Rotational mobility in spherical membranes: The interplay between Saffman-Delbr\"uck length and inclusion size

Eric Lauga; Marco Vona

REVIEW 2 major objections 7 minor 53 references

Rotational mobility in spherical membranes: The interplay between Saffman-Delbr\"uck length and inclusion size

T0 review · 2 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A rotating cap-shaped inclusion in a spherical membrane has its entire rotational drag set by the first rotlet coefficient, so the dimensionless mobility depends only on inclusion size and the Saffman–Delbrück ratio.

desk verdict A genuinely new finite-size curved-membrane mobility calculation with a clean derivation and a useful five-regime map; the main weakness is an unverified truncation error at finite membrane viscosity, plus an obvious manuscript-production glitch. read the letter →

arxiv 2501.05367 v1 pith:PHRQDHHF submitted 2025-01-09 physics.bio-ph physics.flu-dyn

Marco Vona , Eric Lauga This is my paper · ORCID

classification physics.bio-phphysics.flu-dyn MSC 76D0776Z99

keywords membranehydrodynamicsrotationalmobilitySaffman–Delbrücklengthsphericalspherical-capinclusionrotletLegendreexpansionvesicle

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

This paper computes how hard it is to rotate a rigid cap-shaped inclusion embedded in a spherical fluid membrane surrounded by viscous solvent on both sides. It claims that the entire rotational drag is captured by a single Legendre coefficient, the rotlet strength $c_1$, so the dimensionless drag $\Lambda_R(\varepsilon,\theta_p)=\Omega^{-1}\theta_p^{-3}c_1$ depends only on the inclusion's angular size $\theta_p=R_p/R_m$ and the Saffman–Delbrück ratio $\varepsilon=2\mu R_p/\eta$. As the Saffman–Delbrück length is varied, a small cap passes through five asymptotic drag regimes, from rigid whole-vesicle rotation through nearly planar membrane drag to bare three-dimensional solvent drag. The flat Saffman–Delbrück result is recovered only in the intermediate window $R_p \ll \ell_{\mathrm{SD}} \ll R_m^3/R_p^2$, and the paper identifies when planar theory fails for large inclusions or highly viscous membranes. If correct, this provides a route to measuring membrane viscosity from the rotational diffusion of large inclusions.

What carries the argument

The load-bearing object is the rotlet coefficient $c_1$, the first term in a Legendre/rotlet-multipole expansion of the fluid velocity in the outer solvent, inner solvent, and membrane (Eqs. 2.12–2.14). Orthogonality of the Legendre derivatives $P_n'$ reduces the momentum balance to a linear recurrence for the coefficients coupled to an ordinary differential equation for the membrane angular velocity $v(x)$ (Eqs. 2.16–2.19); truncating at $k$ modes gives a finite linear system whose solution yields $c_1$. The identity $G_p=8\pi\mu R_m^3 c_1$ (Eq. 2.21) is what carries the argument, because it shows that all details of the flow and inclusion shape enter the torque only through this single coefficient. The rest of the machinery turns the calculation of $c_1$ into a two-parameter problem depending only on $\varepsilon$ and $\theta_p$.

What would settle it

Recompute $\Lambda_R(\varepsilon,\theta_p)$ at a fixed finite $\varepsilon$ (for example $\varepsilon=1$, $\theta_p=0.1$) with increasing truncation $k=100,200,300,600$; if the values do not converge to within a few percent, the claimed mobility curve and the five-regime boundaries are not established. A complementary check would be to measure the rotational diffusivity of a micrometer-scale domain in a giant unilamellar vesicle with an independently known membrane viscosity and compare the inferred $\Lambda_R$ with Fig. 5.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a spherical membrane of radius $R_m$ containing a rigid spherical cap of half-angle $\theta_p$ rotating with angular velocity $\Omega$, the torque needed to maintain rotation is $G_p = 8\pi\mu R_m^3 c_1$, where $c_1$ is the rotlet (first Legendre) coefficient of the surrounding Stokes flow (Eq. 2.21). All higher-order multipoles reshape the flow but contribute nothing to the torque, so the dimensionless drag coefficient $\Lambda_R(\varepsilon,\theta_p)=\Omega^{-1}\theta_p^{-3}c_1$ is a function of only two parameters: $\varepsilon=2\mu R_p/\eta$ and $\theta_p=R_p/R_m$. In the small-particle limit the paper identifies five asymptotic regimes as $\ell_{\mathrm{SD}}=\eta/\mu$ grows: rigid rotation of the whole vesicle, a transition region, solvent-free two-dimensional disk drag, the planar Saffman–Delbrück regime, and finally three-dimensional disk drag when the membrane effectively disappears. The planar Saffman–Delbrück mobility and the exact rotational drag of a spherical cap in an unbounded fluid are recovered as special limits of the same calculation. The result is obtained by solving the coupled membrane–solvent Stokes equations semi-analytically with truncated Legendre expansions.

