Protein compactness and interaction valency define the architecture of a biomolecular condensate across scales

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   Version published to 10.7554/elife.80038 on eLife

   Jul 20, 2023
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   eLife

   May 15, 2023

   Author Response

   Reviewer #1 (Public Review):

   The strength of the manuscript is highlighted by the application of fractal formalism, which is commonly used in colloidal systems, in conjunction with MD simulation to study the phase separation of an IDP. The weakness lies in the fact that this study does not provide any discussion on how our understanding of the network structure and dynamical behavior of biomolecular condensates and their biological significance improves through this study. The experimental part remains weak, without any measurements of the dynamics of the condensates. Whether and how the formalism can distinguish between phase-separated condensates (WT) and classical protein aggregates (Y to A variant) remains unclear.

   We thank the Reviewer for their careful reading of the manuscript and their appreciation of the …

   Author Response

   Reviewer #1 (Public Review):

   The strength of the manuscript is highlighted by the application of fractal formalism, which is commonly used in colloidal systems, in conjunction with MD simulation to study the phase separation of an IDP. The weakness lies in the fact that this study does not provide any discussion on how our understanding of the network structure and dynamical behavior of biomolecular condensates and their biological significance improves through this study. The experimental part remains weak, without any measurements of the dynamics of the condensates. Whether and how the formalism can distinguish between phase-separated condensates (WT) and classical protein aggregates (Y to A variant) remains unclear.

   We thank the Reviewer for their careful reading of the manuscript and their appreciation of the link between IDP phase separation and colloid chemistry. Establishment of a quantitative framework behind this link, as given by the fractal formalism, and a multiscale model of the spatial organization of a biomolecular condensate, derived from MD simulations in combination with fractal scaling, are indeed two of our main contributions. In particular, to the best of our knowledge, ours is the first atomistically resolved model of the spatial organization of a biomolecular condensate at an arbitrary scale. The key features of the proposed model, as elaborated in the Discussion of the revised manuscript (p. 18, 20-21), are the coexistence of differently sized clusters inside a condensate, and a quantitative prediction of a particular scaling of mass with cluster size (Figure 5A), as further discussed below. Moreover, our results also point to the possible formation of pre-percolation clusters with sizes below the resolution limit of typical microscopy experiments, in agreement with recent observations (https://doi.org/10.1073/pnas.2202222119).

   We agree that the full understanding of biomolecular condensates also requires a detailed treatment of the dynamical aspects. Following the Reviewer’s comments, we have provided significant new results in this regard and included an experimental characterization of fusion behavior (Videos 1, 2) and condensate dynamics by FRAP (Figure 1D, E and Figure 1—figure supplement 2) as well as a detailed analysis of diffusion and viscosity in the simulated systems (Figure 4C and Figure 4—figure supplement 1D-F). The newly performed FRAP experiments provide a direct measure of the condensate dynamics. Importantly, the measured recovery half-times for WT and R>K condensates resemble those of other well-characterized in vitro condensates. We have occasionally observed elongated, amorphous Y>A precipitates, albeit in low number and only at 50-fold higher concentration than the wild-type (45 mM and above, Figure 1C). While this may be consistent with the predictions of the fractal model and hint at the differences in mesoscopic organization between the WT and R>K condensates and the Y>A precipitates, the latter are rare and we are reluctant to draw major conclusions.

   Furthermore, we could show that the WT diffusion coefficient is lower than for either mutant (Figure 4C, and Supplementary File 2). Clearly, this difference is not due to the effect of protein size or a higher solvent viscosity, but primarily indicates protein slow-down due to the more extensive interactions with partners (reflected also in higher average valency, Figure 2D, or probability of interactions Figure 2—figure supplement 1D). The fact that the WT diffusion coefficient drops by about 20% over the last 0.3 µs of the MD trajectory also correlates with the formation of a single percolating cluster in the system (Figure 2C). This is an expected effect on protein diffusion upon crossing the percolation threshold (https://doi.org/10.1038/ncomms11817, https://doi.org/10.1021/acs.jpcb.7b08785). Moreover, the difference in the recovery dynamics observed for WT and R>K mutant can be interpreted using the proposed model. Namely, accurate fitting of FRAP data was only possible if using at least two components (Figure 1—figure supplement 2). According to https://doi.org/10.1016/j.tcb.2004.12.001, these components indicate the contribution of particle diffusion and interaction (binding). Thus, recovery of the centrally bleached condensates is faster for WT than for the R>K mutant, which can be related to the higher compactness of the WT particles across scales as compared to R>K. On the other hand, the FRAP results for the condensates bleached in the peripheral area highlight the contribution of the binding component. Indeed, the recovery is about 3-fold faster for the R>K mutant, which could potentially be related to the lower valency of the interactions and the ease of the replacement of inactivated fluorescent species and/or exchange with proteins in the bulk. A further connection of the developed model and condensate dynamics concerns the multimodal description of diffusion in biomolecular condensates, together with multimodal fitting of FCS and FRAP data as used recently for interpreting single particle tracking results (https://doi.org/10.1016/j.bpj.2021.01.001). Namely, the polydisperse nature of the protein phase as suggested by the model translates to multimodal diffusion, reflecting the dynamics of protein clusters of different size. For instance, regularization fits used for DLS autocorrelation curves assume a multimodal character of the diffusion and are interpreted to reflect a multimodal distribution of cluster sizes in condensates (https://doi.org/10.1073/pnas.2202222119).

