recent experimental result with three components, showing qualitative agreement with an observed lack of buffering, which was unexpected at the time due to the behavior observed for homotypic systems. Here this result is now rationalized via the Ksp, which does plateau despite the monomer concentrations changing.

    • Spatial simulations probe the role of structure and flexibility in impacting phase separation, finding general agreement with previously published experimental and modeling work. These observations about flexibility and matched valency are also relatively intuitive.

    Weaknesses

    • There is no quantitative comparison between the two simulation approaches (spatial and non-spatial), which should be straightforward. By using the same composition and KD in both types of simulations and directly comparing outcomes, it would help explain when and why the spatial simulations differ from the non-spatial ones-see subsequent comments below:

    • A related methodological point: On Line 97 it states that NFSim does not allow intramolecular bonds to form, but this is not true. On one hand, they can be written out explicitly. E.g. A(a1!1, a2).B(b1!1, b2)->A(a1!1, a2!2).B(b1!1, b2!2), would form a second bond between an AB complex that already had one bond. While quite tedious, these could be enumerated, allowing for the zippering effect they see spatially, although the rates would not be bimolecular. This would still leave out intra-complex bonds between proteins without a direct link. However, based on the NFsim website, by default it does in fact allow these types of intra-complex bonds to be formed (http://michaelsneddon.net/nfsim/pages/support/support.html) see "Reactant Connectivity Enforcement". So it is not clear to me which option was used in this paper. According to what is written in the methods, no intra-complex bonds are formed, but this is not the default in NFsim and is indeed allowable.

    • The spatial simulations do not show the bimodal distribution under the fixed concentrations (Fig S9). This is a significant difference from the non-spatial result. They attribute this to a 'dimer trap', but given they see the dense phase in the clamped monomer simulations, this cannot be the only explanation. What about kinetic effects, due to the differences in initial concentrations of monomers in the two simulation approaches? The rate constants are not listed anywhere. They only seem to see large clusters at fixed concentrations for the mismatched sizes (Fig S12B), where the Ksp behavior does not hold. Can they increase monomer flexibility more and start to see bimodal at fixed concentration, or change the rates and see a bimodal distribution?

    • Related-I am surprised that the sterically hindered monomers would not form large clusters at fixed concentration, as it looks like it is impossible for them to 'zipper' up their binding sites and become trapped in dimers. Is the distribution at fixed concentrations bimodal? The data is not shown.

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  4. Reviewer #1 (Public Review):

    The gist of this work is that the simple concept of a solubility product determines a threshold for phase separation, thereby enabling buffering even in systems where phase separation is driven by heterotypic interactions. The solubility product or SP is determined by the number of complementary interaction sites and the coordination number i.e., the number of bonds one can make per site.

    The work appears to be motivated by two questions: Are concentrations buffered in systems where heterotypic interactions drive phase separation thereby negating the presence of a rigorously definable saturation concentration? This question was motivated by work from Klosin et al., showing how phase separation can enable buffering of noise in transcription. They relied on the concept of a saturation concentration. In a paper that followed a few months after, Riback et al., showed that the concept of a saturation concentration ceases to exist, as defined for systems where phase separation is driven purely by homotypic interactions. This was taken to imply that the formation of multicomponent condensates via a blend of homotypic and heterotypic interactions causes a loss of buffering capacity afforded by phase separation. The second question motivating the current work is the apparent absence of a theoretical framework for "varying threshold concentrations" in systems governed by heterotypic interactions.

    Using two flavors of simulations, the authors propose that the SP sets an upper limit on the convolution of concentrations that determine phase separation. They show this via simulations where they follow the formation of clusters formed by linear multivalent macromolecules and monitor the emergence of a bimodal distribution of clusters. In 1:1 mixtures of multivalent macromolecules they find that SP sets a threshold beyond which a bimodal distribution of clusters emerges. The authors further find that SP sets an upper limit even in systems that deviate from the 1:1 stoichiometry.

    The authors proceed to show that the SP is influenced by the valence of multivalent macromolecules. They also demonstrate that short rigid linkers can cause an arrest of phase separation through a so-called "dimer trap" reminiscent of the "magic number" postulate put forth by Wingreen and colleagues.

    Is the work significant, novel, and timely? Effectively the authors propose that the driving forces for phase separation can be distilled down to the concept of a solubility product. Given prior knowledge of the valence, coordination number, and affinities can one predict concentration thresholds for phase separation? The authors suggest that this can be gleaned from either network based simulations, which are very inexpensive, or through more elaborate simulations. They further propose that it is the solubility product that sets the threshold.

    It is worth noting that the authors are quantifying what is known in the physical literature as a percolation threshold. The seminal work of Flory and Stockmayer dating back to the 1940s showed how one can calculate a percolation threshold by taking in prior knowledge of valence, coordination numbers, and affinities whilst ignoring cooperativity. These ideas have been refined and advanced in several theoretical contributions by various labs. While none of the papers in the physical literature use the concept of a solubility product, they rely on the concept of a percolation threshold because the transition to large, system-spanning clusters is a continuous one and it is debatable if this is a bona fide phase transition. Rather it is a topological transition.

    As for novelty, unfortunately the authors disregard prior work that showed how linker length impacts local vs. global cooperativity in phase transitions that combine phase separation and percolation. Ref. 23 is the work in question and it is mentioned in passing, even though the contributions here are entirely a redux.

    The concept of a solubility product, introduced here to model / understand phase behavior of multivalent macromolecules, is an interesting and potentially appealing simple description. It might make the understanding of phase transitions more accessible, but it has problems: (a) it does not define phase separation; rather it defines percolation transitions; (b) without prior knowledge of the relevant quantities, the solubility product cannot be readily inferred, even from simulations, although one can scan parameter space to arrive at predictions regarding the apparent valence and coordination numbers. (c) the solubility product does not tell us much about properties of condensates, interfaces, or the driving forces for phase transitions that are influenced by the collective effects of interaction domains / motifs and spacers.

    Finally, as for the absence of a theoretical explanation for the apparent loss of buffering in systems with heterotypic interactions, the authors would do well to see the work of Choi et al., published in PLoS Comput. Biol. in 2019. Figure 12 in that work clearly establishes that the concentrations of A and B species in the coexisting dilute phase are set by the slopes of tie lines - the lines of constant chemical potential. These slopes are set by the relative strengths of homotypic vs. heterotypic interactions, and to zeroth order, that is the physical explanation.

    Overall, the two interesting observations are that the percolation threshold can be cast as a solubility product and that this product sets an upper limit on joint concentration thresholds for phase separation, even in systems with heterotypic interactions, thereby rescuing the concept of buffering.

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  5. Evaluation Summary:

    Recent experiments have raised questions regarding concentration buffering provided by the formation of multicomponent biomolecular condensates via phase separation driven by heterotypic interactions. In this work, Chattaraja et al., demonstrate that the concept of a solubility product, used to describe the solubility limits of ionic solutions, sets an upper limit on concentration thresholds, even in systems where the driving forces for phase separation are primarily heterotypic in nature. Their work suggests that the concept of a solubility product rescues the concept of buffering via phase separation.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)

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