Load-bearing premise

The numerical truncated-Legendre solution is trusted at the finite membrane viscosities used in the results, even though the paper's stated 2% accuracy check (Section 3(a), with $k=100$–$300$ modes) was performed only in the limit $\eta\to0$; if the truncation error grows at finite $\eta$, the reported drag values and the inferred boundaries between the five regimes would shift.

Editorial extensions

If this is right

For a small inclusion in a vesicle, planar Saffman–Delbrück theory is valid only while $\ell_{\mathrm{SD}} \ll R_m^3/R_p^2$; outside this window the spherical geometry or bare solvent drag dominates and planar fits misestimate membrane viscosity.
Rotational diffusion measurements of large inclusions, where planar theory fails, can be converted into a membrane-viscosity estimate using the computed relation between $\Lambda_R$ and $\varepsilon$.
In the limit of vanishing membrane viscosity the spherical membrane imposes no constraint on purely azimuthal rotational flow, so it simply disappears and the drag tends to that of a rotating spherical cap in an unbounded fluid.
For very viscous membranes the whole vesicle rotates almost rigidly with the cap, producing a torque proportional to $R_m^3$ and a drag coefficient that scales as $\theta_p^{-3}$, a regime absent from flat-membrane theories.
The five-regime sequence is controlled by two dimensionless parameters, so a single plot such as Fig. 5 can be used to read off the expected rotational drag for any vesicle radius, inclusion size, and viscosity ratio.

Reading between the lines

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

A natural extension, which the paper leaves for future work, is the translational analogue: because purely azimuthal flow on a sphere is automatically divergence-free, the regular $\eta\to0$ limit found here is special to rotation, and translational mobility should still feel membrane incompressibility and curvature.
The unsteady version needed for rotational Brownian motion of liquid domains, where random molecular torques act on fast timescales, may not obey the same five-regime ordering because oscillatory flows introduce a new length scale into the membrane–solvent coupling.
Because the torque depends only on $c_1$, a measurement of rotational drag (or rotational diffusion) gives a direct handle on the rotlet amplitude, potentially allowing the prediction of the far-field solvent flow without reconstructing the full membrane velocity field.
The same Legendre machinery could be applied to asymmetric inclusions on a sphere by using vector spherical harmonics, though the loss of axisymmetry would replace the scalar ODE with a more complex coupled system.

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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

This is a real contribution, not a repackaging. The paper solves the rotational drag on a rigid spherical cap in a spherical membrane coupled to Stokes solvents, treating cap size and Saffman-Delbrück length as two independent parameters. The reduction of the torque to the rotlet coefficient c1 is clean (Eqs. 2.20-2.21), the governing equations are derived carefully from surface Stokes balances, and the known limits all check out: rigid-body rotation for large Saffman length, the ε^{-1} two-dimensional drag, the planar Hughes-Pailthorpe-White result, and the Dorrepaal spherical-cap torque in the solvent-dominated limit. The five-regime asymptotic map for small inclusions is the most useful output, and the analytic scalings in regimes (1), (3), and (5) do not depend on the numerics. That is solid, careful work.

The soft spots are proportionate. The stress-test concern is fair: the 2% convergence check is only reported for η→0, and the finite-ε truncation error is not assessed. The boundary-layer structure near the cap edge at intermediate ε suggests the k=300 truncation could in principle be less accurate there than in the solvent-only limit, so the quantitative crossover positions and prefactors in regions (2) and (4) rest on an unverified numerical assumption. I would not call this fatal, because the main physical conclusions would not change if those prefactors moved by a few percent, but the authors should either provide a convergence study at finite ε or release code so readers can check. Second, no code or data is included, which is a practical barrier for a semi-analytical method that others will want to use. Third, there is a page of unrelated text from another paper inserted into the arXiv PDF (the DNA/colloidosome material around section 1). That is plainly a production error and should be fixed before any publication, but it does not affect the science.

