Pith. sign in

REVIEW 5 major objections 3 minor 1 cited by

Depletion forces from molecular crowding can initiate receptor-mediated endocytosis, and the model's kinetic phase diagram puts the optimal virus radius near HIV-1's ~50 nm.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Entropic depletion forces from crowding molecules are modeled as the initiator of receptor-mediated endocytosis, yielding phase diagrams and an optimal virus radius of 30–60 nm that reduces to prior predictions when the new terms are dropped.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A plausible new mechanism (depletion-driven initiation) buried under load-bearing math errors—worth referee attention, not acceptance as written. the 5 major comments →

arxiv 2607.12766 v2 pith:5BSRJJXV submitted 2026-07-14 cond-mat.soft cond-mat.stat-mechphysics.bio-phq-bio.SC

Entropy-Driven Initiation and Cellular Uptake Mediated by Viscoelastic Cytoskeleton: A Kinetic Phase Diagram from Onsager Variational Principle

Jinjie Liu , Zhong-Can Ou-Yang , Hao Wu This is my paper
classification cond-mat.soft cond-mat.stat-mechphysics.bio-phq-bio.SC
keywords receptor-mediated endocytosisdepletion forcesmolecular crowdingOnsager variational principleviscoelastic cytoskeletonengulfment phase diagramoptimal virus sizeHIV-1
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper claims that the long-ignored initiation step of receptor-mediated endocytosis—what brings a virus or nanoparticle close enough to the membrane for ligand–receptor binding—is driven by entropic depletion forces from small crowding molecules. It builds a continuum model on the Onsager variational principle, with engulfment depth as the single generalized coordinate, and combines depletion attraction, ligand–receptor binding, membrane bending and tension, and viscoelastic cytoskeletal resistance. The model yields a kinetic phase diagram with quantitative thresholds: a critical crowder concentration for initiation, a minimum ligand density for complete engulfment, a finite particle-size window, and an optimal radius near 50 nm that shrinks as binding energy increases. If correct, this gives a variational foundation that connects molecular crowding, cell mechanics, and viral size, and explains why HIV-1 sits near the predicted optimum.

Core claim

The central claim is that endocytosis does not need pre-existing virus–membrane contact: crowding agents exert a depletion force that drives the particle to the membrane within about 7×10^-5 seconds, solving the initiation problem, after which ligand–receptor binding sustains wrapping against viscoelastic cytoskeletal resistance. From the free energy landscape the paper derives closed-form predictions: a critical crowder concentration for initiation (Eq. 3), a minimum ligand density for complete engulfment (Eqs. 31–32), a quartic equation that defines the engulfable size window (Eq. 34), and an optimal radius Ropt = sqrt(6κ/(a-2γ)) that decreases with binding energy density, giving roughly 4

What carries the argument

The load-bearing object is the Onsager variational principle—a rule that the system's evolution minimizes the sum of free-energy change and dissipated power—applied to a single generalized coordinate, the engulfment depth h(t). The free energy E(h) adds four terms: depletion energy, which is linear in h and gives a constant driving force; ligand–receptor binding energy; membrane deformation from the Helfrich–Canham Hamiltonian; and cytoskeleton deformation via a viscoelastic Hertz contact built with the elastic–viscoelastic correspondence principle. Minimizing the total Rayleigh action gives the kinetic law ζ(h) dh/dt = F(h), and the Onsager solubility condition—the requirement that the loga

Load-bearing premise

The phase boundaries are obtained by requiring the argument of a logarithm in the wrapping-time formula to be positive and finite—a mathematical existence condition—and the paper treats this as a physical threshold for engulfment; if that interpretive step is not legitimate, the predicted thresholds (critical concentration, minimum ligand density, size window, critical stiffness) do not follow.

What would settle it

Measure uptake of well-characterized nanoparticles as a function of crowder concentration in a cellular or synthetic system: if particles below the predicted Rmin (roughly 22 nm) or above Rmax are internalized at the stated ligand densities, or if uptake persists when crowder concentration falls below the predicted critical value, the phase-boundary predictions are contradicted. A simpler variant: in a vesicle-only assay, check whether increasing crowder concentration produces membrane adhesion even in the complete absence of ligand–receptor binding.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Below a critical crowder concentration, initiation fails: even with high receptor density, no ligand–receptor encounter occurs, so uptake cannot start.
  • Complete engulfment requires ligand density above a minimum value; below it, the driving force cannot overcome membrane bending and cytoskeletal resistance, so the particle is only partially wrapped.
  • Only particles within a finite size window (roughly 20 nm to about 100 nm for typical parameters) can be fully internalized; smaller ones are blocked by bending cost, larger ones by cytoskeletal deformation, and engulfment time diverges at the lower bound.
  • The optimal particle radius is fixed by membrane constants and binding energy density, not by cell stiffness, and it decreases from about 60 nm to about 30 nm as binding strengthens; HIV-1's roughly 50 nm radius sits near the predicted optimum for its ligand–receptor system.
  • Stiffer cells slow engulfment, narrow the size window, and impose a critical stiffness beyond which no particle size can be completely engulfed.

