Transport kinetics across interfaces between coexisting liquid phases
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eLife Assessment
This study offers a valuable theoretical framework for quantifying molecular transport across interfaces between coexisting liquid phases, emphasizing interfacial resistance as a central factor governing transport kinetics. The mathematical derivations are solid. To enhance the paper's relevance and broaden its appeal, it would be helpful to clarify how the key equations connect to existing literature and to elucidate the physical mechanisms underlying scenarios that give rise to substantial interfacial resistance.
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Abstract
Biomolecular condensates provide compartments that organize biological processes in the cell. Spatial organization of condensates relies on transport across phase boundaries. We investigate the kinetics of molecule transport across interfaces in phase-separated mixtures. Using non-equilibrium thermodynamics we derive the interfacial kinetics, describing the movement of the interface, and of molecular transport across the interface. In the limit of a thin interface, we show that breaking local equilibrium at the interface introduces a local transport resistance. Subsequently, using a continuum approach, we explore two physical scenarios where such a resistance emerges: a mobility minimum and a potential barrier at the interface. These scenarios lead to the same effective transport model in the sharp interface limit, characterized by a single interface resistance parameter. We discuss experimental conditions for which interface resistance could be observed and how these scenarios could be distinguished.
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eLife Assessment
This study offers a valuable theoretical framework for quantifying molecular transport across interfaces between coexisting liquid phases, emphasizing interfacial resistance as a central factor governing transport kinetics. The mathematical derivations are solid. To enhance the paper's relevance and broaden its appeal, it would be helpful to clarify how the key equations connect to existing literature and to elucidate the physical mechanisms underlying scenarios that give rise to substantial interfacial resistance.
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Reviewer #1 (Public review):
Summary:
In this manuscript, the authors theoretically address the topic of interface resistance between a phase-separated condensate and the surrounding dilute phase. In a nutshell, "interface resistance" occurs if material in the dilute phase can only slowly pass through the interface region to enter the dense phase. There is some evidence from FRAP experiments that such a resistance may exist, and if it does, it could be biologically relevant insofar as the movement of material between dense and dilute phases can be rate-limiting for biological processes, including coarsening. The current study theoretically addresses interface resistance at two levels of description: first, the authors present a simple way of formulating interface resistance for a sharp interface model. Second, they derive a formula for …
Reviewer #1 (Public review):
Summary:
In this manuscript, the authors theoretically address the topic of interface resistance between a phase-separated condensate and the surrounding dilute phase. In a nutshell, "interface resistance" occurs if material in the dilute phase can only slowly pass through the interface region to enter the dense phase. There is some evidence from FRAP experiments that such a resistance may exist, and if it does, it could be biologically relevant insofar as the movement of material between dense and dilute phases can be rate-limiting for biological processes, including coarsening. The current study theoretically addresses interface resistance at two levels of description: first, the authors present a simple way of formulating interface resistance for a sharp interface model. Second, they derive a formula for interface resistance for a finite-width interface and present two scenarios where the interface resistance might be substantial.
Strengths:
The topic is of broad relevance to the important field of intracellular phase separation, and the work is overall credible.
Weaknesses:
There are a few problems with the study as presented - mainly that the key formula for the latter section has already been derived and presented in Reference 6 (notably also in this journal), and that the physical basis for the proposed scenarios leading to a large interface resistance is not clearly supported.
(1) As noted, Equation 32 of the current study is entirely equivalent to Equation 8 of Reference 6, with a very similar derivation presented in Appendix 1 of that paper. In fact, Equation 8 in Reference 6 takes one more step by combining Equations 32 and 35 to provide a general expression for the interface resistance in an integral form. These prior results should be properly cited in the current work - the existing citations to Reference 6 do not make this overlap apparent.
(2) The authors of the current study go on to examine cases where this shared equation (here Equation 32) might imply a large interface resistance. The examples are mathematically correct, but physically unsupported. In order to produce a substantial interface resistance, the current authors have to suppose that in the interface region between the dense and dilute phases, either there is a local minimum of the diffusion coefficient or a local minimum of the density. I am not aware of any realistic model that would produce either of these minima. Indeed, the authors do not present sufficient examples or physical arguments that would support the existence of such minima.
