Amoeboid propulsion of active solid bodies, vesicles and droplets: a comparison

Annette Zippelius; Reiner Kree

REVIEW 2 major objections 4 minor 18 references

Amoeboid propulsion of active solid bodies, vesicles and droplets: a comparison

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Inside a unified low-Reynolds perturbation theory, a droplet is the fastest shape-changing swimmer, a vesicle second, and a deformable solid third, but efficiency favors the vesicle above a viscosity ratio near 1.35.

desk verdict Careful unified perturbation theory with genuine new droplet results; the droplet-first race ranking is real for prescribed strokes, but physical drivability of the optimal strokes is left open. read the letter →

arxiv 2502.08420 v1 pith:OLZENRTR submitted 2025-02-12 cond-mat.soft physics.flu-dyn

Reiner Kree , Annette Zippelius This is my paper · ORCID

classification cond-mat.softphysics.flu-dyn MSC 76D0776Z1076M45 PACS 47.15.G47.63.mf

keywords amoeboidswimmingStokesflowmicroswimmervesicledropletLighthillefficiencyswimstrokelowReynoldsnumber

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 builds one perturbation scheme for three near-spherical “amoeboid” swimmers—a deformable solid, a vesicle with an incompressible fluid membrane, and a fluid droplet—all driven by periodic, axially symmetric, achiral radial shape changes, and compares their swimming speeds and Lighthill efficiencies at small deformation amplitudes. The central result is a race ranking: when each swimmer is allowed to pick its speed-maximizing stroke, the droplet always wins, the vesicle comes second, and the solid body finishes third, for every value of the viscosity ratio $\lambda$. Efficiency tells a different story: the droplet is more efficient than the vesicle only when internal dissipation is small, with the maximum efficiencies crossing near $\lambda \approx 1.35$, and the solid is typically two orders of magnitude less efficient. The paper also finds that speed-optimal and efficiency-optimal strokes are not the same, and that if all three swimmers execute the same stroke their ranking can change, with some strokes even propelling different swimmers in opposite directions.

What carries the argument

The central machinery is a second-order perturbation expansion about a reference sphere, using vector spherical harmonics to solve the Stokes equations for the outer and, for the droplet and vesicle, inner flows. The swimmer type enters only through boundary conditions: no-slip for the solid, surface incompressibility plus the kinematic condition for the vesicle, and continuity of velocity with tangential-stress balance for the droplet. The coefficients $R_l$ and $S_l$ in the velocity formula Eq. (27) carry the type-specific content, and the two-parameter manifold of harmonic strokes $(F_2, \alpha_2)$ with $l=2,3,4$, constrained by constant volume and constant surface area, lets the paper map speed and Lighthill efficiency over all admissible strokes. Lighthill efficiency is the ratio of the power needed to tow a rigid sphere at the swimmer’s mean speed to the power actually dissipated by the stroke.

What would settle it

Compute the active normal tractions needed to execute the speed-optimal droplet stroke from the normal-traction balance that the paper sets aside in Section 2.3; if the required traction field is negative, singular, or incompatible with a constant-surface-tension interface, the droplet-first ordering is not physically realizable. A complementary experiment would impose identical harmonic strokes on a vesicle and a droplet and compare their mean speeds—the ranking and the crossing behavior near the zero-velocity lines would settle the claim directly.

Watch

Extended reading notes

Core claim

The paper’s central claim is that, within a second-order expansion in small deformation amplitudes, the mean swimming speed of each swimmer type takes the form $U = \sum_{l\ge 2} (R_l f_l \dot f_{l+1} + S_l f_{l+1}\dot f_l)$, with $R_l$ and $S_l$ determined entirely by the boundary conditions that define the swimmer type. For solids and vesicles these coefficients are independent of the viscosity ratio $\lambda$; for droplets they are rational functions of $\lambda$. On the two-parameter manifold of area-preserving strokes built from spherical harmonics $l=2,3,4$, the maximum attainable average speed obeys droplet $>$ vesicle $>$ solid for all $\lambda$. The maximum Lighthill efficiency of the droplet exceeds that of the vesicle only for small internal viscosity, crossing near $\lambda \approx 1.35$; above that, the vesicle is the most efficient despite being slower. The paper states these orderings within the stated model class—radial, axially symmetric, achiral strokes with volume and surface constraints—and shows that the same stroke can yield different speeds and even different directions for the three swimmers.