Where Pith is reading between the lines

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

  • If depletion forces really initiate uptake, then raising the concentration of inert crowding agents (such as dextran or Ficoll) in culture medium should be able to restore or accelerate uptake for particles that otherwise fail—a testable prediction the paper does not state explicitly.
  • Because the optimal radius is independent of cytoskeletal mechanics while the engulfment time is not, cells could modulate uptake speed without changing their size preference, which suggests a separation of control knobs for drug-delivery design.
  • The flat-surface limit recovering the Asakura–Oosawa result suggests the entropic driving force is generic, so the same initiation mechanism may apply to other membrane-adhesion events, such as immune-cell phagocytosis or uptake of lipid nanoparticles, beyond viruses.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 3 minor

Summary. The paper proposes a unified Onsager-variational model of receptor-mediated endocytosis in which crowding-induced depletion forces initiate particle-membrane contact, and viscoelastic creep of the cytoskeleton controls wrapping kinetics. It derives a critical crowder concentration for initiation, a minimum ligand density, an engulfable size window, a critical cell stiffness, and an optimal virus radius claimed to match HIV-1. The main quantitative output is a kinetic phase diagram built from the 'Onsager solubility condition' applied to an expression for the complete engulfment time.

Significance. If sound, the model would fill an acknowledged gap by explaining how a virus and membrane come into proximity before specific binding, and it would make falsifiable predictions about ligand density, particle size, and cell stiffness. The paper also recovers the Asakura-Oosawa flat-surface depletion result in an asymptotic limit, which is a useful consistency check. However, the central Phase-2 kinetic derivation is not the solution of the stated Onsager equation, and the phase boundaries are obtained from the domain of a logarithm rather than from a variational principle. These defects are load-bearing: Eqs. (31)-(34) and Figs. 7-8 rest on them. The claimed optimal-size/HIV match is derived after removing the new entropic and cytoskeleton terms, so it does not validate the proposed mechanism.

major comments (5)
  1. [§II.F-H and Appendix F] Eqs. (22)-(26) adopt the Lee-Radok step-load solution a^3(t)=(3/8)RFΦ(t) for a constant force, but the driving force in Eq. (24) depends explicitly on h through −2γπh and −(√R/D)h^{3/2}. For a time-varying load the correct viscoelastic contact relation is a Boltzmann superposition integral, not the algebraic substitution F(h). Consequently Eq. (26) is not the solution of ζ(h)h˙=F(h), and no Rayleigh dissipation function for Phase 2 is actually minimized. The unphysical nature is visible at h=0: Eq. (26) gives t(0)=−τ ln(1+Ec/Ev)<0, whereas wrapping must start at t=0.
  2. [§III.A and Eqs. (28)-(30)] The 'Onsager solubility condition' is simply the statement that the argument of the logarithm in Eq. (27) is positive and not larger than 1. It is the existence domain of the algebraic formula, not a consequence of the variational principle. The lower bound F(2R)>16√2 EcEvR^2/[3(Ec+Ev)] is the condition that a constant force equal to the final force F(2R) would reach h=2R at infinite time; it is neither necessary nor sufficient for the actual depth-dependent force history. Every downstream result—amin in Eq. (31), ζmin in Eq. (32), the quartic size window in Eq. (34), and the phase diagrams in Figs. 7-8—inherits this unidentified assumption.
  3. [Eq. (3) and Appendix B.3] The critical concentration formula is dimensionally inconsistent. In Eq. (3)/Eq. (B15), ccrit is a number density with dimensions L^{-3}, while the right-hand side evaluates to dimensions L^{-4} (the bracket 4πκ/R+2πγR has dimensions of force, J/L, not energy). Thus the threshold concentration cannot be correct as written, and the claimed initiation condition is not established.
  4. [Appendix A and §II.C] The numerical evaluation of the Phase-1 time is wrong, and the accompanying physical explanation contradicts the formula. Eq. (A11) with Table II gives t1 ≈ 6×10^{-5} s, not 1.0×10^{-9} s as stated in Eq. (A13). Moreover, Appendix A.5 says the depletion force 'becomes singular near contact' and that the 'force diverges as the depletion volume derivative diverges,' but the sub-interval force F1 in Eq. (A9) is constant in h, and V1 is linear in h. This changes tapproach in Eq. (8) and weakens the time-scale argument in §II.C.
  5. [Appendix I and Eq. (36)] The optimal radius Ropt=√(6κ/(a−2γ)) is derived after explicitly dropping the entropic and cytoskeleton terms (F(2R)≈2πRa−4πκ/R−4γπR). The claimed match to HIV-1 therefore does not test the new crowding or viscoelastic physics; it reduces to a previously known membrane-binding balance. This should be acknowledged, and the statement in §IV.C that the match 'validates' the present model is unsupported.
minor comments (3)
  1. [Table II and §IV.C] Table II lists τ=ηm/Em=2×10^3 s, but the text in §IV.C estimates τ≈800 s using ηc=4000 kPa·s and Ec=5 kPa. The notation Em vs Ec is also inconsistent between the table and the main text, and the unused parameter δ=5 nm could be removed or described.
  2. [Fig. 4 caption] The caption states that 'the optimal size depends on the ligand-receptor binding energy density,' but the curves are plotted for different Ec at fixed parameters; the optimum is the same for all curves, as the text correctly notes. The caption should be revised.
  3. [References] Ref. [58] lacks author names and a full journal identification, and Ref. [52] duplicates Ref. [3]. A few equations use '−' unary spacing ambiguously (e.g., Eq. (A11)), making signs hard to parse.