Who is this for? Anyone working on membrane hydrodynamics, especially experimentalists wanting to extract membrane viscosity from rotational diffusion of large inclusions. The paper gives them a concrete prediction and a clear warning about where planar Saffman-Delbrück theory fails. I would cite it if I were writing on curved-membrane mobility. It deserves a serious referee: the derivation is original, the asymptotics are informative, and the numerical concern is addressable. My recommendation is to send it to review, with the request that the authors add a finite-ε convergence check and share their code.

Referee Report

2 major / 7 minor

Summary. This paper develops a semi-analytical theory for the rotational mobility of a rigid spherical-cap inclusion embedded in a spherical incompressible membrane surrounded by Stokes solvents. The membrane and solvent flows are expanded in a Legendre-derived rotlet multipole basis, reducing the problem to a coupled system, Eqs. (2.16)-(2.19), for the expansion coefficients and the membrane angular velocity. The total torque is shown to be controlled solely by the rotlet coefficient c1, Eq. (2.21), so the dimensionless drag Lambda_R(epsilon, theta_p) depends only on the Saffman-Delbruck ratio epsilon=2 mu R_p/eta and the cap half-angle theta_p=R_p/R_m. The paper computes Lambda_R numerically for a range of parameters and identifies five asymptotic regimes for small inclusions, namely rigid whole-vesicle rotation, a crossover, two-dimensional membrane drag, the planar Saffman-Delbruck regime, and three-dimensional solvent-dominated drag, together with a large-inclusion limit that recovers the spherical-cap result of Dorrepaal.

Significance. If the numerical results are converged, this is a substantive contribution. It gives a parameter-free model for rotational drag on finite-size inclusions in spherical vesicles, includes curvature effects that are absent from planar Saffman-Delbruck theory, and offers a concrete route to measuring membrane viscosity from rotational diffusion of large inclusions. The clean reduction of the torque to the rotlet coefficient c1 and the analytic derivation of the asymptotic scalings are clear strengths, and the comparisons with the planar result of Ref. [30] and the spherical-cap result of Ref. [48] are appropriate external benchmarks. The main outstanding issue is numerical: the truncation accuracy is verified only in the eta->0 limit, so the quantitative values in Figs. 5 and 6 and the inferred crossover positions need confirmation before the quantitative claims can be accepted.

major comments (2)

[Section 3(a), Figs. 5 and 6] The stated truncation at k=100-300 modes is validated only in the eta->0 limit, where the spherical-cap mobility [48] is recovered with a relative error no larger than 2%; no convergence test or error estimate is reported for the finite values of epsilon plotted in Fig. 5A (epsilon=0.1-100) and Fig. 6A (epsilon=0.1, 1, 10). Since the quantitative positions of the crossovers in regions (2) and (4) and the values of Lambda_R in Figs. 5 and 6 are numerical, this is a load-bearing gap: if the Legendre truncation error grows at finite epsilon, the regime boundaries and the proposed viscosity-measurement method would shift. Please add a convergence study at representative finite epsilon, for example at theta_p=0.1 with epsilon=0.1, 1, and 10 and k=300, 600, and 1200, and report either the relative change in Lambda_R or an a posteriori residual of Eqs. (2.16)-(2.18).
[Section 3(b)(i), Fig. 6A] The five-regime picture for small inclusions is tested numerically only at theta_p=0.1. While this is a small cap, it is not asymptotically small for all purposes, and the planar-limit comparison in region (4) and the crossover between regions (3) and (5) would be considerably more convincing if a second, smaller value, such as theta_p=0.05, were shown or if the theta_p-dependence of Lambda_R were demonstrated to have converged. Without such evidence, the stated universal boundaries, which are obtained by scaling rather than by exact asymptotics, may not yet represent the theta_p->0 limit accurately.

minor comments (7)

