Interaction between cell membranes and protein inclusions in the large-deformation regime

Gaetano Ferraro; Michele Castellana

REVIEW 4 major objections 5 minor 53 references

Interaction between cell membranes and protein inclusions in the large-deformation regime

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper establishes that beyond small membrane slopes, a protein inclusion feels a non-monotonic restoring force, pairwise interactions decay slower than a power law and keep a charge-like sign rule, and a flow speed v* = κ/(Lη) marks th

desk verdict Useful large-deformation FE results and a clean analytic minimal-surface solution, but the flow threshold v*=κ/(Lη) is a finite-box artifact and the paper overclaims it as an intrinsic scale. read the letter →

arxiv 2601.15477 v2 pith:QU6LIZ5Q submitted 2026-01-21 physics.bio-ph cond-mat.soft

Gaetano Ferraro , Michele Castellana This is my paper · ORCID

classification physics.bio-phcond-mat.soft

keywords lipidmembranesproteininclusionslargedeformationsHelfrichenergyfiniteelementmethodmembrane-mediatedinteractionsconicalmembraneflow

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most theoretical work on proteins bending cell membranes has lived in the small-deformation regime, where the membrane slope is treated as tiny. This paper shows what happens when that assumption breaks: in the large-deformation regime, solved with a nonlinear finite-element method, the membrane's restoring force on a protein inclusion is non-monotonic in the inclusion's vertical displacement, the membrane-mediated interaction between two inclusions decays slower than a power law with distance, and conical inclusions with the same orientation repel while opposite orientations attract, extending the 'charge' picture. It also shows that a background membrane flow only begins to visibly deform the membrane once the flow speed exceeds v* = κ/(Lη), with a deformation wavelength of about κ/(ηv). The payoff is a set of quantitative predictions for strongly deformed membrane-protein systems—relevant to protein sorting, clustering, and membrane trafficking—plus an approximate closed-form membrane profile for quick estimates.

What carries the argument

The engine is the full, nonlinear Helfrich shape equation—the Euler–Lagrange equation of the standard bending-plus-tension energy—solved by finite elements rather than by expanding about a flat membrane. The main analytic device is a change of slope variable ω = ψ/√(1−ψ²); assuming zero mean curvature gives ψ = C/r and a logarithmic membrane profile z(r) = −r0 sinα ln[(r−√(r²−r0² sin²α))/(R−√(R²−r0² sin²α))], a closed-form map from contact angle to spontaneous displacement. The force calculation comes from the boundary terms of the varied Helfrich functional, giving normal and tangential line forces f⊥ = −2κ ∂H/∂r and f∥ = −n_i(2κH² + σ)e_i. The two-protein results come from numerically eval

What would settle it

Solve the flow equations for the same protein and material parameters with the outer box size L doubled and tripled at a fixed inflow velocity; if the onset of deformation does not move as v* ∝ 1/L—meaning a deformation appears at a velocity that was previously sub-threshold—the flow-onset claim is contradicted.

Watch

Extended reading notes

Core claim

The central discovery is that the qualitatively new features of protein–membrane interaction appear only when the membrane is strongly deformed, and they can be captured numerically. Linearized small-deformation theory is adequate only for contact-angle slopes up to about tan α ≈ 0.4; beyond that the full shape equation is needed. In this large-deformation regime the vertical membrane force on a single protein is non-monotonic in the protein's displacement; the two-protein interaction energy decays sub-power-law with separation, and its sign follows the 'charge' rule of the conical contact angles. In the presence of flow, the paper finds a crossover speed v* = κ/(Lη) below which flow has no

Load-bearing premise

The load-bearing premise in the flow section is that the membrane is flat and undisturbed at the outer boundary of the simulation box, so the threshold speed v* = κ/(Lη) shrinks as the box grows and would vanish for an unbounded membrane—if that boundary condition is relaxed, the flow-onset claim may not hold.