Circularity Check

2 steps flagged

Phase boundaries are the log-domain of the paper's own step-load formula, not a consequence of Onsager; the HIV-1 optimal-size match is obtained after dropping the new terms.

specific steps
  1. self definitional [Sec. II.H, Eqs. (26)-(30); Appendix F, Eq. (F3); Appendix G, Eq. (G7)]
    "This expression is the Onsager solubility condition: the argument of the logarithm must be positive, which requires: 1 + Ec/Ev − 16√2EcR2/(3F(2R)) > 0, and finite, which requires: 1 + Ec/Ev − 16√2EcR2/(3F(2R)) ≤ 1."

    Eq. (26) is obtained in Appendix F by solving the constant-force Lee-Radok formula h^{3/2}=3F/(8R^{1/2})Φ(t) for t, not by minimizing ∂E/∂h hdot + (1/2)ζ hdot^2 or by integrating ζ(h) hdot=F(h). The 'solubility condition' is then defined as the requirement 0<X≤1 so that the paper's own tc formula returns a finite positive time. Because Eqs. (31)-(34) are algebraic rearrangements of that same inequality, the predicted amin, ζmin, size window, and critical stiffness are constructed to be exactly the log-domain of the author's approximate formula. The formula is also internally inconsistent with the stated process: at h=0, Eq. (F3) gives e^{-t/τ}=1+Ec/Ev>1, hence t(0)<0. The phase boundaries therefore are not consequences of the Onsager variational principle; they are the existence domain of

  2. renaming known result [Appendix I, Eqs. (I4)-(I7); Sec. III.C, Eq. (36) and Table I]
    "For the simplified case where entropic and cytoskeleton terms are negligible compared to binding and membrane terms, F (2R) ≈ 2πRa − 4πκ/R − 4γπR . ... The optimal condition becomes: R2(a − 2γ) = 6κ. Therefore: Ropt = sqrt(6κ/(a−2γ))."

    The section is titled 'Optimal Virus Size from Variational Principle' and Table I/abstract tie Ropt≈50 nm to HIV-1. But the derivation explicitly drops the depletion and cytoskeleton contributions before differentiating; the surviving balance 2F=R F' with F≈2πRa−4πκ/R−4γπR is the membrane/binding balance already underlying the receptor-diffusion models the paper cites, and the paper itself says the result 'agrees qualitatively with previous predictions from receptor-diffusion models [3,4]'. The claimed match to HIV-1 therefore does not validate the new entropic/viscoelastic mechanism; it is the old energetic optimum obtained by deleting the new physics.

full rationale

Most of the paper's independent algebra—depletion-volume geometry, the R→∞ Asakura-Oosawa limit, and the Phase-1 approach time—is self-contained and not circular. No load-bearing self-citation was found; the Lee-Radok correspondence principle is cited to external solid-mechanics sources and is a genuine external input. The circularity is local but central: the 'Onsager solubility condition' is not derived by minimizing the Onsager action; it is the 0<log≤1 domain of the author's own inversion of a step-load Hertz-creep formula, and Eqs. (31)-(34) are rewrites of that domain. In addition, the HIV-1 optimal-size prediction is obtained after setting the new depletion and cytoskeleton terms to zero, so it is a restatement of the earlier membrane/binding optimum rather than a test of the new mechanism. These two reductions affect the central quantitative claims; the independent Asakura-Oosawa limit and approach-time calculation prevent the whole paper from being merely definitional. Score 6 reflects partial, load-bearing circularity rather than a completely empty derivation.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central claims rest on literature parameter choices (r, c, a, τ) and on several domain assumptions about the continuum model. The most fragile inputs are the ad hoc initiation barrier and the unstated quasi-static approximation in the viscoelastic kinetics. No new entities are postulated.

