A model for organization and regulation of nuclear condensates by gene activity

Condensation by phase separation has recently emerged as a mechanism underlying many nuclear compartments essential for cellular functions. Nuclear condensates enrich nucleic acids and proteins, localize to specific genomic regions, and often promote gene expression. How diverse properties of nuclear condensates are shaped by gene organization and activity is poorly understood. Here, we develop a physics-based model to interrogate how spatially-varying transcription activity impacts condensate properties and dynamics. Our model predicts that spatial clustering of active genes can enable precise localization and de novo nucleation of condensates. Strong clustering and high activity results in aspherical condensate morphologies. Condensates can flow towards distant gene clusters and competition between multiple clusters lead to stretched morphologies and activity-dependent repositioning. Overall, our model predicts and recapitulates morphological and dynamical features of diverse nuclear condensates and offers a unified mechanistic framework to study the interplay between non-equilibrium processes, spatially-varying transcription, and multicomponent condensates in cell biology.

-The results of the simulations strongly depend on the selected parameters. The choice of the parameters should be discussed in much more details in the Supplementary Tables, specifying references for each parameter, or the rational behind the choice in the absence of references; -Connected to the previous point, in several parts of the text the authors refer to experimental observations that are in agreement with the simulations, in one case referring to their previous work. These comments are crucial to show the importance of the findings. The impact of the work would further increase by expanding this comparison, in a quantitative way whenever possible. It is also suggested to incorporate elements of this comparison between experiments and simulations in the figures.
-The model is not described in the main text but only in the supplementary. Several terms are not defined and difficult to follow from the main text only. E.g. line 97: kt, kp, x; line 167 det(J) -Minor: line103: Typo " Figure  Nuclear biomolecular condensates are a rich condensate subclass, with interesting additional factors due to their environment, mostly related to the presence of DNA and active transcription. The authors propose a modelling framework that can qualitatively explain features of nuclear condensates, particularly related to their shape and dynamics. The model itself, to my knowledge, has mostly been proposed previously (Ref. 27, Henninger, Oksuz, Shrinivas et al. 2021, Cell). The present study instead explores the model predictions with respect to condensate dynamics and morphology. The paper is unique in that it makes many predictions, none of which are very surprising, but all of which seem to be broadly consistent with the literature. I see the main value of the paper in providing a guideline to which experimental quantifications would be most useful to test our understanding of nuclear condensate formation and maintenance. -I really like the fact that the code is available in a well-structured repository.
-Everything should be dimensional. The claim is applicability to in vivo, so it's necessary to get dimensional diffusion coefficients, etc., to judge whether this is reasonable. Generally given the number of parameters and the lack of any measurements it's hard to make specific claims. Input = Output? Many parameters, can obviously explain many phenomena.
-I cannot comment on how common shape changes in the nucleus upon inhibition of transcription are and whether this could be a side-effect, rather than a direct consequence. This argument is crucial to the paper and hopefully another reviewer will be able to comment on this.
-Would be good to have a table with realistic rates and references next to the simulation paramters used.
-The discussion is well done and uses a well-balanced language.
-I feel the paper would benefit from highlighting a few key results more prominently, e.g. vacuoles and flows and could be made more concise otherwise. None of this is super surprising, the key strength would be quantitative comparison to data.
-compartmentalization would be a more common term than compartmentation? -l 79/80 is way too strong. 81/82 is great! -How does 2D simulation impact conclusions and in particular time scales? 2D and 3D diffusion are very different, and estimating the right time scales is critical to this study's relevance.