In my view, these two issues limit the general interest of the latter portion of the current manuscript. While point 1 can be remedied by proper citation, point 2 is not so simple to address. The two ways the authors present to produce a substantial interface resistance seem to me to be mathematical exercises without a physical basis. The manuscript will improve if the authors can provide examples or compelling arguments for a minimum of either diffusion coefficient or density between the dense and dilute phases that would address point 2.
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Reviewer #2 (Public review):
Summary:
This work provides a general theoretical framework for understanding molecular transport across liquid-liquid phase boundaries, focusing on interfacial resistance arising from deviations from local equilibrium. By bridging sharp and continuous interface descriptions, the authors demonstrate how distinct microscopic mechanisms can yield similar effective kinetics and propose practical experimental validation strategies.
Strengths:
(1) Conceptually rich and physically insightful interface resistance formulation in sharp and continuous limits.
(2) Strong integration of non-equilibrium thermodynamics with biologically motivated transport scenarios.
(3) Thorough numerical and analytical support, with thoughtful connection to current and emerging experimental techniques.
(4) Relevance to various systems, …
Reviewer #2 (Public review):
Summary:
This work provides a general theoretical framework for understanding molecular transport across liquid-liquid phase boundaries, focusing on interfacial resistance arising from deviations from local equilibrium. By bridging sharp and continuous interface descriptions, the authors demonstrate how distinct microscopic mechanisms can yield similar effective kinetics and propose practical experimental validation strategies.
Strengths:
(1) Conceptually rich and physically insightful interface resistance formulation in sharp and continuous limits.
(2) Strong integration of non-equilibrium thermodynamics with biologically motivated transport scenarios.
(3) Thorough numerical and analytical support, with thoughtful connection to current and emerging experimental techniques.
(4) Relevance to various systems, including biomolecular condensates and engineered aqueous two-phase systems.
Weaknesses:
(1) The work remains theoretical, mainly, with limited direct comparison to quantitative experimental data.
(2) The biological implications are only briefly explored; further discussion of specific systems where interface resistance might play a functional role would enhance the impact.
(3) Some model assumptions (e.g., symmetric labeling or idealized diffusivity profiles) could be further contextualized regarding biological variability.
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Reviewer #3 (Public review):
The manuscript investigated the kinetics of molecule transport across interfaces in phase-separated mixtures. Through the development of a theoretical approach for a binary mixture in a sharp interface limit, the authors found that interface resistance leads to a slowdown in interfacial movement. Subsequently, they extended this approach to multiple molecular species (incorporating both labeled and unlabeled molecules) and continuous transport models. Finally, they proposed experimental settings in vitro and commented on the necessary optical resolution to detect signatures of interfacial kinetics associated with resistance.
The investigation of transport kinetics across biomolecular condensate interfaces holds significant relevance for understanding cellular function and dysfunction mechanisms; thus, the …
Reviewer #3 (Public review):
The manuscript investigated the kinetics of molecule transport across interfaces in phase-separated mixtures. Through the development of a theoretical approach for a binary mixture in a sharp interface limit, the authors found that interface resistance leads to a slowdown in interfacial movement. Subsequently, they extended this approach to multiple molecular species (incorporating both labeled and unlabeled molecules) and continuous transport models. Finally, they proposed experimental settings in vitro and commented on the necessary optical resolution to detect signatures of interfacial kinetics associated with resistance.
The investigation of transport kinetics across biomolecular condensate interfaces holds significant relevance for understanding cellular function and dysfunction mechanisms; thus, the topic is important and timely. However, the current manuscript presentation requires improvement. Firstly, the inclusion of numerous equations in the main text substantially compromises readability, and relocation of a part of the formulae and derivations to the Appendix would be more appropriate. Secondly, the manuscript would benefit from more comprehensive comparisons with existing theoretical studies on molecular transport kinetics. The text should also be written to be more approachable for a general readership. Modifications and sufficient responses to the specific points outlined below are recommended.
