arxiv: 2604.13415 · v1 · submitted 2026-04-15 · ⚛️ physics.bio-ph · math-ph· math.MP

Recognition: unknown

Membrane Tension Governs Particle Wrapping-Unwrapping Transitions and Stalling

Yasin Ranjbar , Yujun Teng , Haleh Alimohammadi , Huajian Gao , Mattia Bacca

Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:36 UTC · model grok-4.3

classification ⚛️ physics.bio-ph math-phmath.MP

keywords membrane tensionparticle wrappingendocytosismembrane energystalling boundaryunwrappingnanoparticles

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The pith

Membrane tension creates a stalling point that decides if particles wrap fully into a cell or get expelled.

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

The paper shows that the energy stored in the unwrapped membrane outside the particle contact zone, controlled by tension, determines the progress of wrapping. Tracking total energy as wrapping increases reveals that adhesion drives coverage while tension and particle size can stop it at an intermediate stage or reverse it. This produces a clear boundary in parameter space separating conditions for particle uptake from those for expulsion. The work supplies an analytical approximation for the outer membrane shape that matches numerical solutions and unifies wrapping and unwrapping under one energy landscape.

Core claim

Including the tension-dependent deformation energy of the non-contact membrane in the total energy budget shows that wrapping degree reaches a stable point or reverses depending on the balance of adhesion strength, membrane tension, and particle radius. The resulting stalling boundary separates regimes of continuous uptake from partial stalling and spontaneous unwrapping.

What carries the argument

the stalling boundary obtained by locating extrema of total membrane energy versus wrapping angle, using a compact analytical approximation for the non-contact membrane shape that reproduces the full numerical solution.

If this is right

Wrapping stalls at a predictable intermediate coverage for tensions above a threshold set by adhesion and size.
Particles above a critical size are expelled rather than taken up at given tension and adhesion.
The same energy function describes both wrapping progression and unwrapping reversal.
Nanoparticle design can target the stalling boundary to control uptake versus release.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Cells with different baseline tensions could selectively internalize or reject particles of specific sizes.
The framework could be extended to time-varying tension during membrane fusion events.
Tuning particle surface adhesion might shift the stalling boundary for targeted drug delivery.

Load-bearing premise

The membrane is treated as having constant tension with bending energy negligible outside the contact zone, and particles are rigid spheres.

What would settle it

An experiment that varies membrane tension and particle radius while measuring the wrapping fraction at which motion stops or reverses, and checks whether the observed transition matches the predicted boundary curve.

read the original abstract

Membrane wrapping underlies nanoparticle uptake during endocytosis, whereas the reverse process of membrane unwrapping accompanies particle expulsion and membrane fusion events. Existing theoretical descriptions typically focus on adhesion and bending energies within the particle-membrane contact region and often neglect the deformation energy of the membrane outside the contact zone. This approximation is valid only in the limit of vanishing membrane tension, where the non-contact membrane assumes a catenoid-like configuration with negligible bending energy. However, at finite tension the deformation of the non-contact membrane becomes a dominant energetic contribution. Here we show that this tension-dependent non-contact energy governs the progression of particle wrapping. By analyzing the variation of the total membrane energy with wrapping degree, we uncover a competition between particle adhesion, membrane tension and particle size that determines whether wrapping proceeds, stalls, or reverses into spontaneous unwrapping. This framework reveals a stalling boundary separating regimes of particle uptake and expulsion. To capture the non-contact deformation efficiently, we derive a compact analytical approximation that accurately reproduces the full numerical solution of the membrane shape. The resulting energetic map provides a unified physical description of particle wrapping and unwrapping, with implications for endocytosis, membrane fusion, and nanoparticle design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds a usable analytical approximation for non-contact membrane energy at finite tension and maps the resulting stalling boundary between wrapping and unwrapping.