   Finally, a way of testing the model prediction, which would merit a study in its own right, would involve static light scattering (SLS): a linearly decreasing scattering intensity as a function of the scattering vector in a log-log representation, as frequently seen for different colloidal systems, is expected by the fractal model. In fact, fractal dimension dF could directly be estimated from SLS experiments (https://doi.org/10.1038/339360a0) from the limiting value of scattering intensity for high values of the product of the scattering vector q and the average cluster size . As a direct test of the predictions of the model, the experimental value of dF could then be compared with the predicted one. Moreover, techniques such as DLS and MALS could be used to measure independently masses and sizes of biomolecular condensates in vitro at different scales in order to test the validity of the particular scaling predicted by the fractal model. Such experiments are not trivial and are out of scope of the present study.

   Reviewer #2 (Public Review):

   A key aspect of the work is to use the simulations to explain differences between (i) dilute and dense phases and (ii) wild-type and mutant variants. Here, it would be important with a clearer analysis of convergence and errors to quantify which differences are significant.

   Following the Reviewer’s suggestion, we now provide an analysis of convergence and statistical significance. Specifically, in Supplementary File 1 “Technical summary” we now report the average value, standard deviation and a block-average measure of convergence for all the key observables analyzed, including radius of gyration (Rg), valency (n), and compactness (), for all modeled systems. Furthermore, in the revised manuscript, we now also include the analysis of protein translational diffusion constants and solution viscosity for all modeled systems to assess the ability of the simulations to capture protein dynamics realistically (Figure 4C, Figure 4—figure supplement 1D-F, Supplementary File 2, see also above). Moreover, we include in the revised version a new figure depicting time evolution of average compactness in the 24-copy systems (Figure 4—figure supplement 1C). Thus, it can be seen that the two key model parameters derived from MD simulations of the 24-copy system – protein valency and compactness – reach a stable plateau over the last 0.3 µs (Figure 2D and Figure 4—figure supplement 1C), which were used for final analyses, with block-averaged deviations of less than 10% throughout (see below for details). All the differences in these parameters between single-copy and 24-copy simulations, as well as those between WT and mutation simulations, were found to be significant with p-values < 2.2 10-16 according to the Wilcoxon rank sum test with continuity correction (details in Supplementary File 1). Finally, considering the sampling limitations implicit in most MD studies, we clearly recognize the possibility that with longer simulation times or more protein copies per simulation box, the simulated systems may show a qualitatively different behavior. However, we emphasize that our derivation of the formalism that links the features of simulated ensembles on the scale of 10s of nanometers with their behavior on the scale of 100s of nanometers and beyond is independent of such limitations. Once longer, larger and more accurate simulations become available, one will be able to apply the formalism without alteration and obtain a model of the spatial organization of the condensate on an arbitrary scale, starting just from the local features of individual proteins. We now discuss these details on pp. 10, 11, 13 of the revised manuscript.

   It would also be useful with a clearer description of how the analytical model is predictive, of which properties, and how they have been/can be validated. Which measurable quantities does the model predict?

   As pointed above, the model predicts the existence and provides a quantitative description of pre-percolation finite-size clusters (https://doi.org/10.1016/j.molcel.2022.05.018, https://doi.org/10.1073/pnas.2202222119). More generally, the model provides the fractal dimension (dF) of protein clusters and enables evaluation of different scale-dependent properties of clusters of arbitrary size, including protein density as a function of cluster size (Figure 5—figure supplement 1C, Figure 5C). Importantly, the fractal dimension can be used in combination with local MD simulations and cluster–cluster aggregation algorithms to derive a detailed model of the 3D organization of fractal clusters of a chosen size at atomistic resolution (Figure 5A, B, and Videos 4, 5, and 6). Such detailed structural understanding of the interior organization of a condensate can, for example, be used to evaluate cavity sizes and interpret partitioning experiments. Since the differences in the morphology of WT and mutant protein clusters propagate across length scales, they can even be qualitatively characterized by the analysis of microscopic images (e.g. circularity, Figure 1—figure supplement 1C, see also discussion above). Finally, static light scattering (SLS) experiments give the possibility to test the model directly, which will be the subject of our future work. Namely, the fractal formalism predicts linear behavior in the log-log representation of the SLS intensity vs. scattering vector curves, while dF, which can directly be evaluated from such experiments, providing a quantitative point of comparison between theoretical predictions and experiment (see above).

   In addition to these overall questions, a number of more specific suggestions follow below.

   Major:

   p. 7, line 120 (Fig. S1B) The proteins do not appear particularly pure based on the presented SDS PAGE analysis. How pure is the protein estimated to be, and is the presence of the other bands expected to affect e.g. the data presented in Fig. 1?