free parameters (5)
  • crowder radius r = 20 nm
    Chosen in Table II; sets depletion-zone thickness and all depletion energies.
  • crowder concentration c = 1.5 × 10^22 m^-3
    Chosen in Table II; controls approach time and depletion force magnitude.
  • binding energy density a = 4.14e-4 J/m^2 in Table II; 2.5e-4 J/m^2 for HIV match
    Input from literature but the headline HIV match depends on the chosen value.
  • membrane initiation barrier = 4πκ/R + 2πγR
    Ad hoc barrier in Eq. (B14) with wrong dimensions; defines ccrit.
  • cytoskeleton relaxation time τ = 2000 s in Table II; 800 s in Discussion
    Inconsistent values used for kinetics.
axioms (6)
  • domain assumption Onsager variational principle with linear response ζḣ = F
    Foundational bracketing of the theory; the paper assumes this principle governs both phases.
  • domain assumption Elastic-viscoelastic correspondence principle applied to Hertz contact with a state-dependent force
    The constant-force solution is used with F(h), requiring an unstated quasi-static approximation.
  • domain assumption Membrane deformation energy truncated to 4πκh/R + γπh²
    Neglects full Helfrich shape equation and higher-order curvature terms.
  • domain assumption Depletion zone geometry with membrane as a flat plane and linear depletion volume in h after contact
    Central to the constancy of the depletion force during wrapping.
  • domain assumption Standard linear solid model for the cytoskeleton
    Chosen rheological model; parameters taken from literature.
  • ad hoc to paper Onsager solubility condition as a physical phase-boundary condition
    The log-argument positivity is treated as a thermodynamic threshold, but it is only the domain of the approximate formula.

reviewed 2026-08-02 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Entropy-Driven Initiation and Cellular Uptake Mediated by Viscoelastic Cytoskeleton: A Kinetic Phase Diagram from Onsager Variational Principle." pith.science (2026). https://pith.science/paper/5BSRJJXV

@misc{pith2026260712766,
  author       = {Pith},
  title        = {Pith review of: Entropy-Driven Initiation and Cellular Uptake Mediated by Viscoelastic Cytoskeleton: A Kinetic Phase Diagram from Onsager Variational Principle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5BSRJJXV}},
  note         = {Machine review of arXiv:2607.12766}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

A fundamental question in receptor-mediated endocytosis remains unanswered: what initial driving force brings ligands and receptors into close proximity? While previous models assume pre-existing contact and overlook this initiation problem, we propose that entropic forces from nanoscale biomolecules in crowded cellular environments provide the essential driving mechanism. We develop a unified continuum model rooted in the Onsager variational principle, where engulfment depth serves as the generalized coordinate and the driving force derives from a free energy landscape of entropic, binding, membrane, and cytoskeleton contributions. The framework naturally incorporates: (i) entropy-driven adhesion as initiation; (ii) ligand-receptor binding as the sustaining force; (iii) membrane deformation via the Helfrich-Canham Hamiltonian; and (iv) cytoskeleton viscoelasticity through the elastic-viscoelastic correspondence principle. The kinetic phase diagram predicts a critical biomolecule concentration for initiation, a lower bound of ligand density for complete engulfment, a finite size window for engulfable particles, and an optimal virus radius of 30--60 nm that decreases with increasing binding energy. The Onsager solubility condition naturally yields the phase boundaries. The model exhibits asymptotic consistency with the classic Asakura-Oosawa result in the large-particle flat-surface limit. Stiffer cells lead to longer engulfment times and narrower size windows. Strikingly, the optimal size matches HIV-1 dimensions under physiologically realistic parameters. This work provides a variational foundation for cellular uptake with implications for virology, nanotechnology, and drug delivery.