read the letter

The main thing to know is that this work treats the energy stored in the membrane outside the particle contact zone as the controlling factor once tension is nonzero. Previous models could drop that term because it vanishes in the zero-tension catenoid limit, but here tension makes the outer deformation cost comparable to adhesion and bending inside the zone. They derive a compact analytical form for the outer shape that they say tracks the full numerical solution, then track total energy versus wrapping fraction to locate the points where the energy slope changes sign. That produces a boundary in the space of tension, adhesion energy per area, and particle radius that separates full uptake, stalling, and spontaneous expulsion. The approach is straightforward and stays within standard Helfrich-Canham energetics without extra fitted constants. The practical payoff is a quick way to sketch regimes without solving the shape equation at every step, which is useful for endocytosis or fusion calculations. The limitation is that the abstract and available description give no error curves or range-of-validity tests for the approximation, so it is not yet clear how much the stalling line shifts when tension rises or when wrapping is only partial. If the match to numerics holds only in a narrow window, the boundary predictions would need adjustment. The logic itself is internally consistent and does not rely on circular fitting. This is aimed at modelers who already work with membrane-particle energetics and want an analytical handle rather than full simulations. A reader who needs to estimate uptake probability versus tension would find the phase diagram directly usable. It has enough new analytical content and a clear physical claim that it deserves a serious referee to check the approximation accuracy and the energy-extremum analysis.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a theoretical model showing that finite membrane tension governs particle wrapping-unwrapping transitions by making the deformation energy of the non-contact membrane a dominant contribution, unlike prior models valid only at vanishing tension. The authors derive a compact analytical approximation for the non-contact membrane shape that reproduces the numerical solution, then analyze the total energy variation with wrapping degree to identify a stalling boundary arising from the competition among particle adhesion energy, membrane tension, and particle size; this boundary separates regimes of uptake, stalling, and spontaneous unwrapping, with implications for endocytosis and membrane fusion.

Significance. If the central results hold, the work supplies a unified, physically grounded description of wrapping dynamics that extends beyond the zero-tension limit and yields testable predictions for stalling conditions. A clear strength is the compact analytical approximation for non-contact deformation that matches full numerical solutions, enabling efficient parameter exploration without repeated numerical solves. The approach is built on standard membrane energy functionals and energy extremization with respect to wrapping degree.

minor comments (2)

Abstract: the statement that the approximation 'accurately reproduces the full numerical solution' would be strengthened by a brief quantitative metric (e.g., maximum relative error or parameter range) already in the abstract or introduction.
Main text: a table or explicit list of symbols for wrapping degree, contact angle, and the separate energy contributions would improve readability and reduce notation ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the recognition of the analytical approximation for non-contact membrane deformation as a clear strength, and the recommendation for minor revision. The referee's description accurately reflects the manuscript's focus on tension-dependent energetics and the resulting stalling boundary.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a compact analytical approximation to the non-contact membrane shape that is explicitly validated by reproducing the full numerical solution of the shape equation. Total energy is then expressed as a function of wrapping degree and minimized to identify the stalling boundary arising from the explicit competition among adhesion, tension, and particle-size terms. No equation reduces to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The framework is a standard variational energy analysis grounded in the Helfrich-Canham model plus tension, with the new contribution being the finite-tension non-contact term whose effect is computed rather than presupposed.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model rests on standard continuum membrane mechanics but extends them to finite tension. No new physical entities are introduced. Free parameters are the physical inputs (tension, adhesion, size) that are varied to delineate regimes.

free parameters (3)

membrane tension
Key control parameter whose finite value makes non-contact deformation dominant; varied to map regimes.
particle adhesion energy
Competes with tension and size in the energy balance; treated as a variable input.
particle size
Affects the wrapping geometry and energy competition; used to define the stalling boundary.

axioms (2)

domain assumption Total membrane energy is the sum of adhesion, bending, and tension contributions from both contact and non-contact regions.
Invoked when analyzing energy variation with wrapping degree.
domain assumption Non-contact membrane deformation energy is negligible at zero tension but becomes dominant at finite tension.
Load-bearing premise that allows tension to govern wrapping progression.

pith-pipeline@v0.9.0 · 5529 in / 1525 out tokens · 35785 ms · 2026-05-10T12:36:38.386922+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

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1 Membrane Tension Governs Particle Wrapping-Unwrapping Transitions and Stalling Yasin Ranjbar1, Yujun Teng2, Haleh Alimohammadi3, Huajian Gao4, Mattia Bacca1, * 1Mechanical Engineering, University of British Columbia, BC V6T 1Z4, Canada 2Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore 3Molecular Biology & Biochem...

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>0.99) closed-form analytical expression for the NC energy as a function of the wrap angle θ (or wrapping degree f=θ/π) and the dimensionless membrane tension σ.=σR

While the C-region energy is straightforward, as it can be described by an analytical formula, the NC-region energy requires solving a complex variational problem, which first involves minimizing an energy integral through delicate ordinary differential equations called the membrane’s shape equation. The solution to these shape equations allows us to find...

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[3]

(𝐻−𝐻$)"+𝐵%𝐾−𝛤<2𝜋𝑟𝑑𝑠&$ +𝜎∆𝑆+𝑝∆𝑉 (1) Here, 𝐵 is the bending rigidity, penalizing deviations of the mean curvature 𝐻=𝐶'+𝐶

𝑊=∫3#"(𝐻−𝐻$)"+𝐵%𝐾−𝛤<2𝜋𝑟𝑑𝑠&$ +𝜎∆𝑆+𝑝∆𝑉 (1) Here, 𝐵 is the bending rigidity, penalizing deviations of the mean curvature 𝐻=𝐶'+𝐶" from the spontaneous curvature 𝐻$. The principal curvatures are 𝐶' and 𝐶", where 𝐶' is along the length of the membrane’s curved profile in the radial plane, and 𝐶" is around the circumference in the rotational direction. 𝐵% is the...