Load-bearing premise

The load-bearing premise is that the prescribed shape histories are physically realizable by active driving mechanisms that are not part of the model; if the optimal strokes require negative, singular, or otherwise unattainable active tractions, the droplet-first ranking could collapse.

Editorial extensions

If this is right

If the paper is correct, an amoeboid swimmer that can choose its stroke will always be fastest as a droplet, next fastest as a vesicle, and slowest as a deformable solid, no matter the viscosity contrast $\lambda$.
The efficiency ranking is not the same as the speed ranking: above $\lambda \approx 1.35$ the vesicle reaches a higher maximum Lighthill efficiency than the droplet even though it is slower.
Speed-optimal and efficiency-optimal strokes differ, so a swimmer optimized for velocity will not be the same as one optimized for energy cost.
Executing the identical stroke on all three swimmers does not preserve the speed ranking; pairs can cross and even reverse direction as stroke parameters vary.
Because the solid body dissipates only in the ambient fluid and still has much lower efficiency, internal fluidity appears necessary for efficient amoeboid propulsion, not merely for speed.

Reading between the lines

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

A designer of synthetic swimmers could treat interior viscosity as a control knob: lower $\lambda$ buys droplet speed, but the efficiency advantage disappears beyond $\lambda \approx 1.35$, so the optimal interior viscosity depends on whether speed or energy cost is the objective.
The ranking is established only for radial, axially symmetric strokes with harmonics $l=2,3,4$; extending the comparison to non-radial or fully three-dimensional strokes, or to strokes involving $l=1$, is the most direct way to test whether the droplet-first ordering is generic.
The zero-velocity lines in the stroke manifold are testable: near them, a vesicle and a droplet driven by the same stroke should show sharply different speeds and opposite directions, a signature an experiment with shape-controlled drops and vesicles could look for.
For biological cells, the results suggest that a cell whose interior behaves more like a low-viscosity fluid may swim faster; measuring swimming speed against intracellular fluidity would be a natural application, though the paper does not make this claim.

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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

Dear colleague,

This one is worth engaging. Kree and Zippelius do the old-school thing—second-order Stokes perturbation theory for solid, vesicle, and droplet swimmers in one notation—and the payoff is real. The new pieces are the droplet velocity and dissipation formulas, the side-by-side comparison over an area-constrained two-dimensional stroke manifold, and the resulting ranking. The solid-body results reproduce Lighthill and Blake; the vesicle result matches Farutin et al. up to a total time derivative. I found no fitting-to-target and no circular argument. The appendices give enough detail that the algebra is checkable, even though the droplet polynomials in Appendix D are long enough that I would not want to verify them by hand.

The soft spots are real but not disqualifying. The race ranking is computed for prescribed shape histories, not for strokes generated by physically plausible active tractions. Section 2.3 explicitly says the normal traction balance is not needed to determine the flow and defers the calculation of active tractions, but that calculation never appears. So “droplet always first, vesicle second, solid third” is established for the kinematic model, not for a real vesicle or droplet whose membrane or interface must actually produce those deformations. The abstract states the ranking without that qualifier, and I think a referee should push the authors to either add the traction check or soften the claim. This is a scope limitation, not an algebraic error.

Two smaller things. Equation (7) is dimensionally inconsistent as printed: surface divergence is equated to a viscous traction, which does not match units, though the surrounding text and the first-order conditions make the intended constraint clear. Also, there is no code or independent numerical check for the long expressions; that is not a fatal flaw, but it makes independent verification slower than it should be.

If I were the editor, I would send this to peer review. The central argument holds up within its stated model class, the droplet results are new, and the comparison is useful to people designing or modeling synthetic amoeboid swimmers. The requested revision would be to qualify the ranking claim and, if feasible, report the active tractions required for the optimal strokes. That is a meaningful revision, not a rejection.

Best,

[You]

Referee Report

2 major / 4 minor

Summary. This paper presents a unified second-order perturbation theory, in the Stokes approximation, for three near-spherical, axially symmetric amoeboid swimmers driven by prescribed radial deformations: a solid deformable body with no-slip boundary conditions, a vesicle with an incompressible fluid membrane, and a Newtonian droplet with a sharp interface and viscosity contrast λ. The authors derive closed-form coefficients R_l and S_l for the swimming velocity (Eq. 27), dissipation formulas, and Lighthill efficiencies. They benchmark against Lighthill/Blake for solids and against the known l=2 vesicle result, then compare velocities and efficiencies over a two-parameter manifold of area-conserving harmonic strokes with l=2,3,4 (Appendix F). The main conclusions are: when each swimmer optimizes its speed on this manifold, the droplet is fastest, the vesicle second, and the solid third for all λ; the droplet's maximum Lighthill efficiency exceeds the vesicle's only for λ below about 1.35; and speed-optimal strokes differ from efficiency-optimal strokes.