   We have quantified the purity of the constructs by densitometry of the Coomassie stained gels and included it in Figure 1—figure supplement 1A: in the case of WT and R>K, we achieve purity higher than 91%. Importantly, the observed LLPS behavior of the constructs is consistent with the simulation and in agreement with other studies on R>K substitutions (https://doi.org/10.1073/pnas.2000223117; https://doi.org/10.1016/j.molcel.2020.01.025; https://doi.org/10.1073/pnas.2200559119; https://doi.org/10.1016/j.jmb.2019.08.008). In the case of Y>A, we have obtained the least pure protein (~65%), and must note that the precipitates observed in the experiments of Figure 1C are only present at the protein concentrations that are 50-fold higher as compared to WT (45 mM and above). Therefore, at such high total protein concentration, we cannot exclude the possibility that there might be some contamination affecting the behavior of this construct.

   p. 7 & 8, lines 138-159: Has the method and energy function used to calculate the interact potential been validated by comparison to experiments, including studying the effect of varying the solvent? I see the computed error bars are very small, but am more interested in the average error when comparing to experiments. The numbers in water appear different from those e.g. reported by Krainer et al (https://doi.org/10.1038/s41467-021-21181-9), though the latter are also not immediately compared to experiments. Thus, it would be useful to know how much to trust these numbers.

   We thank the Reviewer for raising this important point. To the best of our knowledge, the absolute binding free energies between Y-Y, Y-R or Y-K sidechain analogs or complete amino acids have never been determined experimentally, preventing a direct validation of the computed values and/or an evaluation of the average error when comparing to experiments. On the other hand, we did compare our data against the PMF curves presented by Krainer et al. (https://doi.org/10.1038/s41467-021-21181-9) for R-Y and Y-Y and the general trends are largely similar. In particular, in both analyses the R-Y interaction is stronger than the Y-Y interaction across different conditions, except at zero salt in Krainer et al. where the two are similar. When it comes to exact quantitative differences between the studies, it should first be pointed out that Krainer et al. studied capped amino-acids, while we used amino-acid side-chain analogs. The difference in the observed binding strengths is in part certainly related to the contribution of the capped backbone. Second, the values in Krainer et al. refer to the depth of the free energy minimum in the obtained PMFs and not to the resulting G values, as in our method. The latter includes integration over the PMF and an assumption of a standard-state concentration, which could also lead to significant differences. Finally, the differences could also be due to the intrinsic properties of the interaction potentials used. In particular, the prominent free-energy minima for the R-Y pair in the Krainer et al. study could only be obtained after refitting of the original AMBERff03ws charges on the Y bound to R via semi-empirical quantum-chemical calculations. On the other hand, the interaction potential used in our study was not adjusted to the system at hand, but rather comes from a published, widely used force field, the OPLS-AA (https://doi.org/10.1021/ja9621760), that was independently tested and validated experimentally in multiple studies. For example, OPLS-AA exhibits the low average error in absolute hydration free energy of ~0.5 kcal/mol, errors of only ~2% for heats of vaporization and densities (https://doi.org/10.1021/ja9621760), and a close agreement with osmotic coefficients (https://doi.org/10.1021/acs.jcim.9b00552) or a large range of organic compounds. This raises our confidence in the accuracy of the derived binding free energies, which directly or indirectly depend on these fundamental thermodynamic properties.

   Regarding the method to evaluate PMF profiles, we have used a classical all-atom Monte Carlo approach originally developed by Jorgensen and coworkers (see, e.g., https://doi.org/10.1021/ar00161a004 and https://doi.org/10.1021/ja00168a022), as implemented in the widely used BOSS program (v. 4.8) (https://doi.org/10.1002/jcc.20297). This approach has been extensively tested against experimental data on ΔΔG values of various compounds in environments of different polarity (e.g., 2). Moreover, we have previously successfully applied this methodology in studies of the free energy of association of amino acid residues (https://doi.org/10.1021/jp803640e) and other biologically important groups (https://doi.org/10.1021/acs.jcim.9b00193). The results obtained have been compared with the available experimental data and demonstrated a good agreement. As for the small error bars in the plots, the fairly good convergence achieved in our PMF calculations is a result of extensive sampling combined with small system size, although obviously this is not always the case – see, for example, PMFs in our recent work (https://doi.org/10.1021/acs.jcim.9b00193).

   The above points have been discussed on pp 7-8 of the revised manuscript.

   p. 8, lines 149-154: Following up on the above, the authors also write "Importantly, only in the latter case are the R-Y interactions slightly more favorable than the K-Y ones (Figure S1C). While this can potentially contribute to increasing of Csat for the R>K mutant as compared to WT, the estimated thermodynamic effect is not too strong, especially if one considers that these interactions take place in an environment with largely water-like polarity. Therefore, the effect of R>K substitution on LLPS should be further explored in the context of protein-protein interactions." In the absence of estimates of the accuracy of the predictions, these sentences are somewhat unclear. Also, it is unclear what the authors mean by that the effect of R>K should be studied; there are already several examples of this (https://doi.org/10.1016/j.cell.2018.06.006 [already cited], https://doi.org/10.1038/s41557-021-00840-w & https://doi.org/10.1073/pnas.2000223117 come to mind, but there are likely more).