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[4]

4#=∫X'"Y𝜑,)̅+*+,-.̅Z

𝐶'=𝜑,), 𝐶"=*+,-. (3) The integration domain of Eq. (1) goes from 𝑠=0 to 𝑠=∞, encompassing three main regions of the membrane (see Figure 1): the C domain from 𝑠=0 to 𝑠=𝑠/; the NC domain from 𝑠=𝑠/ to 𝑠=𝑠0 and 𝛤=0; and the flat (F) domain from 𝑠=𝑠0 to 𝑠=∞, where also 𝛤=0. Although the latter two domains can effectively be combined into one, we take 𝑠0 as a ...

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large yet finite arc-length

are 𝜑/=𝜃, 𝑟̅/=sin𝜃, 𝑧̅/=1−cos𝜃 (18) at the initial attachment with the NP, for 𝑠̅=𝑠̅/=𝜃. At the F-region, for 𝑠̅→∞, we impose the condition of asymptotic flatness (Deserno 2004a), such that lim)̅→&𝜑(𝑠̅)=lim)̅→&𝜑,)̅(𝑠̅)=0 (19) where the membrane becomes flat, approaching an undeformed state. To ensure the integration domain of Eq. (12) is finite, we replac...

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[6]

or optical methods (Popescu et al. 2003). The reported range is 𝜎=0−4.5×10HM 𝑁/𝑚. Deserno (2004a) cites a representative tension value of 𝜎=2×10HN 𝑁/𝑚. As for the bending rigidity 𝐵, Helfrich (1973) estimated 𝐵≈10H'" 𝑒𝑟𝑔 (~ 23.4 𝑘#𝑇). Subsequent studies have commonly adopted 𝐵=20 𝑘#𝑇 as the reference value (Deserno 2004a, Gao et al. 2005). In the tension-...

2003
[7]

#∗"4#=𝑎𝜎[^ (A3) 𝑓∗=1−'

Notably, wrapping is spontaneous for the intermediate values 𝑓=0.83−0.87. Thus, as unwrapping progresses from 𝑓=1 to 0.87, this process stalls at even such small adhesive energy. This provides a potential explanation for the kiss-and-run where vesicles only reach hemifusion states thereby limiting their delivery efficiency 15 (Long et al. 2012). This cond...

2012
[8]

(A1)-(A5), with a bell width of ∆0"=0∗('H0∗)]H' (A6) The circular markers in Fig

The plot shows a bell shape, which width, from the fitting of Eqs. (A1)-(A5), with a bell width of ∆0"=0∗('H0∗)]H' (A6) The circular markers in Fig. A1 represent the simulated energy values obtained by numerically integrating Eq. (12) for each 𝑓, with a total of 80 data points for each 𝜎[ value. The solid lines showcase the analytical fits from Eq. (22). ...

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[9]

Soft Matter, 18(18), pp.3531-3545

The morphological role of ligand inhibitors in blocking receptor-and clathrin-mediated endocytosis. Soft Matter, 18(18), pp.3531-3545. Auth T, Gompper G. Budding and vesiculation induced by conical membrane inclusions. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics. 2009 Sep;80(3):031901. Bacca, M.,

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Journal of the Royal Society Interface, 19(196), p.20220525

Mechanics of diffusion-mediated budding and implications for virus replication and infection. Journal of the Royal Society Interface, 19(196), p.20220525. Dai J, Sheetz MP. Membrane tether formation from blebbing cells. Biophysical journal. 1999 Dec 1;77(6):3363-70. Deserno M, Bickel T. Wrapping of a spherical colloid by a fluid membrane. Europhysics lett...

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Mechanics of receptor-mediated endocytosis. Proceedings of the National Academy of Sciences, 102(27), pp.9469-9474. Helfrich W. Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift für Naturforschung c. 1973 Dec 1;28(11-12):693-703. Jülicher F, Seifert U. Shape equations for axisymmetric vesicles: A clarification. Physical Re...

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Journal of the Mechanics and Physics of Solids, 145, p.104133

Elastocytosis. Journal of the Mechanics and Physics of Solids, 145, p.104133. Popescu G, Ikeda T, Goda K, Best-Popescu CA, Laposata M, Manley S, Dasari RR, Badizadegan K, Feld MS. Optical measurement of cell membrane tension. Physical review letters. 2006 Nov 24;97(21):218101. Rangamani P. The many faces of membrane tension: Challenges across systems and ...

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