(1) The authors introduced a theoretical framework to study the kinetics of molecules across an interface between two coexisting liquid phases and found that interface resistance leads to a slowdown in interfacial movement in a binary mixture and a decelerated molecule exchange between labeled and unlabeled molecules across the phase boundary. However, these findings appear rather expected. The work would be strengthened by a more thorough discussion of the kinetics of molecule transport across interfaces (such as the physical origin of the interface resistance and its specific impact on transport kinetics).
(2) The formulae in the manuscript should be checked and corrected. Notably, Equation 10 contains "\phi_2\ln\phi_2" while Eq. 11b shows "n^{-1}\ln\phi_2", suggesting a missing factor of "n^{-1}". Similarly, Equation 18 obtained from Equation 11: the logarithmic term in Eq.11a is "n^{-1}\ln phi_1-\ln(1-\phi)" but the pre-exponential factor in Equation 18a is just "\phi_1/(1-\phi*)", where is "n^{-1}"? Additionally, there is a unit inconsistency in Equation 36, where the unit of \rho (s/m) does not match that of the right-hand side expression (s/m^2).
(3) The authors stated that the numerical solutions are obtained using a custom finite difference scheme implemented in MATLAB in the Appendix. The description of numerical methods is insufficiently detailed and needs to be expanded, including specific equations or models used to obtain specific figures, the introduction of initial and boundary conditions, the choices of parameters and their reasons in terms of the biology.
(4) The authors claimed that their framework naturally extends to multiple molecular species, but only showed the situation of labeled and unlabeled molecules across a phase boundary. How about three or more molecular species? Does this framework still work? This should be added to strengthen the manuscript and confirm the framework's general applicability.
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Author response:
Reviewer #1 (Public review):
Summary:
In this manuscript, the authors theoretically address the topic of interface resistance between a phase-separated condensate and the surrounding dilute phase. In a nutshell, "interface resistance" occurs if material in the dilute phase can only slowly pass through the interface region to enter the dense phase. There is some evidence from FRAP experiments that such a resistance may exist, and if it does, it could be biologically relevant insofar as the movement of material between dense and dilute phases can be rate-limiting for biological processes, including coarsening. The current study theoretically addresses interface resistance at two levels of description: first, the authors present a simple way of formulating interface resistance for a sharp interface model. Second, they …
Author response:
Reviewer #1 (Public review):
Summary:
In this manuscript, the authors theoretically address the topic of interface resistance between a phase-separated condensate and the surrounding dilute phase. In a nutshell, "interface resistance" occurs if material in the dilute phase can only slowly pass through the interface region to enter the dense phase. There is some evidence from FRAP experiments that such a resistance may exist, and if it does, it could be biologically relevant insofar as the movement of material between dense and dilute phases can be rate-limiting for biological processes, including coarsening. The current study theoretically addresses interface resistance at two levels of description: first, the authors present a simple way of formulating interface resistance for a sharp interface model. Second, they derive a formula for interface resistance for a finite-width interface and present two scenarios where the interface resistance might be substantial.
Strengths:
The topic is of broad relevance to the important field of intracellular phase separation, and the work is overall credible.
Weaknesses:
There are a few problems with the study as presented - mainly that the key formula for the latter section has already been derived and presented in Reference 6 (notably also in this journal), and that the physical basis for the proposed scenarios leading to a large interface resistance is not clearly supported.
(1) As noted, Equation 32 of the current study is entirely equivalent to Equation 8 of Reference 6, with a very similar derivation presented in Appendix 1 of that paper. In fact, Equation 8 in Reference 6 takes one more step by combining Equations 32 and 35 to provide a general expression for the interface resistance in an integral form. These prior results should be properly cited in the current work - the existing citations to Reference 6 do not make this overlap apparent.