   As pointed above, the free-energy values were obtained using well-established computational techniques and are expected to reflect realistic trends. However, considering that there exist no equivalent experimental results to assess the accuracy of the predicted free energies, they indeed must clearly be understood as predictions. This is now stated on pp. 7-8 of the revised manuscript. Furthermore, it seems that the vague phrasing on our part in the above paragraph resulted in a misunderstanding. Namely, when we talk about “further exploration”, we only meant it in relation to our study, i.e. a connection with the MD part, and not in relation to a wider literature on the topic. In other words, we simply wanted to refer to the fact that our binding free energies for individual residues do not provide sufficient information about interactions between Lge11-80 protein chains. Following the Reviewer’s comment, we have rephrased this part and included additional references on the known role of R and K residues on phase separation.

   p. 8, lines 161-162: The authors perform MD simulations of Lge1 and variants using 24 copies and a box that gives them protein concentrations "in the mM concentration range". I realize that there's a concern about what is computationally feasible, but it would be important with an argument for this choice. Why is 24 expected to be enough to represent a condensate (I expect that there could be substantial finite-size effects)? What is the exact protein concentration in the simulations of the 24 chains [and of the 1-chain simulations]? How does this protein concentration compare to that in the condensates? The authors performed simulations in the NPT ensemble; how stable were the box dimensions?

   The effective protein concentration for different 24-copy systems is 6-7 mM, depending on the system (Figure 2—figure supplement 1A). This concentration range was selected in order to get a reasonable system size for microsecond all-atom MD simulations, while still being approximately one order of magnitude lower than the semi-dilute regime of the protein at hand. As a testament to the internal consistency of our framework, the fractal model predicts the concentration inside WT condensates of the size observed in the experiment to indeed be in the mM range. Moreover, as seen in many other systems, the concentration inside the observed droplets is expected to be significantly higher than Csat (https://doi.org/10.1101/2020.10.25.352823). Here, we should again emphasize that we did not aim to model the process of phase separation in our all-atom MD. We rather use multicopy simulations for the analyses of the organization of the protein crowded phase and specifically, the mode of intermolecular interactions, and then use the fractal scaling to derive a model of the internal organization of condensates at arbitrary scales.

   Regarding the experimental determination of the protein concentration in the condensates, we have used different approaches to estimate Csat and CD values: spin-down analyses (https://doi.org/10.1126/science.aaw8653), volumetry analysis (https://doi.org/10.1038/nchem.2803), estimation of concentration by fluorescent intensity of the condensates (https://doi.org/10.1016/j.molcel.2018.12.007; https://doi.org/10.1016/j.cell.2019.08.008), FCS (https://doi.org/10.1038/nchem.2803; https://doi.org/10.1016/j.cell.2019.10.011; https://doi.org/10.1126/science.aaw8653). However, different approaches yield values that vary by several orders of magnitude. That is the reason why we did not report definitive numbers. In general, there are uncertainties in the field about how to reliably measure protein concentrations in a condensate, necessitating the development of new approaches (https://doi.org/10.1101/2020.10.25.352823).

   With regard to the convergence and potential finite-size effects, we agree that this is an important issue and have addressed it in the revised version. In general, the convergence of our observables such as valency or compactness (Figure 2C, D and Figure 4—figure supplement 1C) gives confidence that the simulations are at least in a local equilibrium, especially when it comes to short-range properties such as contact preferences as further elaborated in our reply to the Reviewer’s specific comment about convergence below (please, see also above for our response to Editor’s comment #5). Importantly, in all 24-copy systems, the average separation between protein images lies in the 12-15 nm range, and no instances of self-interaction between images due to PBC were observed (Supplementary File 1). Finally, analysis of fluctuations in box dimensions shows that they are all in the range of picometers and largely negligible when it comes to the analysis at hand (Supplementary File 1).

   In order to highlight the realistic behavior of the simulated systems in the revised version, we now also report a detailed analysis of protein translational diffusion in MD simulations (Figure 4C and Figure 4—figure supplement 1D-F and Supplementary File 2). According to this analysis, single-molecule translational diffusion coefficients of Lge11-80 variants obtained from fitting of MSD curves with applied finite-size PBC correction and rescaling by the solvent viscosity (see Methods for details) are typically in the range of 100 µm2/s (Figure 4C and Supplementary File 2), which corresponds to experimentally measured values for different proteins of similar size. Importantly, the requisite finite-size corrections applied in the case of 24-copy systems are relatively small and amount to about 35-60%, while this is almost an order of magnitude higher (450-530%) for the single-copy simulations (Supplementary File 2). Please, see also the reply above to the Editor’s statements above for more details.

   Also, did the authors include the Strep- and His-tags in the simulations? If not, why not?

   We did not simulate the constant part of the constructs in order to: 1. expedite computation and 2. more directly expose the effect of different mutations. Since our comparison between simulation and experiment concerned largely qualitative observables, we have primarily focused on the relative differences between the three Lge11-80 variants. Importantly, the effect of mutations on the full-length protein and its different variants was analyzed in vivo in a previous publication (https://doi.org/10.1038/s41586-020-2097-z).

   Throughout: One of my major concerns about this work is the general lack of analysis of convergence of the simulations. The authors must present some solid analysis of which results are robust given the relatively short simulations and potential for bias from the chosen starting structures.