Abstract and Intro ------------------Well written, but I feel uncomfortable with the strong claims based on molecular biology terms. None of these things are 'shown', since you can only 'show' molecular biology results by doing molecular biology, not theory. E.g, line 17 "We show that spatial clustering of active genes enables" could be "We show that in our model spatial clustering of active genes enables..." -k_p depends directly on space, so genome organization is not mobile, correct? -the equation part of the figure is copied over from reference 27 ( fig. 4c), also by the authors, this needs to be indicated! Figure 2: ---------The in-figure legend of 2a seems wrong (Gene compartment and Uniform cannot be found in panel) 2c: the overall changes are pretty modest, this would presumably change if system size was larger and larger sigma could be achieved? 2f: why is the boundary so wiggly? Is this a numerical artefact? Boundary conditions? Figure 4: ---------2b/c: plotting Ldr/r vs peak velocity would be more useful to see the transition from no to flow regime. Also, is this an actual jump or a continuous increase in flow velocity? If the latter, then the threshold needs to be discussed, since it is somewhat arbitrary.
Is flow coupled to a significant increase in condensate volume? This would be a potential experimental check for this mechanism. 1

Summary of reviewer response
We thank the reviewers for their helpful feedback and suggestions. Reviewers 2 and 3 note that our work is "... of high quality and physically sound", that we "develop an elegant model" "... several physical insights that could possibly explain many biological observations", "The paper is unique in that it makes many predictions ...", and "of high interest to the wider community". In their critique to improve the study, reviewers bring up concerns regarding model parameterization, connection and motivation to specific biological parameters and experiments, as well as suggestions to better present figures and text. We have incorporated the reviewers' feedback and worked to address their concerns in this revised manuscript, and this has greatly improved our paper. A point-by-point description of changes are provided (in blue text) in the reviewer response.

Point-by-point response to reviewers
Reviewer #1 (Remarks to the Author): In the present manuscript, the author aims to explore the relationship between biomolecular condensation, transcription, and genome organization through physics-based modeling. This is a biophysical problem of great importance and a formidable task which necessitates a multi-scale approach to combine the physical nature of genome architecture with models of phase separation and gene expression. 2. The model lacks any features that one could associate with the genome spatial structure: e.g., the hierarchical organization of loops and domains, epigenetic patterning, and polymeric compartmentalization.
3. The model is not data-driven; it does not incorporate any Hi-C or imaging experiments, and it is not validated against any experimental data, which has become routine in most contemporary models of genome architecture. See, for instance, papers that couple nonequilibrium kinetic processes with genome architecture derived from Hi-C data: 4. Finally and predictably, most of the results from numerical solutions are represented via cartoonish graphs which lack any realism for genome architecture and, contrary to claims, do 4 not provide any insight into the stochastic molecular origin of non-equilibrium coupling between gene expression, genome architecture, and transcription factor phase-separation.
Reviewer 1 suggests that our model is "fundamentally flawed" predicated on their assumption that the goal of this paper is to predict genome structure. This is incorrect. We do not study genome architecture or long-range effects on the scale of the entire genome (Reviewer 1's major concerns).
Rather, our paper models how nuclear condensates are organized and regulated by spatially varying gene expression on length scales shorter than the entire genome, which to our knowledge, has not been done before. Further, the references Reviewer 1 provides to support their arguments do not model RNA/protein condensates, re-entrant phase behavior, or active RNA synthesis -features that are central and essential to our study. Also, because we study specific phenomena concerning gene regulation on shorter length scales and not genome architecture, we do not employ multi-scale models. Rather, we use phase-field models which have been demonstrated to describe cellular phenomena i.e., nuclear condensate phenomenology that are of interest to us as attested to by comparisons with experiments (Henninger*, Oksuz*, Shrinivas* et al., Cell 2021). As Reviewer 2 notes, we "develop an elegant model".
To minimize any potential further confusion, we have revised the text to eliminate any statements that might mislead readers into thinking that we are modeling genome structure.