We agree and will make the overlap explicit, acknowledging priority and clarifying what is new here. The initial version of the preprint of Zhang et al. (2022) (https://www.biorxiv.org/content/10.1101/2022.03.16.484641v1) lacked the derivation (it referenced a Supplementary Note not yet available); it was added during the eLife submission. We worked from the preprint and missed this update, which we will now correct.
(2) The authors of the current study go on to examine cases where this shared equation (here Equation 32) might imply a large interface resistance. The examples are mathematically correct, but physically unsupported. In order to produce a substantial interface resistance, the current authors have to suppose that in the interface region between the dense and dilute phases, either there is a local minimum of the diffusion coefficient or a local minimum of the density. I am not aware of any realistic model that would produce either of these minima. Indeed, the authors do not present sufficient examples or physical arguments that would support the existence of such minima.
We respectfully disagree with the reviewer on the physical plausibility of these scenarios there is both concrete experimental and theoretical evidence for the scenarios we discussed.
Experimental: Strom et al. (2017) (our reference 11) describes a substantially reduced protein diffusion coefficient at an in vivo phase boundary, while Hahn et al. (2011a) and Hahn et al. (2011b) (our references 27 and 28) describe transient accumulation of molecules at a phase boundary, which they attribute to the Donnan potential, but conceivably a lowered mobility could play a role.
Theoretical: Recent work (e.g., Majee et al. (2024)) shows that charged layers could form at phase boundaries, which could either repel or attract incoming molecules, depending on their charge, thus altering the local volume fraction, resulting in a trough or peak. Arguably, the model put forth by Zhang et al. (2024) could be mapped to a potential wall, where particles are reflected, unless in a certain state. We will add sentences to the corresponding results section, as well as the discussion to make this plausibility more apparent.
In my view, these two issues limit the general interest of the latter portion of the current manuscript. While point 1 can be remedied by proper citation, point 2 is not so simple to address. The two ways the authors present to produce a substantial interface resistance seem to me to be mathematical exercises without a physical basis. The manuscript will improve if the authors can provide examples or compelling arguments for a minimum of either diffusion coefficient or density between the dense and dilute phases that would address point 2.
We believe we will be able to address both issues.
Reviewer #2 (Public review):
Summary:
This work provides a general theoretical framework for understanding molecular transport across liquid-liquid phase boundaries, focusing on interfacial resistance arising from deviations from local equilibrium. By bridging sharp and continuous interface descriptions, the authors demonstrate how distinct microscopic mechanisms can yield similar effective kinetics and propose practical experimental validation strategies.
Strengths:
(1) Conceptually rich and physically insightful interface resistance formulation in sharp and continuous limits.
(2) Strong integration of non-equilibrium thermodynamics with biologically motivated transport scenarios.
(3) Thorough numerical and analytical support, with thoughtful connection to current and emerging experimental techniques.
(4) Relevance to various systems, including biomolecular condensates and engineered aqueous two-phase systems.
Weaknesses:
(1) The work remains theoretical, mainly, with limited direct comparison to quantitative experimental data.
We agree with the reviewer, an experimental manuscript is in progress.
(2) The biological implications are only briefly explored; further discussion of specific systems where interface resistance might play a functional role would enhance the impact.
We thank the reviewer for this comment. We will add several such scenarios to the discussion, including the possibility to use interface resistance as a way of ordering biochemical reactions in time, as well as their potential to exclude molecules from condensates for long time periods, which, while not effective in the long-time limit, could help on cellular timescales of minutes to hours to respond to transient events.
(3) Some model assumptions (e.g., symmetric labeling or idealized diffusivity profiles) could be further contextualized regarding biological variability.
The treatment of labelled and unlabelled molecules as physically identical is well supported by our experiments. Droplets under typical experimental conditions, i.e. when bleaching is not too strong, do not markedly change size or volume fraction of molecules, which would be expected if the physical properties like molecular volume or interaction strength were significantly changed. However, we do agree that in more extreme bleaching regimes the bleach step itself will change the droplet properties, but this can be avoided by tuning the FRAP laser power and dwell times accordingly.