   First, we would like to emphasize that we did not attempt to capture the process of phase separation or characterize two coexisting phases, for which much larger ensembles and/or simulation times would be needed. Rather, our aim was to study the conformational behavior of individual protein chains in the context of a crowded protein mixture, taken as a model for the dense phase, and then use fractal scaling to provide a model of spatial organization of a condensate at an arbitrary length scale. Having said this, it is absolutely important to address how converged the key observables are, given the finite size of the all-atom simulation setup and the limited sampling used. In the revised manuscript, we have included an additional analysis of convergence of our simulations and could show that both key MD-derived parameters required by the fractal model, protein compactness and valency, display convergent behavior over the last third of 0.3 µs MD in the 24-copy systems (Figure 4—figure supplement 1C) and all analyses were performed over this region. In particular, the block averages of compactness and valency exhibit a standard deviation of only 2-4% and 4-8%, respectively, over the last 0.3 µs of MD simulations. Moreover, since we are interested in single-chain features in the context of a crowded mixture, our sampling corresponds effectively to 24 x 0.3 µs = 7.2 µs. Finally, a detailed analysis of convergence in conformational sampling was performed for single-copy simulations using calculations of configurational entropy as evaluated by the MIST formalism (Figure 4—figure supplement 1B). For instance, in the case of the weakly self-interacting Y>A, we do observe a close convergence in terms of the configurational entropy between two independent replicas on 1 µs MD trajectory (Figure 4—figure supplement 1B). However, we still recognize the possibility that with longer simulation times and/or more protein copies per simulation, the simulated systems may show a qualitatively different behavior, as discussed on pp. 10, 11, and 13 of the revised manuscript. Finally, we would like to reiterate the point that our derivation of the formalism that links the features of simulated ensembles on the scale of 10s of nanometers with their behavior on the scale of 100 s of nanometers and beyond is independent of such limitations. Once longer, larger and more accurate simulations become available, one will be able to apply the formalism without alteration and obtain a model of the spatial organization of the condensate on an arbitrary scale, starting just from the local features of individual proteins. We now discuss these details on pp. 10, 11, and 13 of the revised manuscript.

   As an example, on p. 8 the authors discuss a potential asymmetry between the interactions found in the dilute (single-copy) and dense (24-mer) phases. These observations are somewhat in contrast to other observations in the field, namely that it is the same interactions that drive compaction of monomers as those that drive condensate formation.

   Obviously, both the results in the literature and those presented here could be true. But in order to substantiate the statements made here, the authors should show some substantial statistical analyses to make it clear which differences are robust. The above holds for all parts of the computational/simulation work (e.g. other aspects of Fig. 2)

   Note: this comment by the Reviewer echoes in several respects the comment 7 by the Editor. Because of this, our reply in some parts is identical to that given above to the Editor. We have decided to include it here for the ease of reading and completeness.

   An expectation of the symmetry between intra- and intermolecular modes of interaction emerged from the background of polymer theory, which was primarily aimed to describe the behavior of homopolymers. In the case of heteropolymers such as proteins, the asymmetry in the aforementioned modes is rather intuitive. For instance, if there is only a single Y in a protein, then Y-Y contacts will not be possible in the intramolecular context, but could occur in multichain interactions. However, we agree with the Reviewer that this is an important issue and have deepened the analysis of this phenomenon in the revised manuscript.

   First, our analysis shows that the observed asymmetry between intra- and intermolecular contexts is statistically significant and is likely not a consequence of limited sampling (pp. 10-11, Figure 3—figure supplement 1B-C). Moreover, the observed symmetry breaking is in line with the recent studies by Bremer et al. (https://doi.org/10.1038/s41557-021-00840-w) and Martin et al. (https://doi.org/10.1126/science.aaw8653), which have delineated the key requirements for the symmetry between single-chain and collective phase behavior to hold. Specifically, we have compared in detail the sequence composition of Lge11-80 with that of A1-LCD variants studied by Bremer et al. When it comes to aromatic composition, Lge1 is most similar to the -12F+12Y mutant of A1-LCD, and by this token, i.e. the high frequency of stickers tyrosines, should exhibit a strong coupling between single-chain and phase behavior. However, the net charge per residue (NCPR) in Lge11-80 of 0.075 is greater than that of A1-LCD (0.059) and this could contribute to the extent of decoupling, as suggested by Bremer et al. Moreover, Lge1 is extremely abundant in Arg (13.5 % as compared to 7.4 % in A1-LCD), and is in this sense most similar to the +7R A1-LCD mutant, which showed the greatest degree of decoupling between single-chain and phase behavior in Bremer et al., in agreement with what we see here. While these authors have demonstrated that NCPR is the primary determinant of decoupling in the case of A1-LCD mutants, their analysis showed that the nature of positive and negative residues involved also makes a significant difference. In particular, the significant excess of Arg residues, as context-dependent auxiliary stickers, could create the asymmetry between interactions that determine single-chain dimensions vs. collective phase behavior.

   Furthermore, Martin et al. (https://doi.org/10.1126/science.aaw8653) have shown that an approximately uniform distribution of stickers along the sequence is required for the correspondence between the driving forces behind coil-to-globule transitions and phase separation to hold. We have analyzed the patterning of Tyr residues along the Lge11-80 sequence using Waro parameter used by Martin et al. (note that Tyr is the only aromatic in the Lge11-80 sequence). Interestingly, Lge11-80 exhibits a highly non-uniform patterning of Tyr residues, with Waro of the native Lge1 sequence (0.47) falling in the middle of the distribution for its shuffled variants (p=0.57). This is in contrast to the highly patterned sequences such as that of A1-LCD with p>0.99. Taken together, in addition to the relatively high NCPR, symmetry breaking in the case of Lge11-80 could be a consequence of its complex sequence composition, including both the non-uniform patterning of tyrosines and a high abundance of arginines. Provided that our simulations are long enough to provide an equilibrium picture and are on the length-scale of a single protein not strongly influenced by finite-size effects (these potential artifacts cannot be discounted), they actually can be seen as a demonstration of such symmetry breaking in a heteropolymer.