[Section 1, after Fig. 1] The manuscript contains several paragraphs that appear to be reprinted from Refs. [6], [7], and [39], including passages titled 'Unbinding of DNA from Cationic Membranes', 'Coverage of the emulsified droplets by colloidal particles', and 'Diffusion of Liquid Domains in Lipid Bilayer Membranes'. These paragraphs do not belong to the paper's narrative and should be removed before submission; as printed, they disrupt the introduction and create a provenance concern.
[Appendix C, Eq. (A7)] Equation (A7) has a spurious trailing '=0' after the explicit non-zero traction sum; this is inconsistent with Eq. (2.17) and with the preceding line (A6), and should be removed.
[Section 3(b)(i) and Table 2] The region-(1) boundary condition is written inconsistently: the text gives epsilon << theta_p^3, which is equivalent to l_SD >> R_m^3/R_p^2, while Table 2 and the following sentence write l_SD >> R_m^3/R_p^3. The exponents should be harmonized, with R_m^3/R_p^2 being the version consistent with the text.
[Table 1 caption] The phrase 'rotlet 2n-pole' should presumably be 'rotlet 2^n-pole', since the n-th derivative of the rotlet produces a 2^n-pole singularity.
[Abstract and Section 2(a)] The phrase 'the solid angle formed by the particle' is used to mean the polar half-angle theta_p; consider using 'solid angle' only for the actual quantity 2*pi*(1-cos(theta_p)), and referring to theta_p as the cap half-angle.
[References] Several reference entries have questionable or mismatched DOIs; for example, Ref. [14] gives a DOI from a Biophysical Journal article that does not appear to correspond to Brochard and Lennon 1975. The reference list should be checked systematically.
[Section 3(a)] The sentence 'These values of k ensure that, in the limit eta->0, we recover the mobility of a rotating spherical cap with a relative error no larger than 2% (see § ii)' points to Section 3(b)(ii), which is the large-particle asymptotic section, not the truncation check; the pointer should be to Eq. (3.7) and Fig. 6C, where the spherical-cap comparison is made.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central mobility result is derived from the Stokes equations with no fitted parameters, and the cited planar and spherical-cap results are external benchmarks used for validation.

full rationale

The paper's central claim, ΛR(ε, θp) = Ω^{-1}θ_p^{-3} c1 with c1 obtained from the coupled system (2.16)-(2.18), is derived from the governing Stokes equations and membrane force balance, not imposed or fitted. The torque expression Gp = 8πµR_m^3 c1 (Eq. 2.21) follows by direct integration and orthogonality of the Legendre basis. The asymptotic regimes in Section 3(b) are obtained analytically from the same equations via boundary-layer scalings (Eqs. 3.2-3.6), and the comparisons with planar Saffman-Delbrück results [30] and the spherical-cap solution [48] are external validations used only in the limits η→0, small ε, or small θp. The only self-citation is ref. [47] for the Legendre/squirmer expansion form, which is a standard mathematical basis and not load-bearing for the physical result. The numerical truncation check is performed in the η→0 limit, but that is a convergence/correctness concern, not circularity. No fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation. The derivation is self-contained and the circularity burden is minimal.

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

No free parameters are fitted to data; ε and θ_p are rescalings of physical inputs (η, µ, Rm, Rp), not adjustable numbers. The central derivation relies on the standard surface-Stokes model of membranes and on the adequacy of the truncated Legendre expansion. No new physical entities are introduced.

assumptions (5)

domain assumption Membrane is a two-dimensional incompressible, impermeable, Newtonian fluid with purely tangential velocity and constant surface viscosity η.
Invoked in Section 2(b), Eqs. (2.1)-(2.4); this is the standard model but neglects normal deformation, bending rigidity, and surface-tension gradients.
domain assumption Solvents are Newtonian Stokes fluids of equal viscosity µ on both sides, with no-slip at the membrane.
Eqs. (2.2), (2.5) and the single-coefficient Legendre expansion (2.12)-(2.14) require equal viscosities; unequal viscosities would need two coefficient sets.
standard math Stokes flow with negligible inertia in membrane and solvents, so force balance is linear and pressure gradients from rotation vanish by symmetry.
Stated in Section 2(b) and Appendix A; this is the low-Reynolds-number limit.
ad hoc to paper Truncation of the Legendre expansion at k=100-300 modes gives accurate results for the plotted parameters, with the stated 2% check only in the η→0 limit.
Section 3(a); no convergence or error analysis is given for finite η, which is a gap for the central numerical results.
domain assumption Particle is a rigid spherical cap, flush with the membrane, with no-slip angular velocity Ω imposed on the cap.
Eq. (2.18); real proteins may protrude or slip, which would change the boundary condition.