Figures

Figures reproduced from arXiv: 2607.12766 by Hao Wu, Jinjie Liu, Zhong-Can Ou-Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic illustration of depletion-driven engulfment. Macromolecules (pink [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Two-phase engulfment model. (a) Phase 1: Depletion-driven approach until [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: displays the engulfment depth h(t) as a function of time for different host cell Young’s moduli (Ec = 1, 2, 3, 4, 5 kPa) at fixed virus radius R = 50 nm, calculated from Eq. (13). Softer cells allow faster engulfment, with complete wrapping occurring within seconds to tens of seconds depending on local mechanical properties. This timescale is consistent with experimental observations of clathrin-mediated e… view at source ↗
Figure 4
Figure 4. Figure 4: illustrates the complete engulfment time tc as a function of virus radius for dif￾ferent host cell Young’s moduli, calculated from Eq. (27). The optimal size is clearly visible as the minimum of each curve. Notably, this optimal size is independent of cell stiffness, although the magnitude of the minimum engulfment time increases with cell stiffness. This non-monotonic behavior arises from the competition … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Minimum ligand-receptor binding energy density [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the complete engulfment time tc as a function of Young’s modulus of host cells for different cell radii. A critical stiffness exists for each radius beyond which engulfment becomes impossible; larger viruses are more sensitive to cell stiffness and exhibit lower critical stiffness values. This result has important implications for understanding how changes in cellular mechanical properties—such as th… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Engulfment phase diagram in the plane of virus radius [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Engulfment phase diagram in the plane of virus radius [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Variational formulation for the dynamics of soft matter including inertia

    cond-mat.soft 2026-07 conditional novelty 4.0

    The proposed generalization of Onsager's variational principle to include inertia reduces to the standard Euler-Lagrange equation with Rayleigh dissipation and reproduces known under-damped DDFT equations.

Reference graph

Works this paper leans on

82 extracted references · cited by 1 Pith paper

  1. [1]

    The depletion zone around the virus (radius R + r) overlaps with the depletion zone of the membrane

    A.1 Geometric Setup for the Approach Phase During Phase 1, the virus has not yet contacted the membrane ( h ≤ 0). The depletion zone around the virus (radius R + r) overlaps with the depletion zone of the membrane. The depletion volume is the overlap volume between the spherical depletion shell of the virus and the half-space excluded by the membrane. The...

  2. [2]

    A.2 Sub-interval 1: h ∈ [−2r, −r] In this regime, the depletion volume is given by: V0(h) = π(h + 2r) [h2(r + R) + 2Rr(2r + 3R) − h(2r2 + 5Rr + 3R2)] 3(r + R) . (A1) This expression is derived by calculating the volume of the spherical cap of the virus depletion zone that extends below the membrane surface, accounting for the fact that the cap height is h...

  3. [3]

    A.3 Sub-interval 2: h ∈ [−r, 0] In this regime, the depletion volume is: V1(h) = πr [h(−r2 + 3Rr + 6R2) + r(2r2 + 13Rr + 15R2)] 3(r + R) . (A7) This expression is obtained by calculating the volume of the intersection of the spherical depletion shell with the half-space, valid when the cap height ish+r and the overlap becomes a simple cap. The correspondi...

  4. [4]

    A.4 T otal Approach Time The total time for the virus to approach the membrane from an initial separation of 2 r down to contact is: tapproach = t0 + t1. (A12) 32 Using the parameter values from Table II ( R = 50 nm, r = 20 nm, c = 1.5 × 1022 m−3, T = 300 K, µ = 10−3 Pa·s), we evaluate: t0 ≈ 7.136 × 10−5 s, t 1 ≈ 1.0 × 10−9 s, (A13) so that: tapproach = t...

  5. [5]

    This is because the depletion force becomes singular near contact due to the divergence of the depletion volume derivative

    A.5 Physical Interpretation The fact that t1 ≪ t0 indicates that the final stage of approach (from h = −r to h = 0) is practically instantaneous compared to the initial stage. This is because the depletion force becomes singular near contact due to the divergence of the depletion volume derivative. The total approach time is dominated by the early stage w...

  6. [6]

    Smaller crowding agents of radius r are excluded from a depletion zone of thickness r around both the virus and the membrane

    B.1 Geometric Setup We consider a spherical virus of radius R approaching a planar cell membrane. Smaller crowding agents of radius r are excluded from a depletion zone of thickness r around both the virus and the membrane. The engulfment depth h is measured from the membrane surface. The depletion volume is the volume of the intersection of the depletion...

  7. [7]

    The height of the small cap of the depletion zone that enters the membrane is: Hq2 = R − (R + r) cosψ, (B6) and its radius is: rq2 = p (R + r)2 − [r + (R + r) cosψ]2

    B.2 Depletion V olume After Contact After the virus contacts the membrane and engulfs to depth h, the geometry changes. The height of the small cap of the depletion zone that enters the membrane is: Hq2 = R − (R + r) cosψ, (B6) and its radius is: rq2 = p (R + r)2 − [r + (R + r) cosψ]2. (B7) The volume of this small cap is: Vq2 = πHq2(H 2 q2 + 3r2 q2) 6 . ...