   Furthermore, analysis of pairwise contacts suggests that intra- and intermolecular interactions rely on a similar pool of contacts by amino-acid type, but differ significantly if one analyzes specific sequence location of the interacting residues involved (Figure 2—figure supplement 1B and C). For example, one observes a high correlation between the frequencies of different contacts by amino-acid type when comparing intramolecular contacts in single-copy simulations and intermolecular contacts in 24-copy simulations (Figure 3—figure supplement 1B). This correlation is completely lost (Figure 3—figure supplement 1C) if one analyzes position-resolved statistics (2D pairwise contacts maps) or statistically defined interaction modes (Figure 3A, and Figure 3—figure supplement 1A). For example, although Tyr-Tyr interactions dominate in both cases, in single-copy simulations of WT Lge11-80 the C-terminal Tyr80 barely participates in any intramolecular interactions with other residues (Figure 3—figure supplement 1A), while in 24-copy simulations it is one of the most intermolecularly interactive residues (Figure 3). In other words, while the symmetry between intra- and intermolecular interactions can be observed at the level of pairwise contact types (similar type contact used for both), the distribution of these contacts along the peptide sequence is clearly different in the two cases. Finally, it should be mentioned that the parallel between single-copy and phase behavior in both homopolymers and heteropolymers is observed primarily at the level of thermodynamic variables such as LLPS critical temperature (Tc), coil-to-globule transition temperature (Tq) or the Boyle temperature (TB). It is possible that the noted correspondence extends primarily to such and similar thermodynamic variables, while and more structural, topological features of the globule in the single-molecule case and the network in the collective phase case remain uncoupled.

   Interestingly, the core of intramolecular interactions observed for a single molecule at infinite dilution and in the crowded context remain approximately the same as reflected in the high correlation between intramolecular modes obtained in single and multichain simulations. Namely, proteins keep core self-contacts and establish new ones with neighbors, but do not donate everything to the intermolecular network losing “self-identity”, as in homopolymer melts. Similar effects have also been observed elsewhere: https://doi.org/10.1073/pnas.2000223117, https://doi.org/10.1073/pnas.1804177115.

   Similarly, how were the errors of the radius of gyration for WT, R>K and Y>A mutants calculated? Is the Rg for WT significantly smaller than the values for the two mutants? And are the differences in Rg between single-copy and multi-copy simulations statistically significant? I am asking since converging the Rg of IDPs of this length in all-atom MD is not easy.

   The errors for Rg values correspond to the standard deviations of the underlying distributions and are reported in Figure 4A and B, together with the corresponding means and an assessment of statistical significance of the difference. In particular, the character of the distributions (especially, for 24-copy systems) also suggests significant differences. In order to deepen this part in the revised version, we have added a new supplementary table (Supplementary File 1 “Technical summary) where we have included the average values of Rg together with the standard deviations for all modeled systems. Due to distributions being non-Gaussian, we have estimated the significance of the differences in Rgs between single-copy and multicopy simulations, as well as WT and mutants, using Wilcoxon rank sum test with continuity correction, with the resulting p-values < 2.2 10-16 for all cases.

   p. 12, line 251: Has the MIST formalism been validated for IDPs; if so please provide a reference.

   In the present work, we have evaluated the configurational entropy using a mutual information expansion approach with maximum-spanning-tree (MIST) approximation in internal-coordinate (bond-angle-torsion) representation. The latter is particularly well-suited for the analysis of IDPs as it allows one to avoid a number of artifacts (e.g., due to fitting of disordered ensembles to the average structure) associated with the more widely-used Cartesian-coordinate-based quasi-harmonic approaches. In particular, the MIST approach was used previously for the analysis of disordered protein ensembles (https://doi.org/10.1021/acs.jctc.8b00100). Here it should also be noted that, since intramolecular couplings are in general lower in IDPs, this makes them even better suited for MIST as compared to globular proteins. We have highlighted these points on p. 13 of the revision.

   p. 5, line 105, p. 16 line 334 and p. 18 line 283: It is not completely clear what the predictions are and what/which experiments they are compared to. On p. 16, exactly what does the analytical model predict? As far as I understand, the results from the MD simulations are input to the model, but I am probably missing something. Which concrete and testable predictions does the model enable?

   A key contribution of the present work is the development of a quantitative model that treats the spatial organization of a biomolecular condensate across scales using two key properties of individual polymer chains in the condensate - their average valency and compactness. The main predictions of the model concern the presence of a particular scaling of condensate mass with its radius, M(R), as captured by the fractal dimension, and the consequences this has on condensate morphology across scales. In the present manuscript, we have taken the first steps in testing these predictions in four different contexts. First, we could show that the MD simulations indeed match the predictions of fractal scaling for the three smallest clusters, which relates to the discussion on p. 16 that the Reviewer refers to. Here, it is important to understand that MD simulations in the first instance just give the average valency and compactness of individual chains in the dense phase. These values are then input into the fractal scaling formalism, which is conceptually fully independent from MD simulations, to obtain the dependence of condensate mass on its radius, M(R), at any desired length scale. The analysis presented in Figure 5—figure supplement 1B and discussed on p. 16 shows that the predictions of fractal scaling for the first three smallest clusters indeed correspond to what is seen in MD. This is a non-trivial correspondence and can be taken as direct evidence that fractal organization is present even at the shortest scale, i.e. at the level of MD simulation boxes.