Editorial extensions

If this is right

At large vertical displacements the membrane's restoring force on a protein stops growing and falls, so strongly invaginated inclusions are pulled back less forcefully; this changes estimates of the forces needed for endocytic uptake or membrane tube formation.
Membrane-mediated interactions between two inclusions decay more slowly than a power law at the distances studied, so at separations of a few protein radii the interaction is stronger than small-deformation theory would predict, affecting clustering and pattern formation.
The charge analogy survives large deformations: two conical inclusions with the same orientation repel, opposite orientations attract, so a mixture of differently oriented curvature-generating proteins will tend to segregate by orientation.
A flow speed above v* = κ/(Lη) deforms the membrane with wavelength ~κ/(ηv); for typical protein diffusion velocities this wavelength can be microns, comparable to inter-protein spacings, so flow-induced deformation can affect protein mobility and organization in dense arrays.
The approximate zero-mean-curvature profile provides a ready formula for the spontaneous displacement of a protein with a given contact angle, letting experimentalists estimate displacements without solving the full equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Reading the boundary conditions literally, the flow threshold v* = κ/(Lη) scales inversely with the simulation-box size L, so for an unbounded membrane the threshold would tend to zero; the robust physical scale is the deformation wavelength λ = κ/(ηv), and the apparent 'onset' is likely a finite-domain crossover rather than an intrinsic biological speed.
Since small-deformation studies already show non-pairwise forces among multiple inclusions, the sub-power-law two-body potential found here suggests that clusters of three or more proteins in the large-deformation regime may develop many-body ordering not captured by pairwise sums—an untested but natural extension.
The non-monotonic force curve offers a concrete experimental target: a micropipette or optical-tweezer measurement of force versus displacement on a vesicle with a protein inclusion should reveal a force peak that shifts with the patch size, directly testing the large-deformation prediction.
The closed-form logarithmic profile could serve as a cheap building block for coarse-grained or multi-protein screening models, letting researchers explore LD effects without a finite-element solve for every configuration.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news here is the finite-element work: first full solutions for finite-size protein inclusions in the large-deformation regime, with a non-monotonic force–displacement curve and a two-inclusion interaction potential that decays slower than a power law. Those results look plausible and are a genuine step beyond the small-deformation literature. The analytic zero-mean-curvature solution (Eq. 14) is exact when H=0 and gives a useful closed-form approximation for large contact angles. The force profiles in Fig. 5 are the strongest part, and the charge analogy surviving beyond small deformations is a nice confirmation. I would not be surprised if the pair-interaction curves become a reference for future work.

The soft spots are mostly in the flow section, and the stress-test note is right. The threshold v*=κ/(Lη) is defined with L as the simulation box size, and the boundary conditions fix the membrane flat at that distance. So v* is not a material property; as L→∞ it goes to zero, and in an infinite membrane any nonzero flow has a deformation wavelength λ=κ/(ηv). The paper's own explanation—deformations with wavelength larger than L are suppressed by the boundary—is circular. That claim in the abstract, about a characteristic velocity separating bending- and viscous-dominated regimes, needs to be re-scoped as a finite-size threshold or dropped. The rest of the paper does not depend on this threshold, so the force and interaction results stand.

Other issues are minor by comparison. The derivation of Eq. (10) is not shown, and the numerical curves in Figs. 5, 8, and 11 have no error bars or convergence metrics; the authors defer to the companion IRENE paper, which is acceptable but should be made explicit. The “sub-power-law” claim lacks a quantitative baseline—what power law is it being compared to? A fitted exponent or a comparison to the known small-deformation scaling would settle it. The citation pattern looks fine, and using their own software is not a problem if it is public and validated.

Bottom line: this is a solid contribution on the mechanics side, with one overstated claim. A serious referee should engage with it and ask for the flow section to be revised or relabeled, and for numerical convergence data. It deserves peer review, not desk rejection.