  8. [8]

    (B13) The membrane deformation barrier at initial contact is approximately: Ebarrier ∼ 4πκ R + 2πγR

    B.3 Initiation Criterion At h = 0, the entropic free energy is: Edep(0) = − cπr2 3(r + R) (2r2 + 13rR + 15R2)kBT. (B13) The membrane deformation barrier at initial contact is approximately: Ebarrier ∼ 4πκ R + 2πγR. (B14) The initiation condition Edep(0) > Ebarrier yields: c > ccrit = 3(r + R) πr2(2r2 + 13rR + 15R2)kBT 4πκ R + 2πγR . (B15) Appendix C: Deri...

  9. [9]

    (D3) For a virus of radiusR indenting a cell of radius Rcell, in the limit Rcell ≫ R and Ec ≪ Ev, the combined elastic modulus is: D = 3 4 1 − σ2 c Ec + 1 − σ2 v Ev

    D.1 Elastic Contact For two elastic spheres in contact [71], the Hertz theory gives: F = 4 3 E∗p Reff h3/2, (D1) where 1 E∗ = 1 − σ2 1 E1 + 1 − σ2 2 E2 , (D2) and 1 Reff = 1 R1 + 1 R2 . (D3) For a virus of radiusR indenting a cell of radius Rcell, in the limit Rcell ≫ R and Ec ≪ Ev, the combined elastic modulus is: D = 3 4 1 − σ2 c Ec + 1 − σ2 v Ev . (D4)...

  10. [10]

    The contact radius for a constant force F is: a3(t) = 3 8 RF Φ(t)

    D.2 Viscoelastic Correspondence Principle For a viscoelastic medium, the elastic modulus is replaced by the creep compliance oper- ator. The contact radius for a constant force F is: a3(t) = 3 8 RF Φ(t). (D7) 37 Since h = a2/R: h(t) = F 2 R 1/3 3Φ(t) 8 2/3 . (D8) This is the Onsager-consistent kinetic law for the wrapping phase: the contact radius evolves...

  11. [11]

    (D9) Appendix E: Derivation of the Driving F orce The total energy is: E(h) = − cπr 3(r + R) h(−r2 + 3rR + 6R2) + r(2r2 + 13rR + 15R2) kBT − 2πRha + 4πκh R + γπh 2 + 2 √ R 5D h5/2

    D.3 Cytoskeleton Deformation Energy The cytoskeleton deformation energy is obtained by integrating the force with respect to displacement: Ecyto(h) = Z h 0 F (h′) dh′ = Z h 0 √ R D h′3/2 dh′ = 2 √ R 5D h5/2. (D9) Appendix E: Derivation of the Driving F orce The total energy is: E(h) = − cπr 3(r + R) h(−r2 + 3rR + 6R2) + r(2r2 + 13rR + 15R2) kBT − 2πRha + ...

  12. [12]

    G. J. Doherty and H. T. McMahon, Annu. Rev. Biochem. 78, 857 (2009)

    2009

  13. [13]

    Zhang, H

    S. Zhang, H. Gao, and G. Bao, ACS Nano 9(9), 8655 (2015)

    2015

  14. [14]

    H. Gao, W. Shi, and L. B. Freund, Proc. Natl. Acad. Sci. USA 102, 9469 (2005)

    2005

  15. [15]

    W. Shi, J. Wang, X. Fan, and H. Gao, Phys. Rev. E 78, 061914 (2008)

    2008

  16. [16]

    Mercer, M

    J. Mercer, M. Schelhaas, and A. Helenius, Annu. Rev. Biochem. 79, 803 (2010)

    2010

  17. [17]

    Marsh and A

    M. Marsh and A. Helenius, Cell 124, 729 (2006)

    2006

  18. [18]

    Mammen, S.-K

    M. Mammen, S.-K. Choi, and G. M. Whitesides, Angew. Chem. Int. Ed. 37, 2754 (1998)

    1998

  19. [19]

    Verma and F

    A. Verma and F. Stellacci, Small 6, 12 (2010)

    2010

  20. [20]

    Nelson, Biological Physics (WH Freeman, New York, 2004)

    P. Nelson, Biological Physics (WH Freeman, New York, 2004)

    2004

  21. [21]

    Asakura and F

    S. Asakura and F. Oosawa, J. Chem. Phys. 22(7), 1255 (1954)

    1954

  22. [22]

    Asakura and F

    S. Asakura and F. Oosawa, J. Chem. Phys. 33(126), 183 (1958)

    1958

  23. [23]