While we do not specifically model genome architecture, Reviewer 1 (in their third concern) and other reviewers bring up the need for stronger connection between the model we propose and specific biological parameters/experiments. Towards addressing that, we have (this point is repeated in Reviewer 2.1's concerns) developed and now provide a combination of extensive new simulations, analyses, and references to specific biological variables that are summarized below. First, we have performed an extensive literature survey and included a Supplementary Table (Table   S1) describing biologically relevant ranges of parameters used in this study. We note that the purpose of our paper is to describe new mechanisms rather than quantitatively recapitulate specific data, which are both incredibly hard to measure precisely and vary widely for complex biological systems. Therefore, we have attempted to constrain the parameters to be in a biologically relevant range (as the reviewer states).
Second, we justify the choices of the free energy parameters used in this study through figure S2.
Specifically, we look at the free energy parameters and , as they capture the cross-interactions between the RNA and the protein species which couples their dynamics together. We find that both and have to be positive to give rise to a re-entrant phase diagram of the RNA-protein system, which is consistent with experiments (Henninger et al., 2021). Increasing the magnitude of and leads to a qualitatively similar re-entrant phase diagram with the differences largely being in the extent of 5 protein partitioning to the dense phase and the RNA:Protein ratios at which the partition ratio starts to decrease. Given these observations, we used a value of = 1.0 and = 10.0 for the rest of this study, which is consistent with a re-entrant phase transition observed in experiments and qualitatively recapitulates this phase diagram.
We also investigated the impact of using free energy expressions with parameters that are inconsistent with a re-entrant phase transition: by setting = 0 or = 0. For the case with = 0, there are no attractive interactions between the RNA and the protein species whatsoever, and the protein partition ratio monotonically decreases with the RNA:protein ratio ( Figure S2C). In this artificial scenario, our dynamic model predicts that RNA transcription does not lead to condensate nucleation ( Figure S3H), vacuole formation ( Figure S4I), or flow ( Figure S5E). For the case with = 0, the RNA and protein species are always sticky and there are no electrostatic or entropic penalties that prevent phase separation at large concentrations. In this artificial scenario, our dynamic model predicts that RNA transcription can aid condensate nucleation, but does not lead to any condensate dissolution (Figures S3H) or vacuole formation ( Figure S4I). We have summarized the same in the below figure (Rev Fig 1).
Third, we investigate the impact of the different dynamical parameters on condensate nucleation, vacuole formation, and flow. For example, the RNA degradation rate, which is a measure of RNA 6 Additionally, our model predicts that condensate nucleation, vacuole formation, and flow do not occur in a parameter regime where the RNA mobility is much larger than the protein mobility i.e. / >> 1 ( Figures S3F, S4G, S5A). This reflects the observed biological constraints that RNA often diffuse much slower than proteins, including in part, due to tethering to chromatin or increased bulk. We have summarized the same below (Rev Fig 3).
Finally, we have added text to reflect that the results of the simulations depend on parameters (Section 1 in Results,[167][168][169][170][171][172][195][196][197][203][204][205][206][207][208], which in turn, are motivated by biological observations. 7 Reviewer #2 (Remarks to the Author): In this work Schede H.H. et al. develop an elegant model to describe several aspects of the spatiotemporal interplay between condensation, active transcription, and genome organization. The simulations provide several physical insights that could possibly explain many biological observations, and will be of high interest to the wider community. The paper is very well written. I have a few general remarks to further improve the presentation of the work and the impact for a more biological readership. The work is of high quality and physically sound. The biological relevance of the simulations can be further strengthened.
We thank the reviewer for their positive assessment of our paper as well as the specific feedback for increasing biological relevance and improving presentation. Addressing the comments of this reviewer has greatly strengthened our paper.
1. The results of the simulations strongly depend on the selected parameters. The choice of the parameters should be discussed in much more details in the Supplementary Tables, specifying references for each parameter, or the rational behind the choice in the absence of references; We thank the reviewer for bringing up this point and have addressed their concern through a combination of extensive new simulations and analyses.
