Quantitative theory for the diffusive dynamics of liquid condensates
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Evaluation Summary:
This paper will be of broad interest to chemists and biologists studying complex coacervate systems, including biomolecular condensates. Its model provides a new way of obtaining diffusion properties inside and outside the condensates without the necessity of nontrivial assumptions. The model's capability is well presented by applying to experimental data and through further investigating the model through simulations.
(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. Reviewer #1 agreed to share their name with the authors.)
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Abstract
Key processes of biological condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels. However, to date, a physics-based, quantitative framework for the dynamics of labeled condensate components is lacking. Here, we derive the corresponding dynamic equations, building on the physics of phase separation, and quantitatively validate the related framework via experiments. We show that by using our framework, we can precisely determine diffusion coefficients inside liquid condensates via a spatio-temporal analysis of fluorescence recovery after photobleaching (FRAP) experiments. We showcase the accuracy and precision of our approach by considering space- and time-resolved data of protein condensates and two different polyelectrolyte-coacervate systems. Interestingly, our theory can also be used to determine a relationship between the diffusion coefficient in the dilute phase and the partition coefficient, without relying on fluorescence measurements in the dilute phase. This enables us to investigate the effect of salt addition on partitioning and bypasses recently described quenching artifacts in the dense phase. Our approach opens new avenues for theoretically describing molecule dynamics in condensates, measuring concentrations based on the dynamics of fluorescence intensities, and quantifying rates of biochemical reactions in liquid condensates.
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Author Response:
Reviewer #2 (Public Review):
[...] Despite the success in the primary purpose of the model, we note concerning issues that we recommend the authors address.
There is a recurrent lack of clarity in many sections of text and how the author's claims are supported by the evidence shown. Though we were able to fully understand the study, interpretation was needlessly difficult at times. Outstanding, but not exhaustive, examples are listed below:
- Overstatement of capabilities of this model that are weakly or not supported by the study. At various points throughout the article, the authors speak of the applicability of their model to nuanced conditions that we believe is either only indirectly supported, or not supported at all by their evidence.
a. On line 62 the authors claim that this model uses non-equilibrium …
Author Response:
Reviewer #2 (Public Review):
[...] Despite the success in the primary purpose of the model, we note concerning issues that we recommend the authors address.
There is a recurrent lack of clarity in many sections of text and how the author's claims are supported by the evidence shown. Though we were able to fully understand the study, interpretation was needlessly difficult at times. Outstanding, but not exhaustive, examples are listed below:
- Overstatement of capabilities of this model that are weakly or not supported by the study. At various points throughout the article, the authors speak of the applicability of their model to nuanced conditions that we believe is either only indirectly supported, or not supported at all by their evidence.
a. On line 62 the authors claim that this model uses non-equilibrium thermodynamics to capture the diffusion across the droplet interface. This implies that the model would be applicable to dynamic processes in which detailed balance is not preserved. While exchange of photobleached and unphotobleached fluorescently labelled components is a dynamic process, the authors explicitly assume that the volume fraction of condensate components within a droplet (Φtot) remains either at equilibrium or quasi-equilibrium when building their model.
We agree that in our manuscript we focus on the case where the total volume fraction (composed of bleached and unbleached molecules) is at thermodynamic equilibrium leading to partial_t phi_tot = 0. Please note that our theory can be applied to non-equilibrium situations, i.e., in the presence of fluxes. For example, see e.g. Eq. (6) in Bo et al. (reference in manuscript), where we use this coarse-grained theory to derive a single molecule description also away from thermodynamic equilibrium.
Based on the reviewer’s remark, we have revised the paragraph around Eq. (2) and now stress more clearly that we focus on the cases where detailed balance holds.
b. On line 272 where the authors claim that their model is applicable to non-spherical droplets, referencing Fig. 3 as evidence. However, Fig. 3 and the accompanying text sections starting on lines 163 and 188 describe effects of different environments on an explicitly spherical droplet. In particular, the distance to a coverslip (h) and between neighbouring droplets (d) never drops below the spherical droplet radius (r). We believe this data would constitute evidence of the model's applicability to a non-spherical droplet. Another concern is that the dynamic boundary condition could be dependent on the ratio between the bleaching spot radius and the condensate radius. Thus the authors should discuss the applicability of their theory when Rbleaching spot << Rcondensate and when Rbleaching spot >> Rcondensate.
We were referring to the theoretical derivation of Eq. (6), which is general and does not depend on the specific droplet shape or boundaries. We revised the paragraph to improve clarity. While the fitting and imaging procedure would get more involved, the theory is still applicable to a non-spherical scenario. Regarding the bleach spot radius, we now mention the required bleach geometry in the first results section, to indicate that a full bleach is necessary, if this data analysis framework should be applied. Please note that a larger bleaching area does not alter the procedure pointed out in Fig. 1, while bleaching less than droplet size is not allowed within the framework of Eq. (1), due to breaking spherical symmetry of fluorescent protein within the droplet. It would be allowed if we were to drop spherical symmetry in the data analysis.
c. Having claimed that not only Din, but also Dout and P can be determined, in principle, from analyzing a single FRAP experiment, it is unclear why they do not show this capability using the experimental data they have. It is especially obscure because the cost function and how it is calculated is not described at all.
