The interplay between biomolecular assembly and phase separation

Reviewer #1 (Public Review):

Summary:

The authors present a mean-field model that describes the interplay between (protein) aggregation and phase separation. Different classes of interaction complexity and aggregate dimensionality are considered, both in calculations concerning (equilibrium) phase behavior and kinetics of assembly formation.

Strengths:

The present work is, although purely theoretical, of high interest to understanding biological processes that occur as a result of a coupling between protein aggregation and phase separation. Of course, such processes are abundant, in the living cell as well as in in-vitro experiments. I appreciate the consideration of aggregates with various dimensionality, as well as the categorization into different "interaction classes", together with the mentioning of experimental observations from biology. The model is convincing and underlines the complexity associated with the distribution of proteins across phases and aggregates in the living cell.

Weaknesses:

There are a few minor weaknesses.

Reviewer #2 (Public Review):

This work deals with a very difficult physical problem: relating the assembly of building blocks on a molecular scale to the appearance of large, macroscopic assemblies. This problem is particularly difficult to treat, because of the large number of units involved, and of the complex way in which these units-monomers-interact with each other and with the solvent. In order to make the problem treatable, the authors recur to a number of approximations: Among these, there is the assumption that the system is spatially homogeneous, i.e., its features are the same in all regions of space. In particular, the homogeneity assumption may not hold in biologically relevant systems such as cells, where the behavior close to the cell membrane may strongly differ from the one in the bulk. As a result, this hypothesis calls for a cautious consideration and interpretation of the results of this work. Another notable simplification introduced by the authors is the assumption that the system can only follow two possible behaviors: In the first, each monomer interacts equally with the solvent; no matter the size of the cluster of which it is part. In the second case, monomers in the bulk of a cluster and monomers at the assembly boundary interact with the solvent in a different way. These two cases are considered not only because they simplify the problem, but also because they are inspired by biologically relevant proteins.

With these simplifications, the authors trace the phase diagram of the system, characterizing its phases for different fractions of the volume occupied by the monomers and solvent, and for different values of the temperature. The results qualitatively reproduce some features observed in recent experiments, such as an anomalous distribution of cluster sizes below the system saturation threshold, and the gelation of condensed phases above such threshold.

Reviewer #3 (Public Review):

Summary:

The authors combine classical theories of phase separation and self-assembly to establish a framework for explaining the coupling between the two phenomena in the context of protein assemblies and condensates. By starting from a mean-field free energy for monomers and assemblies immersed in solvent and imposing conditions of equilibrium, the authors derive phase diagrams indicating how assemblies partition into different condensed phases as temperature and the total volume fraction of proteins are varied. They find that phase separation can promote assembly within the protein-rich phase, providing a potential mechanism for spatial control of assembly. They extend their theory to account for the possibility of gelation. They also create a theory for the kinetics of self-assembly within phase separated systems, predicting how assembly size distributions change with time within the different phases as well as how the volumes of the different phases change with time.

Strengths:

The theoretical framework that the authors present is an interesting marriage of classic theories of phase separation and self-assembly. Its simplicity should make it a powerful general tool for understanding the thermodynamics of assembly coupled to phase separation, and it should provide a useful framework for analyzing experiments on assembly within biomolecular condensates.

The key advance over previous work is that the authors now account for how self-assembly can change the boundaries of the phase diagram.

A second interesting point is the explicit theoretical consideration for the possibility that gelation (i.e. self-assembly into a macroscopic aggregate) could account for widely observed solidification of condensates. While this concept has been broadly discussed, to date I have yet to see a rigorous theoretical analysis of the possibility.

The kinetic theory in sections 5 and 6 is also interesting as it extends on previous work by considering the kinetics of phase separation as well as those of self-assembly.

Weaknesses:

A key point the authors make about their theory is that it allows, as opposed to previous research, to study non-dilute limits. It is true that they consider gelation when the 3D assemblies become macroscopic. However, dilute solution theory assumptions seem to be embedded in many aspects of their theory, and it is not always clear where else the non-dilute limits are considered. Is it in the inter-species interaction \chi_{ij}? Why then do they never explore cases for which \chi_{ij} is nonzero in their analysis?

The connection between this theory and biological systems is described in the introduction but lost along the main text. It would be very helpful to point out, for instance, that the presence of phase separation might induce aggregation of proteins. This point is described formally at the end of Section 3, but a more qualitative connection to biological systems would be very useful here.

Building on the previous point, it would be helpful to give an intuitive sense of where the equations derived in the Appendices and presented in the main text come from and to spell out clear physical interpretations of the results. For example, it would be helpful to point out that Eq. 4 is a form of the law of mass action, familiar from introductory chemistry.

It would be useful to better explain how the current work extends on existing previous work from these authors as well as others. Along these lines, closely related work by W. Jacobs and B. Rogers [O. Hedge et al. 2023, https://arxiv.org/abs/2301.06134; T. Li et al. 2023, https://arxiv.org/abs/2306.13198] should be cited in the introduction.

The results discussed in the first paragraph of Section 3 on assembly size distributions in a homogeneous system are well-known from classic theories of self-assembly. This should be acknowledged and appropriate references should be added; see for instance Rev. Mod. Phys. 93, 025008 and Statistical Thermodynamics Of Surfaces, Interfaces, And Membranes by Sam Safran.

Equation 14 for the kinetic of volume fractions is given with a reference to Bauermann et al 2022, but it should be accompanied by a better intuitive interpretation of its terms in the main text. In particular, how should one understand the third term in this equation? Why does the change in volume impact the change of volume fraction in this way?

The discussion in the last paragraph of Section 6 should be clarified. How can the total amount of protein in both phases decrease? This would necessarily violate either mass or volume conservation. Also, the discussion of why the volume is non-monotonic in time is not clear.