First, we have performed an extensive literature survey and included a Supplementary Table   (Table S1) describing biologically relevant ranges of parameters used in this study. We note that the purpose of our paper is to describe new mechanisms rather than quantitatively recapitulate specific data, which are both incredibly hard to measure precisely and vary widely for complex biological systems. Therefore, we have attempted to constrain the parameters to be in a biologically relevant range (as the reviewer states).
Second, we justify the choices of the free energy parameters used in this study through figure   S2. Specifically, we look at the free energy parameters and , as they capture the crossinteractions between the RNA and the protein species which couples their dynamics together.
We find that both and have to be positive to give rise to a re-entrant phase diagram of the 8 RNA-protein system, which is consistent with experiments (Henninger et al., 2021).
Increasing the magnitude of and leads to a qualitatively similar re-entrant phase diagram with the differences largely being in the extent of protein partitioning to the dense phase and the RNA:Protein ratios at which the partition ratio starts to decrease. Given these observations, we used a value of = 1.0 and = 10.0 for the rest of this study, which is consistent with a re-entrant phase transition observed in experiments and qualitatively recapitulates this phase diagram.
We also investigated the impact of using free energy expressions with parameters that are inconsistent with a re-entrant phase transition: by setting = 0 or = 0. For the case with = 0, there are no attractive interactions between the RNA and the protein species whatsoever, and the protein partition ratio monotonically decreases with the RNA:protein ratio ( Figure   S2C). In this artificial scenario, our dynamic model predicts that RNA transcription does not lead to condensate nucleation ( Figure S3H), vacuole formation ( Figure S4I), or flow ( Figure   S5E). For the case with = 0, the RNA and protein species are always sticky and there are no electrostatic or entropic penalties that prevent phase separation at large concentrations. In this artificial scenario, our dynamic model predicts that RNA transcription can aid condensate nucleation, but does not lead to any condensate dissolution (Figures S3H) or vacuole formation ( Figure S4I). We have summarized the same in the below figure (Rev Fig 1). proportionally larger values of ( Figures S3E, S4F) and results in flow with a lower flow velocity ( Figure 4E), arising from weaker gradients in RNA concentrations. We have summarized the same in the below figure (Rev Fig 2).
Additionally, our model predicts that condensate nucleation, vacuole formation, and flow do not occur in a parameter regime where the RNA mobility is much larger than the protein mobility i.e. / >> 1 (Figures S3F, S4G, S5A). This reflects the observed biological constraints that RNA often diffuse much slower than proteins, including in part, due to tethering to chromatin or increased bulk. We have summarized the same below (Rev Fig 3).
Finally, we have added text to reflect that the results of the simulations depend on parameters (Section 1 in Results,[167][168][169][170][171][172][195][196][197][203][204][205][206][207][208], which in turn, are motivated by biological observations. We thank the reviewer for this remark and have addressed this point in the revised study by incorporating references of the various biological parameters and phenomena for comparison. First, we have added several references (Table S1)  "Further, we find that the dimensionless flow velocity predicted by our model (SI Dimensionless flow velocity) in Figure 4B corresponds approximately to an intracellular velocity (SI Table 1 We thank the reviewer for pointing this out and have addressed the concern by revising the text and modifying the figures to improve clarity.  We thank the reviewer for their feedback and for highlighting the uniqueness of our model. We are glad that the reviewer finds our model's predictions easy to follow, which although not surprising to the reviewer, have never previously been unified under a single physical framework and we thus believe represents a worthwhile contribution (as the reviewer notes). In addition, we agree that a significant value of this paper is to emphasize guidelines to test or refine this model. We have now included connections to previous experiments as well as specific tests of model predictions. Further, by addressing the many thoughtful suggestions the reviewer provides below, we believe we have significantly strengthened our model and connection to biological experiments. Note that the equations are now found in Fig S1 to improve clarity of presentation.
2. I really like the fact that the code is available in a well-structured repository.
Thank you! 13 3. Everything should be dimensional. The claim is applicability to in vivo, so it's necessary to get dimensional diffusion coefficients, etc., to judge whether this is reasonable. Generally given the number of parameters and the lack of any measurements it's hard to make specific claims.