   Second, the model was used to reconstruct the spatial organization of clusters of arbitrary size at the atomistic level (Figure 5A and B, Videos 4, 5, and 6), enabling a structural understanding of the organization of condensate interior. One direct practical application of such understanding concerns the nature of cavity sizes and interpretation of dextran partitioning experiments (p. 20). Moreover, as pointed above, differences in morphology of protein clusters propagate across scales, and can be qualitatively characterized by the analysis of microscopic images (see also discussion above). In particular, the model correctly predicts the difference in the behavior of WT and R>K as opposed to Y>A variants, solely based on the predicted fractal dimension they exhibit. Ultimately, however, static light scattering experiments would give the best possibility to test the model directly and will be the topic of our future work. In particular, the fractal formalism predicts significant regions of linear behavior in such curves in log-log representation, while the fractal dimension df, provides a quantitative point of comparison between theoretical predictions and experimental measurements (Figure 5C). These points have been further discussed on p. 21 of the revised manuscript.

   p. 19, lines 408-411: The authors find that when building clusters of Y>A from the simulations they find filamentous structures that they suggest explain the aggregation of the Y>A variant at high concentrations. While that sounds like an intriguing suggestion, it would be useful with a bit more detail about the robustness of this observation. For example, the simulations of Y>A appear similar to that of R>K; are the differences in topology really significantly different?

   Fractal dimension, dF, is the key parameter that defines self-similar organization of differently sized protein clusters according to the fractal model. Consequently, the difference in morphology between R>K and Y>A mutants is reflected in different values of dF for the two. In particular, with a dF of 1.63, the Y>A mutant is predicted to form low-dimensional clusters, straddling the range between a linear (1-dimensional) and a planar (2-dimensional object), unlike WT and R>K variants, which both exhibit dF values greater than 2. The qualitative behavior of the three variants, whereby WT and R>K result in spherical condensates and Y>A does not, is consistent with this. Notably, we have observed sporadic precipitates at high protein concentration in the Y>A mutant, which may be consistent with the predictions of the fractal model. However, the material properties and possible influence of sample impurities in the Y>A case at high concentrations remain unclear. Moreover, the sporadic nature of Y>A precipitates prevents an adequate statistical analysis. Hence, in the revised manuscript we refrain from commenting on these infrequently observed precipitates.

   Regarding MD simulations, the morphological differences between Y>A and R>K proteins can already be seen at the level of individual proteins in multicopy simulations, highlighted by the significantly different distribution of Rg (Figure 4B). This distribution in the case of Y>A has a prominently long tail, which indicates the possibility of adopting significantly more elongated configurations. Due to the self-similarity principle, such differences in morphology may propagate across length scales. Importantly, a recent publication included the experimental study of the possibility of IDRs to form low dimensional fractal systems upon disruption of the LLPS tendency by polyalanine insertion in synthetic elastin-like polypeptides (Roberts et al., Nature Materials, 2018).

   Finally, I would suggest that the authors make their code and data available in electronic format.

   All sharable data has been made available as part of the article package. Due to the heterogeneous character of our analysis, we do not have a single master code to be shared, but rather a collection of different scripts in combination with different software packages as indicated in the Methods section of the manuscript (GROMACS, MATLAB, R, FracVAL).

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3. 

   eLife

   May 12, 2023

   eLife assessment

   This study reports a joint experimental and computational investigation of the structural features of biomolecular condensates formed by a specific intrinsically disordered protein. The authors also adapt emerging rules to discuss and physico-chemical determinants of these structures of condensates. Specifically, the authors adapt the analysis of fractal structures, co-opted from the field of colloidal chemistry / physics, and generate important insights regarding the network-like organization of disordered proteins within in silico facsimiles of condensates. The usage of these analyses in the context of studying all atom models for multi-chain assemblies intended to mimic the internal organization of condensates is very interesting. The work is of relevance to cell biology and structural biology.

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4. 

   eLife

   May 12, 2023

   Reviewer #1 (Public Review):

   The strength of the manuscript is highlighted by the application of fractal formalism, which is commonly used in colloidal systems, in conjunction with MD simulation to study the phase separation of an IDP. The weakness lies in the fact that this study does not provide any discussion on how our understanding of the network structure and dynamical behavior of biomolecular condensates and their biological significance improves through this study. The experimental part remains weak, without any measurements of the dynamics of the condensates. Whether and how the formalism can distinguish between phase-separated condensates (WT) and classical protein aggregates (Y to A variant) remains unclear.

   Read the original source
5. 

   eLife

   May 12, 2023

   Reviewer #2 (Public Review):

   A key aspect of the work is to use the simulations to explain differences between (i) dilute and dense phases and (ii) wild-type and mutant variants. Here, it would be important with a clearer analysis of convergence and errors to quantify which differences are significant.