We now describe both cost functions in the flow charts of figures 1 and 4. We have also made a clearer distinction between the theoretically possible determination of all parameters and the experimental feasibility, including a new section heading. Other than effects that are unaccounted for in our theory, such as a potential interfacial resistance, current experiments are also hampered by several effects, such as the presence of a coverslip (which makes P appear artificially large), non-uniform bleaching in the bulk, imaging artefacts at the droplet boundary etc. This means that the global minimum which we find in the cost function when comparing experimental data to our model is not reliable and cannot be used to extract both P and D_out given current limitations.
d. In the Discussion section, the authors claimed that their model can be generally applied to study the diffusive properties of biomolecular condensates. However, recent literature (e.g. Biophys. J. 2019, 117, 1285-1300, Nature 2017, 547, 241-245, etc.) reported that the diffusion at the biomolecular condensate interface cannot be treated with local equilibrium due to interfacial resistance. This work did not take these interfacial effects into consideration, and the authors should explain if they expect these phenomenological effects to hamper the application of their theory.
We have now included a brief discussion of a possible interfacial resistance. While there has been a discussion in the field about this possibility, this has not been shown conclusively to the best of our knowledge. Strom et al. measure diffusivity across the boundary. However, it is unclear what the resolution of their FCS derivative is, and whether these effects could potentially be explained by correlated movement near the boundary. This correlated movement is expected due to a molecule’s tendency to stay inside a droplet, which is expected even without an interfacial resistance. The only work we are aware of (now brought to our attention by Reviewer #1), that investigates resistance at the boundary, are the papers by Hahn et al. and Gebhard et al., which initially found no evidence of a mass transfer resistance (J. Phys.: Condens. Matter 23 (2011) 184107 (8pp)). Later some evidence was found for a mass transfer resistance for proteins (Soft Matter, 2021, 17, 3929). However, these papers investigate a three-component system, where the investigated molecules adsorb to a liquid-liquid interface, which we find no evidence for. We thus remain cautious about the potential role of an interfacial resistance in our simple two-component set-up.
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Evaluation Summary:
This paper will be of broad interest to chemists and biologists studying complex coacervate systems, including biomolecular condensates. Its model provides a new way of obtaining diffusion properties inside and outside the condensates without the necessity of nontrivial assumptions. The model's capability is well presented by applying to experimental data and through further investigating the model through simulations.
(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. Reviewer #1 agreed to share their name with the authors.)
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Reviewer #1 (Public Review):
In this work, the authors develop and apply a physical theory for interpreting fluorescence recovery after photobleaching (FRAP) data in the context of phase separation. Physical theories have been developed for interpreting and fitting FRAP data in other contexts, but such an effort has been missing in the context of phase separation.
The authors introduce a new dynamic boundary condition and show how this can be applied practically to extract diffusion coefficients for labeled molecules within condensates. Following this, the authors separate the molecules into bleached and unbleached species, apply mass balance, and rewrite diffusion equations, deriving the chemical potentials from a Flory-Huggins free energy functions, in terms of the temporal and spatial evolution of unbleached molecules. The authors …
Reviewer #1 (Public Review):
In this work, the authors develop and apply a physical theory for interpreting fluorescence recovery after photobleaching (FRAP) data in the context of phase separation. Physical theories have been developed for interpreting and fitting FRAP data in other contexts, but such an effort has been missing in the context of phase separation.
The authors introduce a new dynamic boundary condition and show how this can be applied practically to extract diffusion coefficients for labeled molecules within condensates. Following this, the authors separate the molecules into bleached and unbleached species, apply mass balance, and rewrite diffusion equations, deriving the chemical potentials from a Flory-Huggins free energy functions, in terms of the temporal and spatial evolution of unbleached molecules. The authors demonstrate that if the internal diffusion coefficient is fixed, the free parameters become the diffusion coefficient outside the condensate, and the partition coefficient.
Their analysis shows that the external diffusion coefficient and the partition coefficient are related to one another. This suggests that their measurement of the internal diffusion coefficient combined with an independent measurement of external diffusion coefficients should enable the extraction of partition coefficients from FRAP data. Their approach avoids the use of empirical fitting functions, and the suggestion is that FRAP data, analyzed using a pre-determined internal diffusion coefficient, based on the dynamic boundary condition, can enable the extraction of transport properties outside condensates and thermodynamic partitioning coefficients.
This work is likely to be of broad interest, providing the numerical apparatus is made more transparent to the average reader and to users of FRAP, of which there are many. The three shortcomings are the lack of a direct connection to the interfacial tension, and the absence of analytical solutions, at least in asymptotic limits, and the description of the so-called cost function, which leads to the finding that the external diffusion coefficient scales essentially linearly with the partition coefficient in the limit of large P.