We thank the reviewer for their comment and believe we have largely addressed their concerns through a combination of revised text and the addition of a supplementary constrain key parameters of the free energy functional ( Figure S2) as well as the dynamic parameters to broadly be in the range of biophysical observations (Table S1)." To address the concern regarding applicability in vivo, we have included references for phenomena occurring in vivo in order to link experimental observations in a more quantitative manner to our findings from model simulations (Supplementary Table 1 "Further, we find that the dimensionless flow velocity predicted by our model (SI Figure 4B corresponds approximately to an intracellular velocity (SI Table 1 S4G-H, S5A). This is summarized below (Reviewer Figure 3). Finally, our model also sheds light on the necessary conditions required for condensate nucleation, vacuole formation and flow. We have added the following lines to the main text to clearly highlight these points [205][206][207][256][257][258][266][267][268]: "Consistent with this, we find that increasing RNA mobility or increasing RNA degradation, both causing weaker gradients through distinct mechanisms, leads to slower condensate motion ( Figures 4D-E)."

Dimensionless flow velocity) in
These are specific statements that can be used to design experiments that test our model.
For example, we predict that disrupting the RNA-protein attractive interactions should prevent de novo nucleation of condensates and flow. Increasing the RNA mobility by having shortlength RNA species for example should prevent condensate nucleation and vacuole formation. Faster degradation of the RNA species should cause a reduction in the flow velocity.
4. Input = Output? Many parameters, can obviously explain many phenomena.
We thank the reviewer for this remark. To address the concern, we have performed extensive  Figure S2). Similarly, the ratio of RNA and protein mobilities are < 1.
Subject to these constraints on the parameters, we show that the model can exhibit diverse phenomena. This is very different from a neural network for example, where the parameters are relatively unconstrained by any mechanistic reasoning and lack any physical meaning. To demonstrate this, we have added Figure S2 to show that the re-entrant phase separation is supported under only specific regimes of parameters as well as Figures S3H, S4I, S5E that indicate that the model is no longer capable of predicting emergent phenomena when parameters do not reflect the underlying biology. The same is summarized below (Rev Fig 1): 16 Additionally, we also performed simulations to assess the sensitivity of the results to the RNA degradation rate, which is a measure of RNA stability, does not affect the qualitative nature of our results (figures S3E, S4F). Increasing the RNA degradation rate results in condensate nucleation and vacuole formation at proportionally larger values of and results in flow with a lower flow velocity, arising from weaker gradients in RNA concentrations ( Figure 4E). The same is summarized in the below figure (Rev Fig 2).
5. I cannot comment on how common shape changes in the nucleus upon inhibition of transcription are and whether this could be a side-effect, rather than a direct consequence.
This argument is crucial to the paper and hopefully another reviewer will be able to comment on this.
We thank the reviewer for this comment and have added citations as well as modifications to the text in the results section to provide instances of condensate shape changes in the nucleus in response to alterations in transcription activity. Our model predicts similar changes in response to transcription inhibition. We predicted that active transcription from multiple sites has the potential to stretch the condensate and cause them to split ( Figure 6, Figure S5). Under specific conditions it would hence be plausible that  (Table S1) and rationale for choice of parameters and pertinent references (SI figures S2, S3E-H, S4F-I).
7. The discussion is well done and uses a well-balanced language Thank you! 8. I feel the paper would benefit from highlighting a few key results more prominently, e.g.
vacuoles and flows and could be made more concise otherwise. None of this is super surprising, the key strength would be quantitative comparison to data.
We thank the reviewer for pointing this out. We agree with the reviewer that the results are sensible and hence not surprising. This fact allows us to be more confident about the validity of the model which would otherwise be questionable. In cis was supposed to refer to spatially proximal effects, which we now clarify using simplified language.