Our diffusivity profiles are chosen in the simplest possible way to handle typical experimental constraints (large D outside, lower D inside, potentially lowered D at the boundary) and allow for a mean-field treatment. To the best of our knowledge, the precise make-up and concentration profiles of phase boundaries in biomolecular condensates are not currently known, due to limitations in optical resolution.
Reviewer #3 (Public review):
The manuscript investigated the kinetics of molecule transport across interfaces in phase-separated mixtures. Through the development of a theoretical approach for a binary mixture in a sharp interface limit, the authors found that interface resistance leads to a slowdown in interfacial movement. Subsequently, they extended this approach to multiple molecular species (incorporating both labeled and unlabeled molecules) and continuous transport models. Finally, they proposed experimental settings in vitro and commented on the necessary optical resolution to detect signatures of interfacial kinetics associated with resistance.
The investigation of transport kinetics across biomolecular condensate interfaces holds significant relevance for understanding cellular function and dysfunction mechanisms; thus, the topic is important and timely. However, the current manuscript presentation requires improvement. Firstly, the inclusion of numerous equations in the main text substantially compromises readability, and relocation of a part of the formulae and derivations to the Appendix would be more appropriate. Secondly, the manuscript would benefit from more comprehensive comparisons with existing theoretical studies on molecular transport kinetics. The text should also be written to be more approachable for a general readership. Modifications and sufficient responses to the specific points outlined below are recommended.
(1) The authors introduced a theoretical framework to study the kinetics of molecules across an interface between two coexisting liquid phases and found that interface resistance leads to a slowdown in interfacial movement in a binary mixture and a decelerated molecule exchange between labeled and unlabeled molecules across the phase boundary. However, these findings appear rather expected. The work would be strengthened by a more thorough discussion of the kinetics of molecule transport across interfaces (such as the physical origin of the interface resistance and its specific impact on transport kinetics).
We thank the reviewer for this comment and will discuss possible mechanisms and how they map to our meanfield model in more detail, both in the corresponding results section, and in the discussion, as also outlined in our response to Reviewer #1.
(2) The formulae in the manuscript should be checked and corrected. Notably, Equation 10 contains "\phi_2\ln\phi_2" while Eq. 11b shows "n^{-1}\ln\phi_2", suggesting a missing factor of "n^{-1}". Similarly, Equation 18 obtained from Equation 11: the logarithmic term in Eq.11a is "n^{-1}\ln phi_1-\ln(1-\phi)" but the pre-exponential factor in Equation 18a is just "\phi_1/(1-\phi*)", where is "n^{-1}"? Additionally, there is a unit inconsistency in Equation 36, where the unit of \rho (s/m) does not match that of the right-hand side expression (s/m^2).
We thank the reviewer. We identified that the error originates in the inline definition of the exchange chemical potential, already before equation 11. We inadvertently dropped a prefactor of n, which then shows up in the following equation as an exponent to (1-phi^*). Very importantly this means the main result eq. 25 still holds, and in the revised manuscript we will correct the ensuing typographical mistakes.
(3) The authors stated that the numerical solutions are obtained using a custom finite difference scheme implemented in MATLAB in the Appendix. The description of numerical methods is insufficiently detailed and needs to be expanded, including specific equations or models used to obtain specific figures, the introduction of initial and boundary conditions, the choices of parameters and their reasons in terms of the biology.
We will substantially expand the Appendix for the numerical solutions and add an explanatory file to the repository to make clear how the code can be run, as well as its dependencies.
(4) The authors claimed that their framework naturally extends to multiple molecular species, but only showed the situation of labeled and unlabeled molecules across a phase boundary. How about three or more molecular species? Does this framework still work? This should be added to strengthen the manuscript and confirm the framework's general applicability.
We have shown in Bo et al. (2021) that the labelling approach can be carried over to multi-component systems. Each species may, for example, encounter its own interface resistance. We will discuss this in more detail in the revised manuscript.
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