Overall, the authors seem to have achieved the goals they set for themselves, and given the immense interest in using FRAP measurements, there is a clear need for quantitative approaches to analyzing these data. On a semantic note, it is high time that we dispensed with the insistence that these condensates have to be liquids. Nothing about FRAP measurements or the underlying data stipulates that any of these systems have to be liquids. The prefix of liquid or liquid-liquid is, given all we now know, not really relevant or accurate.
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Reviewer #2 (Public Review):
This paper establishes a novel theoretical model for diffusion in a liquid-liquid phase separation (LLPS) system to be used in fluorescence microscopy experiments for determination of diffusion and partition coefficients of the system via FRAP data. The crux of the model presented in this study is that it is the first to be derived using a theoretical framework tailored for a LLPS system. The starting points of the model are bulk diffusion theory combined with solution theory, adapted for diffusion across a droplet interface. As the authors note citing Taylor et. al. 2019, previous analyses relying on various phenomenological fitting schemes result in significant discrepancies between models due to underlying assumptions about the system. As LLPS and complex coacervation are the predominant frameworks that …
Reviewer #2 (Public Review):
This paper establishes a novel theoretical model for diffusion in a liquid-liquid phase separation (LLPS) system to be used in fluorescence microscopy experiments for determination of diffusion and partition coefficients of the system via FRAP data. The crux of the model presented in this study is that it is the first to be derived using a theoretical framework tailored for a LLPS system. The starting points of the model are bulk diffusion theory combined with solution theory, adapted for diffusion across a droplet interface. As the authors note citing Taylor et. al. 2019, previous analyses relying on various phenomenological fitting schemes result in significant discrepancies between models due to underlying assumptions about the system. As LLPS and complex coacervation are the predominant frameworks that currently underlie the study of biomolecular condensates, a model that enables accurate and consistent determination of the physical properties of such condensates is of great interest to the field. The authors use both experimental and simulated datasets to successfully show that the diffusion constant inside the condensate can be precisely determined from a single FRAP experiment. The biggest strength here is that diffusion within the droplets can be independently determined without regarding any other parameter or situation outside the condensate. Additionally, when given specific data, their model finds a strong relationship between the diffusion constant in the dilute phase and the partition coefficient, such that if one is known, the other can be determined. Furthermore, they show, in principle, a global cost minimum exists and determine both of these parameters simultaneously. The model presented here will be a strong starting point for improved physical characterization of coacervates and protein condensates to be further adapted to complex systems. Despite the success in the primary purpose of the model, we note concerning issues that we recommend the authors address.
There is a recurrent lack of clarity in many sections of text and how the author's claims are supported by the evidence shown. Though we were able to fully understand the study, interpretation was needlessly difficult at times. Outstanding, but not exhaustive, examples are listed below:
1. Overstatement of capabilities of this model that are weakly or not supported by the study. At various points throughout the article, the authors speak of the applicability of their model to nuanced conditions that we believe is either only indirectly supported, or not supported at all by their evidence.
a. On line 62 the authors claim that this model uses non-equilibrium thermodynamics to capture the diffusion across the droplet interface. This implies that the model would be applicable to dynamic processes in which detailed balance is not preserved. While exchange of photobleached and unphotobleached fluorescently labelled components is a dynamic process, the authors explicitly assume that the volume fraction of condensate components within a droplet (Φtot) remains either at equilibrium or quasi-equilibrium when building their model.
b. On line 272 where the authors claim that their model is applicable to non-spherical droplets, referencing Fig. 3 as evidence. However, Fig. 3 and the accompanying text sections starting on lines 163 and 188 describe effects of different environments on an explicitly spherical droplet. In particular, the distance to a coverslip (h) and between neighbouring droplets (d) never drops below the spherical droplet radius (r). We believe this data would constitute evidence of the model's applicability to a non-spherical droplet. Another concern is that the dynamic boundary condition could be dependent on the ratio between the bleaching spot radius and the condensate radius. Thus the authors should discuss the applicability of their theory when Rbleaching spot << Rcondensate and when Rbleaching spot >> Rcondensate.
c. Having claimed that not only Din, but also Dout and P can be determined, in principle, from analyzing a single FRAP experiment, it is unclear why they do not show this capability using the experimental data they have. It is especially obscure because the cost function and how it is calculated is not described at all.
d. In the Discussion section, the authors claimed that their model can be generally applied to study the diffusive properties of biomolecular condensates. However, recent literature (e.g. Biophys. J. 2019, 117, 1285-1300, Nature 2017, 547, 241-245, etc.) reported that the diffusion at the biomolecular condensate interface cannot be treated with local equilibrium due to interfacial resistance. This work did not take these interfacial effects into consideration, and the authors should explain if they expect these phenomenological effects to hamper the application of their theory.
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