Referee Report

4 major / 5 minor

Summary. The manuscript studies membrane-protein interactions in the large-deformation (LD) regime using finite-element simulations performed with the authors' IRENE software. It claims four main results: (i) the membrane force on a single protein inclusion is non-monotonic in the vertical displacement; (ii) for two inclusions the membrane-mediated potential decays slower than a power law with distance; (iii) conical inclusions with equal and opposite orientations repel and attract, respectively, confirming the 'charge analogy' beyond the small-deformation limit; and (iv) in the presence of membrane flow, a characteristic velocity v* = κ/(Lη) separates a no-deformation regime from a flow-deformation regime. The paper also presents an approximate zero-mean-curvature analytic solution and compares it with numerical solutions.

Significance. If the results hold, the paper would extend membrane-mediated protein interactions beyond the well-studied small-deformation perturbative regime, and it provides concrete quantitative predictions for force-displacement curves, pair potentials, and flow-induced shape changes. Strengths of the paper include the use of a full nonlinear shape equation, the explicit zero-mean-curvature analytic solution (exact when H = 0), and the fact that the numerical results are in principle reproducible through the companion IRENE software [28]. The GUV/bacteriorhodopsin example makes the predictions biologically concrete. However, the numerical claims are not fully supported by in-paper convergence/error analysis, and the flow-velocity threshold is, as discussed below, tied to the computational domain size.

major comments (4)

[§III.B, Eq. (37), Eqs. (34)-(35)] The characteristic velocity v* is defined as v* = κ/(Lη), where L is the simulation-box side. The outer boundary conditions (34)-(35) impose z = 0 and zero normal slope at distance L. As the manuscript itself explains at the end of §III.B, a flow-induced deformation with wavelength λ = κ/(ηv) can appear only when λ < L; otherwise it is suppressed by the boundary. Consequently, v* depends explicitly on the computational domain and v* → 0 as L → ∞. For a truly infinite membrane, any nonzero v would produce a deformation with finite wavelength, so the claimed 'no-deformation' regime and the onset threshold are finite-size artifacts rather than an intrinsic physical scale. This undermines central claim (iv) of the abstract and the corresponding discussion in §IV. The section should be re-scoped as a finite-size effect or supplemented by an L-dependence study and a physical argument for a fin
[§III.A.3, Fig. 5] The non-monotonic force is one of the paper's central predictions, but Figure 5 shows numerical curves without error bars or mesh-convergence metrics. The text states that a convergence analysis for the radially symmetric case is reported in the companion paper [28], but the present manuscript does not provide it. For a quantitative claim about the magnitude, sign, and zero crossings of the force, representative convergence data or an error estimate should be included in this paper, or the claim should be explicitly labeled as preliminary.
[§III.A.4, Fig. 8A] The claim that the two-protein interaction potential exhibits 'sub-power-law decay' is based on visual curvature in a log-log plot. No fit to a power law is shown, no slope is computed, and no alternative decay form (e.g., logarithm, stretched exponential) is tested. Since this is a central quantitative result, the manuscript should define 'sub-power-law' quantitatively and demonstrate that the data are inconsistent with a power-law over the presented range, accounting for numerical uncertainty.
[§III.A.2, Eq. (10), Appendix A, Eq. (A1)] The derivation of Eq. (10) from Eq. (1) is not shown; the text simply states that substituting Eq. (A1) into Eq. (1) gives Eq. (10). A reader cannot verify the algebra, and the regime of validity of the approximate equation is therefore unclear. In addition, Eq. (11) defines H = ∂ψ/∂r + ψ/r, while Appendix A gives H = (1/2)(∂ψ/∂r + ψ/r). The factor 1/2 is immaterial for the zero-mean-curvature solution ψ = C1/r, but the inconsistency is confusing and should be clarified.

minor comments (5)