    Vrij, Pure Appl

    A. Vrij, Pure Appl. Chem. 48, 471 (1976)

    1976

  24. [24]

    Dinsmore, A

    A. Dinsmore, A. Yodh, and D. Pine, Phys. Rev. E 52, 4045 (1995)

    1995

  25. [25]

    Imhof and J

    A. Imhof and J. Dhont, Phys. Rev. Lett. 75, 1662 (1995)

    1995

  26. [26]

    Steiner, A

    U. Steiner, A. Meller, and J. Stavans, Phys. Rev. Lett. 74, 4750 (1995)

    1995

  27. [27]

    S. M. Ilett, A. Orrock, W. Poon, and P. Pusey, Phys. Rev. E 51, 1344 (1995)

    1995

  28. [28]

    Dinsmore, D

    A. Dinsmore, D. Wong, P. Nelson, and A. Yodh, Phys. Rev. Lett. 80, 409 (1998)

    1998

  29. [29]

    Irajizad, N

    E. Irajizad, N. Walani, S. L. Veatch, A. P. Liu, and A. Agrawal, Soft Matter 13, 1455 (2017)

    2017

  30. [30]

    H. Wan, D. Xu, L. Gao, and L.-T. Yan, Small Sci. 4, 2300078 (2024). 42

    2024

  31. [31]

    A. P. Minton, Biophys. J. 63, 1090 (1992)

    1992

  32. [32]

    A. P. Minton, Biophys. J. 68, 1311 (1995)

    1995

  33. [33]

    R. J. Ellis, Trends Biochem. Sci. 26, 597 (2001)

    2001

  34. [34]

    H. Wu, H. Shiba, and H. Noguchi, Soft Matter 9, 9907 (2013)

    2013

  35. [35]

    Wu and H

    H. Wu and H. Noguchi, AIP Conf. Proc. 1518, 649 (2013)

    2013

  36. [36]

    M. R. K. Mofrad and R. D. Kamm, Cytoskeletal Mechanics: Models and Measurements(Cam- bridge University Press, 2009)

    2009

  37. [37]

    Rotsch, K

    C. Rotsch, K. Jacobson, and M. Radmacher, Proc. Natl. Acad. Sci. USA 96, 921 (1999)

    1999

  38. [38]

    T. A. Ryan, S. J. Smith, and H. Reuter, Proc. Natl. Acad. Sci. USA 93(11), 5567 (1996)

    1996

  39. [39]

    W. Liou, H. J. Geuuze, M. J. Geelen, and J. W. Slot, J. Cell Biol. 136(1), 61 (1997)

    1997

  40. [40]

    S. R. Elkin, A. M. Lakoduk, and S. L. Schmid, Wiener Medizinische Wochenschrift 166(7), 196 (2016)

    2016

  41. [41]

    S. X. Sun and D. Wirtz, Biophys. J. 90, L10 (2006)

    2006

  42. [42]

    Matarrese and W

    P. Matarrese and W. Malorni, Cell Death Differ. 12, 932 (2005)

    2005

  43. [43]

    B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, Nano Lett. 6, 662 (2006)

    2006

  44. [44]

    J. Wang, L. Li, and Y. Zhou, Chin. Sci. Bull. 59(19), 2277 (2014)

    2014

  45. [45]

    Kruse, T

    E. Kruse, T. Abdalrahman, P. Selhorst, and T. Franz, Biomech. Model. Mechanobiol. 22(6), 1847 (2023)

    2023

  46. [46]

    Onsager, Phys

    L. Onsager, Phys. Rev. 37, 405 (1931)

    1931

  47. [47]

    Onsager, Phys

    L. Onsager, Phys. Rev. 38, 2265 (1931)

    1931

  48. [48]

    Doi, Soft Matter Physics(Oxford University Press, Oxford, 2013)

    M. Doi, Soft Matter Physics(Oxford University Press, Oxford, 2013)

    2013

  49. [49]

    Arroyo and A

    M. Arroyo and A. DeSimone, Phys. Rev. E 79(3), 031915 (2009)

    2009

  50. [50]

    E. H. Lee and J. R. M. Radok, J. Appl. Mech. 27, 438 (1960)

    1960

  51. [51]

    J. R. M. Radok, Q. Appl. Math. 15, 198 (1957)

    1957

  52. [52]

    Helfrich, Z

    W. Helfrich, Z. Naturforsch. C 28, 693 (1973)

    1973

  53. [53]

    P. B. Canham, J. Theor. Biol. 26, 61 (1970)

    1970

  54. [54]

    Ou-Yang and W

    Z.-C. Ou-Yang and W. Helfrich, Phys. Rev. A 39, 5280 (1989)