   It would also be useful with a clearer description of how the analytical model is predictive, of which properties, and how they have been/can be validated. Which measurable quantities does the model predict?

   In addition to these overall questions, a number of more specific suggestions follow below.

   Major:

   p. 7, line 120 (Fig. S1B)
   The proteins do not appear particularly pure based on the presented SDS PAGE analysis. How pure is the protein estimated to be, and is the presence of the other bands expected to affect e.g. the data presented …

   Reviewer #2 (Public Review):

   A key aspect of the work is to use the simulations to explain differences between (i) dilute and dense phases and (ii) wild-type and mutant variants. Here, it would be important with a clearer analysis of convergence and errors to quantify which differences are significant.

   It would also be useful with a clearer description of how the analytical model is predictive, of which properties, and how they have been/can be validated. Which measurable quantities does the model predict?

   In addition to these overall questions, a number of more specific suggestions follow below.

   Major:

   p. 7, line 120 (Fig. S1B)
   The proteins do not appear particularly pure based on the presented SDS PAGE analysis. How pure is the protein estimated to be, and is the presence of the other bands expected to affect e.g. the data presented in Fig. 1?

   p. 7 & 8, lines 138-159:
   Has the method and energy function used to calculate the interact potential been validated by comparison to experiments, including studying the effect of varying the solvent? I see the computed error bars are very small, but am more interested in the average error when comparing to experiments. The numbers in water appear different from those e.g. reported by Krainer et al (https://doi.org/10.1038/s41467-021-21181-9), though the latter are also not immediately compared to experiments. Thus, it would be useful to know how much to trust these numbers.

   p. 8, lines 149-154:
   Following up on the above, the authors also write "Importantly, only in the latter case are the R-Y interactions slightly more favorable than the K-Y ones (Figure S1C). While this can potentially contribute to increasing of Csat for the R>K mutant as compared to WT, the estimated thermodynamic effect is not too strong, especially if one considers that these interactions take place in an environment with largely water-like polarity. Therefore, the effect of R>K substitution on LLPS should be further explored in the context of protein-protein interactions."
   In the absence of estimates of the accuracy of the predictions, these sentences are somewhat unclear. Also, it is unclear what the authors mean by that the effect of R>K should be studied; there are already several examples of this (https://doi.org/10.1016/j.cell.2018.06.006 [already cited], https://doi.org/10.1038/s41557-021-00840-w & https://doi.org/10.1073/pnas.2000223117 come to mind, but there are likely more).

   p. 8, lines 161-162:
   The authors perform MD simulations of Lge1 and variants using 24 copies and a box that gives them protein concentrations "in the mM concentration range". I realize that there's a concern about what is computationally feasible, but it would be important with an argument for this choice. Why is 24 expected to be enough to represent a condensate (I expect that there could be substantial finite-size effects)? What is the exact protein concentration in the simulations of the 24 chains [and of the 1-chain simulations]? How does this protein concentration compare to that in the condensates? The authors performed simulations in the NPT ensemble; how stable were the box dimensions?

   Also, did the authors include the Strep- and His-tags in the simulations? If not, why not?

   Throughout:
   One of my major concerns about this work is the general lack of analysis of convergence of the simulations. The authors must present some solid analysis of which results are robust given the relatively short simulations and potential for bias from the chosen starting structures.

   As an example, on p. 8 the authors discuss a potential asymmetry between the interactions found in the dilute (single-copy) and dense (24-mer) phases. These observations are somewhat in contrast to other observations in the field, namely that it is the same interactions that drive compaction of monomers as those that drive condensate formation.

   Obviously, both the results in the literature and those presented here could be true. But in order to substantiate the statements made here, the authors should show some substantial statistical analyses to make it clear which differences are robust.

   The above holds for all parts of the computational/simulation work (e.g. other aspects of Fig. 2)

   Similarly, how were the errors of the radius of gyration for WT, R>K and Y>A mutants calculated? Is the Rg for WT significantly smaller than the values for the two mutants? And are the differences in Rg between single-copy and multi-copy simulations statistically significant? I am asking since converging the Rg of IDPs of this length in all-atom MD is not easy.

   p. 12, line 251:
   Has the MIST formalism been validated for IDPs; if so please provide a reference.

   p. 5, line 105, p. 16 line 334 and p. 18 line 283:
   It is not completely clear what the predictions are and what/which experiments they are compared to. On p. 16, exactly what does the analytical model predict? As far as I understand, the results from the MD simulations are input to the model, but I am probably missing something.
   Which concrete and testable predictions does the model enable?

   p. 19, lines 408-411:
   The authors find that when building clusters of Y>A from the simulations they find filamentous structures that they suggest explain the aggregation of the Y>A variant at high concentrations. While that sounds like an intriguing suggestion, it would be useful with a bit more detail about the robustness of this observation. For example, the simulations of Y>A appear similar to that of R>K; are the differences in topology really significantly different?

   Finally, I would suggest that the authors make their code and data available in electronic format.

   Read the original source
6. 

   Version published to 10.1101/2022.02.18.481017v2 on bioRxiv

   May 4, 2023
7. 

   Version published to 10.1101/2022.02.18.481017v1 on bioRxiv

   Feb 19, 2022