[Fig. 6 caption] The caption says '0 ≤ x2 ≤ h' but the domain side length is presumably L, not h. This appears to be a typo.
[Eq. (33) and Fig. 10] The parameter t is introduced in Eq. (33) and used in Fig. 10, but it is not defined in Table I or in the text. Please define t (presumably tanα at the protein boundary).
[Eq. (36), Fig. 9] The notation ∂Ω in Eq. (36) as a union of three boundaries is confusing because the same symbol ∂Ω is also used for the outer boundary in Eqs. (34)-(35). Distinct symbols for the inner and outer boundaries would improve readability.
[Eq. (15) and Eq. (16)] Eq. (16) is stated as the R≫r0 limit of Eq. (15), but the derivation is not shown; expanding Eq. (15) gives the logarithmic terms, but the cross-check of signs and the domain of validity of the expansion should be stated.
[Table I] The viscosity unit 'Pa m sec' should be 'Pa·m·s' to indicate a 2D membrane viscosity; this is not a substantive issue but would avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

Flow threshold v*=κ/(Lη) is defined by simulation box size and enforced by clamped boundaries, making the claimed onset self-definitional; force and pair-potential claims are independent.

self definitional [Section III.B, Eqs. (34)-(35) and Eq. (37), Figs. 10-11 and discussion]
"v∗ ≡ κ/(Lη) (37) ... if v≲v∗, the flow-induced deformation wavelength would be larger than the domain size L: Given that we assumed that no membrane deformations propagate to a distance equal or larger than L from the PI, see Eqs. (34) and (35), for these velocities no deformation can appear, see Fig. 11."

The 'characteristic velocity' is introduced as v* ≡ κ/(Lη), i.e. it is the velocity at which the estimated deformation wavelength λ ≈ κ/(ηv) equals the simulation box size L. The no-deformation regime v < v* is then obtained by imposing flat membrane boundary conditions at distance L (Eqs. 34-35), which by construction forbid deformations with wavelength ≥ L. Thus the numerical observation that v0 < v* shows no deformation, while v0 > v* does, restates the definition of v* plus the clamped outer boundary; it is not an emergent physical threshold. As L→∞, v*→0 and the onset disappears, confirming that the claimed regime separation is equivalent to the finite-domain input rather than an intrinsic membrane/flow property.

full rationale

The central claims in the flow-free sections are not circular. The non-monotonic force (Sec. III.A.3), the sub-power-law pair potential (Sec. III.A.4), and the orientation-dependent attraction/repulsion are numerical results computed from the Helfrich shape equation with stated boundary conditions and no fitted parameters; the zero-mean-curvature solution (Sec. III.A.2) provides an independent analytic check. The only circularity I can exhibit by the paper's own equations is in Sec. III.B: the flow threshold v* is defined via the simulation-domain length L, and the absence of deformation below v* is guaranteed by the same finite-domain boundary conditions (34)-(35). Hence the predicted onset of flow-induced deformation is a construction of the finite box, not a scale-independent result. I do not count the repeated citation to the authors' own IRENE software [28] as a circular step because the numerical data are displayed in the paper and the analytic solution gives partial independent support; this is a reproducibility concern, not a definitional reduction. Overall, because the non-flow results are self-contained but one abstract-level prediction reduces to its finite-size input by construction, the score is 6.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The model introduces no de novo physical entity and fits no material constants; the input parameters in Table I are taken from prior experimental literature. The main free/ad hoc choice is the computational domain size L in the flow section, which controls v*. The central numerical claims also presuppose the Helfrich model, Monge gauge, finite-domain boundary conditions, and correctness of the IRENE solver.

free parameters (1)

computational domain size L in flow section = L = 100 r0 (≈ 10^-6 m)
The threshold velocity v* = κ/(Lη) is inversely proportional to L, so the claimed flow regimes depend on a freely chosen simulation-box size. This is not a fitted material constant but an ad hoc computational length that determines a central claim.