    1989

  55. [55]

    Wu and Z.-C

    H. Wu and Z.-C. Ou-Yang, Membranes 15, 182 (2025)

    2025

  56. [56]

    K. L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985)

    1985

  57. [57]

    J. Hu, R. Lipowsky, and T. R. Weikl, Proc. Natl. Acad. Sci. USA 110, 15283 (2013)

    2013

  58. [58]

    H. Yuan, J. Li, G. Bao, and S. Zhang, Phys. Rev. Lett. 105, 138101 (2010). 43

    2010

  59. [59]

    M. I. Chang, P. Panorchan, T. M. Dobrowsky, Y. Tseng, and D. Wirtz, J. Virol. 79(23), 14748 (2005)

    2005

  60. [60]

    C. Chen, Y. Zhou, C. Chen, S. Zhu and X. Yan, ACS Nano, 16(4), 6886 (2022)

    2022

  61. [61]

    Catenacci, R

    L. Catenacci, R. Rossi, F. Sechi, D. Buonocore, M. Sorrenti, S. Perteghella, M. Peviani, and M.C. Bonferoni, Pharmaceutics 16(12), 1521 (2024)

    2024

  62. [62]

    R. Shi, X. Liu, Y. Wang, M. Pan, S. Wang, L. Shi, and B. Ni, Hum. Vaccin. Immunother. 20(1), 2342592 (2024)

    2024

  63. [63]

    H. Gao, W. Shi, and L. B. Freund, Proc. Natl. Acad. Sci. USA 102(27), 9469 (2005)

    2005

  64. [64]

    L. Li, X. Liu, Y. Zhou, and J. Wang, Biophys. J. 102, 2230 (2012)

    2012

  65. [65]

    X. Song, R. Lai, H. Liu, K. Liang, S. Wang, H. Liu and S. Zhao, Biomacromolecules 26(12), 8858 (2025)

    2025

  66. [66]

    F. Frey, F. Ziebert, and U. S. Schwarz, Phys. Rev. E 100, 052403 (2019)

    2019

  67. [67]

    P. D. Kaplan, J. L. Rouke, A. G. Yodh, and D. J. Pine, Phys. Rev. Lett. 72, 582 (1994)

    1994

  68. [68]

    T. M. Muenker, B. E. Vos, and T. Betz, eLife 13, RP97416 (2024)

    2024

  69. [69]

    Substrate Stiffness and Particle Properties Influence Cellular Uptake of Nanoparticles and Viruses from the Ventral Side, RCS Eng. (2023)

    2023

  70. [70]

    Huang, Y

    C. Huang, Y. Zhang, H. Yuan, and H. Gao, Nano Lett. 13(9), 4546 (2013)

    2013

  71. [71]

    Maiti, K

    Nividha, A. Maiti, K. Parihar, R. Chakraborty, P. Agarwala, D. K. Sasmal, R. Radhakrishnan, T. Ghosh, B. Sinha, D. Bhatia, and K. K. Dey, Small e74049 (2026)

    2026

  72. [72]

    H Wu, M Thi´ ebaud, WF Hu, A Farutin, S Rafai, MC Lai, P Peyla, and C Misbah, Physical Review E 92(5), 050701(R) (2015)

    2015

  73. [73]

    J. Y. Tinevez, U. Schulze, G. Salbreux, J. Roensch, J. F. Joanny, and E. Paluch, Proc. Natl. Acad. Sci. USA 106(44), 18581 (2009)

    2009

  74. [74]

    H Wu, A Farutin, WF Hu, M Thi´ ebaud, S Rafai, P Peyla, MC Lai, and C Misbah, Soft Matter 12(36), 7470 (2016)

    2016

  75. [75]

    Wang and H

    Q. Wang and H. Wu, Phys. Biol. 18(4), 045001 (2021)

    2021

  76. [76]

    Levchenko and P

    A. Levchenko and P. A. Iglesias, Biophys. J. 82(1), 50 (2000)

    2000

  77. [77]

    Farutin, H

    A. Farutin, H. Wu, W.-F. Hu, S. Safai, P. Peyla, M.-C. Lai, and C. Misbah, J. Fluid Mech. 881, 365 (2019)

    2019

  78. [78]

    P. J. M. van Haastert, J. Cell Sci. 123(18), 3031 (2010)

    2010

  79. [79]

    H. Wu, M. A. P. de Le´ on, and H. G. Othmer, J. Math. Biol. 77, 595 (2018). 44

    2018

  80. [80]

    Jacobson, P

    K. Jacobson, P. Liu, and B. C. Lagerholm, Cell, 177(4), 806 (2019)

    2019

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.