Molecular sorting on a fluctuating membrane

This article builds on a previously published work (ref. 14 in the manuscript) that deals with the sorting of molecules in biomembranes under the combined effects of molecule adsorption, phase separation into domains and self-extraction of domains above a critical radius. In this paper, the authors study the effect of long-range fluctuation forces (Casimir) between individual molecules and domains (only short-range interactions between molecules were previously taken into account).

The analytical part is based on a calculation of the fluctuation-induced interaction between a molecule, treated as a pointlike modification of the membrane bending and Gaussian rigidity parameters, and a large domain (an aggregate of molecules), assumed close enough to be treated as a semi-infinite plane, and also treated as a region having, with respect to the background membrane, different bending and Gaussian rigidities. The model is well explained and the calculations are sound. Using analytical calculation and numerical calculations, the authors show then that the Casimir-like forces favor the formation of domains and significantly reduce the critical radius for their formation.

In my opinion, this is an interesting work and I recommend its publication in SciPost, provided the authors take into account the following points.

1) Casimir-like forces, due to rigidity contrasts, are not the only membrane-mediated forces. The authors are aware of that and mention this point at the very end of the manuscript. The complete subject would be molecular sorting in the presence of long-range membrane-mediated forces. The authors neglect the intrinsic curvature of the molecules and the intrinsic curvature of the domains, that would yield long-range interactions even at the mean-field level (in the absence of fluctuations). It is acceptable to focus on Casimir forces, especially because they are ubiquitous, but I would prefer to see this important restriction to be stated right in the abstract and the introduction.

2) The physics behind eq. (8) is somewhat hidden. I am guessing that the authors simply perform a first-order Taylor expansion in 1/R with coefficient named R*, and only invoke line-tension as a possible origin ? Is that so, please make this more clear.

3) The domain extraction process is not discussed. Some biophysical background as to how vesicles are formed when R>R_c and how they are extracted would be welcomed. It is actually a complicated process, involving other molecules, that may be depleted, and some intrinsic fluctuations that are neglected here. A word of caution would be welcome.

4) In a real biological membrane, there is a cytoskeleton attached to the membrane. This changes drastically the fluctuation spectrum, so the results obtained in this paper might not apply to biological membranes. The authors could make this statement, probably both in the introduction and in the discussion as a source of possible future investigations.

Publish (meets expectations and criteria for this Journal)

The manuscript "Molecular sorting on a fluctuating membrane" by Andreghetti and co-authors proposes an extension of previous calculations by the same group, modeling far-from-equilibrium molecular sorting in biological membranes. It is based both on analytical scaling arguments and Monte-Carlo simulations of point-like proteins diffusing on a fluctuating, discretized membrane. The simulations support the analytical findings, at least qualitatively.

The background idea is that attractive forces can lead to protein condensation in small clusters, before they are extracted from the membrane by some active cellular machinery when their size goes above some threshold. The novelty of this work, compared to the previous ones, is that in addition to shorter-range forces of entropic, hydrophobic or molecular origin, it takes into account "long-range" thermal Casimir attraction between proteins, which is well-known to contribute to their condensation in the membrane. As it can be anticipated, the authors demonstrate that taking into account these additional attractive forces facilitates condensation and thus molecular sorting as described above.

1/ The authors present their results as "new" and they claim that the "role of [Casimir-like interactions] in molecular sorting remains unexplored". Generally speaking, this paper follows a long series of works, some of them dating back over twenty years, and based upon similar physical ingredients (more or less short-range forces, some of them mediated by the membrane, and active membrane recycling) to describe the far-from-equilibrium formation of small protein clusters having some biological function, as clusters. See, e.g., Foret, EPL 71, (2005) ; Truong Quang et al, Curr. Biol. 23, 2197 (2013) ; Fan et al, Phys. Rev. Lett. 104, 118101 (2010) ; or Berger et al, J. Phys. Chem. B 120 10588 (2016). It would be useful to place the present work in this bibliographic context. In order to situate the present model among the previously published ones, the description of the cluster-size distribution obtained through the Monte Carlo simulations could be helpful. It is generally discussed in these anterior works.

2/ When out-of-equilibrium membrane processes are simulated through Monte-Carlo dynamics, it is not trivial to set in the simulations the two different time-steps involved in protein diffusions in the membrane plane on the one side, and transverse membrane fluctuations on the other side. A thorough discussion of this issue has for example recently been proposed in Cornet et al., Soft Matter 20, 4998 (2024).

3/ More critically, the authors do not discuss the order of magnitude of the parameter A in eq 4. According to eq A.10, this parameter remains close to unity when alpha is large. Thus it grows from 0 to ~1 when alpha grows from 1 to large values. For example, A=0.36 if alpha=5. Thus U in eq 4 is at best on the order of kBT. To which extend can an interaction lower than kBT can play a significant role? For example, in eq 10, if |U|<<kBT, U has virtually no effect on Phi. The simulations seem to somewhat contradict what I write here, at least for large values of alpha. Why? Above which value of alpha? Are these values of alpha realistic in the experimental context? Generally speaking, the simulations should also be compared more quantitatively to the analytical findings.

4/ Even more critically: even if U is a bit larger than kBT at short range, it decays rapidly so that the Casimir interaction can effectively be seen as a short-range interaction. Then to what extent its effect is not "trivially" to modify the value of the short-range attraction intensity W? Inspection of figures 2 and 3 indicates that the plots for alpha>1 seem to be essentially just translations of the alpha=1 plot. This supports the hypothesis that the whole effect of the Casimir force can just be absorbed in W.

Several points must be clarified before the paper could be published in Scipost.

1/ Membrane tension (sigma) is ignored. However, it dominates the curvature beyond the length xi=sqrt(kappa/sigma), see e.g. the recent work Fournier, EPJ E 47, 64(2024) and references therein where Casimir forces between proteins are studied in the cell membrane context. xi~10 to 100 nm in plasma membranes. It would be helpful to discuss in the last conclusive section how modifying the long-range shape of U(r) affects (or not) the conclusion of this work.

2/ The calculations leading to eqs 7 and 8 would largely benefit from being detailed, to ease the manuscript reading.

3/ In the legend of figure 2, it would be nice to recall the values of the parameters used in the simulation because they are scattered throughout the paper and somewhat difficult to find. Maybe I’m wrong, but I did not find the value of the lattice spacing used in the simulations, neither the value of the time step (both in real units).

4/ On page 4, the second paragraph compares the diffusion and membrane relaxation time scales, but it is not clear where the result is used later in the manuscript.

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