assumptions (5)

domain assumption Helfrich-Canham free energy with constant surface tension governs membrane equilibrium (Eq. 1)
All results rely on this elastic model; it neglects lipid tilt (explicitly for d>r0), spontaneous curvature, thermal fluctuations, and compositional heterogeneity.
domain assumption Monge gauge representation z(x1,x2) is valid for the membrane surface
The membrane is assumed to be a graph over a plane, excluding overhangs or vesicle-like re-entrant shapes; this is used throughout the analytical and numerical treatment.
ad hoc to paper Finite membrane patch with pinned/flat outer boundary (z=0, ni∇iz=0 at R or L)
Boundary conditions (6), (8), (19), (21), (34), (35) suppress long-wavelength deformations; the flow threshold v* directly inherits this finite-size choice.
domain assumption Steady incompressible membrane hydrodynamics described by Eqs. (23)-(25) with no-slip at the protein
The flow section assumes Stokes-like membrane flow, constant viscosity η, negligible inertia/thermal noise, and immobilization of lipids at the protein boundary.
domain assumption The IRENE finite-element solver correctly and convergently solves these PDEs
All numerical claims depend on IRENE [28]; convergence analysis is not reproduced here and the software is by the same authors, providing no independent verification inside this paper.

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Cite this review

Pith. "Pith review of Interaction between cell membranes and protein inclusions in the large-deformation regime." pith.science (2026). https://pith.science/paper/QU6LIZ5Q

@misc{pith2026260115477,
  author       = {Pith},
  title        = {Pith review of: Interaction between cell membranes and protein inclusions in the large-deformation regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QU6LIZ5Q}},
  note         = {Machine review of arXiv:2601.15477}
}

read the original abstract

Biological membranes are dynamic surfaces whose shape and function are critically influenced by protein inclusions (PIs). While membrane deformations induced by PIs have been extensively studied in the small-deformation regime, a variety of processes involves strong membrane deformations. We investigate the interaction between lipid membranes and PIs in the large deformation (LD) regime, with the finite-element method. We develop an approximate analytical solution that captures key features of the LD regime. We show that the force exerted by the membrane on a PI displays a non-monotonic behavior with respect to the PI vertical displacement. The qualitative features of this force appear to be independent of the protein geometry. For two interacting PIs, the membrane-mediated potential exhibits sub-power-law decay with inter-protein distance, reflecting the complex nature of the elastic medium. The interaction potential shows that conical PIs with identical and opposite orientations repel and attract, respectively, confirming the analogy between PI orientation and electric charge, in the LD regime. In the presence of membrane flows, we identify a characteristic velocity that separates two regimes in which bending rigidity and viscous effects dominate, respectively, implying the onset of flow-induced deformations above such velocity threshold. Overall, our results provide quantitative predictions for membrane-protein systems in biologically relevant scenarios involving LDs, with implications for protein sorting, clustering, and membrane trafficking.

Figures

Figures reproduced from arXiv: 2601.15477 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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Indeed, if the membrane shape is radially symmetric, Eq

Linearized equation Equation (1) can be simplified in the case of a circular inclusion. Indeed, if the membrane shape is radially symmetric, Eq. (1) reduces to an ordinary differential equation (ODE). Moreover, Eq. (1) can be linearized for small values of ω≡ ∂z ∂r ,(3) and it reduces to [38] 1− r2 ℓ2 ω−r 1 + r2 ℓ2 dω dr + 2r2 d2ω dr2 + (4) r3 d3ω dr3 = 0...

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Solution with zero mean curvature Analytical insights about the LD regime for the membrane can be obtained considering the follow- ing approximate solution. In order to simplify the equation, it is convenient to introduce the quantityψ, by the relation ω= ψp 1−ψ 2 .(9) In Section A, we show all geometrical quantities expressed in terms of ψ and its deriva...

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Forces Another important feature in the interaction be- tween the protein and the membrane is the force exerted by the membrane on the PI boundary. The expression for this force is derived in Section B, and its tangential and normal components read f⊥ = 2κni∇iH(17) =−2κ ∂H ∂r , f∥ =−n i(2κH 2 +σ)e i (18) In the LD regime, the force exerted by the mem- bra...

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We consider a square domain Ω with side L, containing two circular holes, which represent the two PIs, see Fig

Potential energy in a system with two proteins In what follows, we will focus on the membrane- mediated interaction between the two PIs. We consider a square domain Ω with side L, containing two circular holes, which represent the two PIs, see Fig. 6. In this case, the rotational symmetry of Fig. 1 no longer holds. As a result, the partial differential eq...

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