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Pith. "Pith review of Rotational mobility in spherical membranes: The interplay between Saffman-Delbr\"uck length and inclusion size." pith.science (2026). https://pith.science/paper/PHRQDHHF

@misc{pith2026250105367,
  author       = {Pith},
  title        = {Pith review of: Rotational mobility in spherical membranes: The interplay between Saffman-Delbr\"uck length and inclusion size},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHRQDHHF}},
  note         = {Machine review of arXiv:2501.05367}
}

read the original abstract

The mobility of particles in fluid membranes is a fundamental aspect of many biological processes. In a 1975 paper [1], Saffman and Delbr\"uck demonstrated how the presence of external Stokesian solvents is crucial in regularising the apparently singular flow within an infinite flat membrane. In the present paper, we extend this classical work and compute the rotational mobility of a rigid finite-sized particle located inside a spherical membrane embedded in Stokesian solvents. Treating the particle as a spherical cap, we solve for the flow semi-analytically as a function of the Saffman-Delbr\"uck (SD) length (ratio of membrane to solvent viscosity) and the solid angle formed by the particle. We study the dependence of the mobility and flow on inclusion size and SD length, recovering the flat-space mobility as a special case. Our results will be applicable to a range of biological problems including rotational Brownian motion, the dynamics of lipid rafts, and the motion of aquaporin channels in response to water flow. Our method will provide a novel way of measuring a membrane's viscosity from the rotational diffusion of large inclusions, for which the commonly used planar Saffman-Delbr\"uck theory does not apply.

Figures

Figures reproduced from arXiv: 2501.05367 by the authors.

**Figure 1.** Fluorescence microscopy phase diagram of DOPC/DPPC/ cholesterol and corresponding vesicle images at 20 °C. Semiquantitative dashed tie lines cross the LR -Lo coexistence region.16 Some vesicles studied have a continuous LR (bright) phase (a,b), whereas others have acontinuousLo (dark) phase (c,d). One composition (e) has a continuous dark Lo phase which may contain both Lo and gel (So) phase lipids.17 Vesicle compos… view at source ↗

**Figure 8.** Semidilute solution of λ-DNA confined to a lipid membra 7. (a-c) Micrographs taken in 30 min time intervals. The arrows in diidf 1 h(d) Iilif Figure 1. sm stu ext [PITH_FULL_IMAGE:figures/full_fig_p003_8.png] view at source ↗

**Figure 2.** (A): Schematic of the experimental system under consideration, the rotation of a particle (orange structure) located inside a curved lipid bi-layer membrane (dark blue) that is embedded in a viscous solvent on both sides (light blue). (B): schematic depiction of the mathematical problem. The rigid particle is modelled as a spherical cap of halfangle θ = θp and curvilinear radius Rp inside a spherical membrane (vesi… view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: Flows associated with (A) a rotlet, (B) a rotlet dipole, (C) a rotlet quadrupole located at the origin and oriented long the z axis (dashed white line). In each case, the flow rotates around the z axis, so a single slice is plotted corresponding the the xz plane. The b…

**Figure 4.** Figure 4: Flows within the membrane and the solvents for different inclusion sizes and relative Saffman lengths ε = 2Rp/ℓSD. (A) Normalised flow in the membrane for ε = 0.05, θp = 0.4 (blue line) and its relative solid-body motion component (purple line). Inset: 3D plot of the m…

**Figure 5.** Figure 5: Dependence of the exerted torque on the particle size and the membrane viscosity: (A) torque Gp exerted on the particle to maintain rotation (non-dimensionalised by the torque Gm on a rigid sphere of radius Rm rotating with the same angular velocity in a solvent with v…

**Figure 6.** Figure 6: Asymptotic behaviour of rotational drag coefficient, ΛR(ε, θp), in the limits of small and large particles. A: Small particle limit, Rp ≪ Rm (or θp ≪ 1) for many values of ε = 2Rp/ℓSD (numerics run with θp = 0.1, k = 300), with details in § i. The numerical values of Λ…

**Figure 7.** Figure 7: Force balance sketch: the membrane viscous forces exerted on the boundary of the area patch A with unit normal n, boundary ∂A and boundary unit normal p should balance the viscous forces generated by the solvents. Exploiting the arbitrariness of A yields the membrane S…

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