Abstract and Intro ------------------Well written, but I feel uncomfortable with the strong claims based on molecular biology terms. None of these things are 'shown', since you can only 'show' molecular biology results by doing molecular biology, not theory. E.g, line 17 "We show that spatial clustering of active genes enables" could be "We show that in our model spatial clustering of active genes enables..." We thank the reviewer for their suggestion and have suitably softened the language in the text. We have modified words like "show" or "shown" to "our model predicts" or "our model indicates" or "our simulations show".
Specifically, we have changed the wording in the following lines: Previous: We show that spatial clustering of active genes enables precise localization and de novo nucleation of condensates  -k_p depends directly on space, so genome organization is not mobile, correct?
Yes, that is correct. We have clarified this in Figure 1B.
-the equation part of the figure is copied over from reference 27 ( fig. 4c), also by the authors, this needs to be indicated! We thank the reviewer for pointing this out and have cited the reference in the figure caption. We thank the reviewer for having pointed this out. We have changed the in-figure legend.
2c: the overall changes are pretty modest, this would presumably change if system size was larger and larger sigma could be achieved?
Yes. In addition to this, the change in condensate radius between small and large sigma is also going to depend on the initial concentration of the protein/amount of protein in the system.
2f: why is the boundary so wiggly? Is this a numerical artefact? Boundary conditions?
It reflects that the finite sampling of a discrete set of and sigma, which we now clearly state in our caption of Figure 2F. Also, is this an actual jump or a continuous increase in flow velocity? If the latter, then the threshold needs to be discussed, since it is somewhat arbitrary.
We thank the reviewer for their comment and have addressed their question by significantly revising our presentation of the related text and figures. Following the reviewer's suggestion, we first plotted / versus peak velocity as we change distance and mobility as shown below and observe the change from flow to no-flow.:

22
We find that the increase in flow velocity is a continuous change but happens rather quickly and thus appears abrupt. To better understand the origin of this phenomenon, we developed a new theoretical analysis method to better approximate the RNA gradient (SI Theory of flow). We derive an analytical Is flow coupled to a significant increase in condensate volume? This would be a potential experimental check for this mechanism.
We thank the reviewer for their comment. The size of the condensate at steady state is set by the parameters and of the gene locus. If we start with a nucleated droplet of a larger size than this, the condensate will shrink. If we start with a droplet that is smaller than this size, it will expand.
Therefore, we would like to be careful and not make any statements about size changes during directed flow. We have changed the caption of figure 4A to avoid this confusion. 1

Summary of reviewer response
We thank the reviewers for their response and positive assessment of our revised study. We have addressed the remaining minor concerns of Reviewer 2 and outline them point-by-point below.
Reviewer #2 (Remarks to the Author): The Minor: - Table S1: Da>>>1: it is rather vague, this can be any number.
We now specify the ranges we simulated explicitly in Table S1.
-Caption Fig. 4B: "Compare this to figure 2D in Kim et al". This is unclear. I would suggest to make the comparison explicit here, as in the main text.
The comparison is now made explicit as it was in the main text.
- Figure S3d: why nucleation time does not increase monotonically in the range of kr between 100 and 500?
We thank the reviewer for making this observation. When we looked at the condensate radius on the accompanying left panel in S3D, one clearly notes a monotonically decreasing size with gene activity.
This discrepancy led us to double check the original panel and we identified that the nucleation time panel (on the right) was incorrectly plotted by accident. We corrected this (including through validation by re-running simulations again) error -as shown in the updated version of the current Figure S3SD, the nucleation time monotonically decreases with increasing transcription activity as expected from the condensate size. This finding is intuitive, as the rate of transcription increases, the RNA concentration required to nucleate a dense phase is achieved quicker and thus nucleation times decrease with increasing activity.
-I leave this decision to the authors, but I would suggest adding in the title something along the line "Modelling organization and regulation of..." We propose the following title that we believe addresses the reviewer's concern: "A model for organization and regulation of nuclear condensates by gene activity"