Significance. If the results are taken with the qualifiers of the model, this is a useful contribution: it unifies three swimmer types in one Stokes boundary-value framework, reproduces known benchmark results, provides new explicit algebraic results for vesicles and droplets, and gives a clean graphical comparison over a well-defined stroke manifold. The analytic formulas for R_l, S_l and the dissipation coefficients, together with the benchmark checks, make the core calculation credible. The main qualification is that the headline ranking is kinematic: it has not been shown that the optimal strokes are realizable by active physical mechanisms.

major comments (2)

[Secs. 2, 2.3, 5.4; Abstract] The race ranking stated in the Abstract and Sec. 6 ('the droplet always comes in first...') is established only for prescribed shape histories. In Sec. 2 the authors write 'we consider the time-dependent deformations as given and compute the corresponding propulsion,' and in Sec. 2.3 the normal traction balance for the droplet is explicitly deferred ('not needed to determine the flow at given fl(t), but it can be used to calculate the active tractions ... if necessary'). That calculation is never reported. Consequently the optimal strokes of Sec. 5.4 are not checked for physical realizability: one does not know whether the required active tractions are non-singular, of admissible sign, or compatible with the tangential-stress condition assumed for a Marangoni-type droplet (or with membrane tension for the vesicle). Since the abstract presents the ranking without this qualifier, the manuscript should either provide the active-traction computation for the optimal strokes or restate the ranking as a property of the prescribed-deformation kinematic model.
[Sec. 5.4 and Appendix F] The 'always' ranking is also restricted to the specific two-parameter manifold of time-harmonic, area-conserving strokes composed of l=2,3,4 harmonics (Appendix F). The abstract and Sec. 6 state the ranking without this restriction, and no argument is given that the ordering persists for other l, for non-harmonic time courses, or for non-radial three-dimensional deformations. Please either add these qualifiers to all summary statements or provide evidence or proof that the ranking is generic within the broader model class.

minor comments (4)

[Sec. 2.2, Eq. (7)] Equation (7) writes '∇s·v = t_vis·n = 0', which is dimensionally inconsistent and appears to conflate two separate statements (surface incompressibility and a viscous-traction condition). The surface-divergence condition ∇s·v=0 is the one actually used in the first-order solution; please correct or remove the erroneous equality and state the traction condition, if any, with the proper factor of η.
[Introduction and Sec. 5.5] There are several typos: 'Dicties' should be 'Dictyostelium'; 'e.t.c.' should be 'etc.'; reference markers 40-42 appear as plain numbers in the text; and 'as shown in In Fig. 10' has a duplicated preposition.
[Appendix D, Eq. (78)] In the linear system for the droplet, the term '−8η +a − 4η −d = Iσ' has lost the labels distinguishing the internal and external viscosities; please restore the η+ and η− notation (or equivalent) so that Eq. (82) can be checked by a reader.
[Fig. 7 and Sec. 5.4] The caption of Fig. 7 asserts a ranking 'for all values of λ' but does not state the λ range over which the curves were evaluated; please specify the range and, if the statement is exact, provide the underlying inequalities or clearly state that it is a numerical observation over the plotted range.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivations are self-contained second-order Stokes calculations; the droplet-first ranking is not circular, though it is conditional on prescribed deformations.

full rationale

The paper's central quantities—swimming velocities, dissipation, and Lighthill efficiencies—are computed by explicit perturbative solution of the Stokes boundary-value problem. The velocity coefficients for the solid and vesicle appear as closed-form expressions (Eqs. 28–29 and 36–37), and the droplet coefficients are given as explicit rational functions of the viscosity contrast λ (Eqs. 44–45 with Appendix D). The ranking in Sec. 5.4 is obtained by optimizing these independently derived expressions over a fully parameterized two-parameter stroke manifold; no parameter is fitted to reproduce the ranking or any target velocity. The calculation is benchmarked against external results: Lighthill and Blake for the solid (Sec. 3.1) and Farutin et al. (2013) for the vesicle (Secs. 5.1 and 3.2). Self-citations (refs. 20–22 and 33) supply a vector-spherical-harmonic solution basis and examples of active drives, but the basis is written out in Eqs. (14)–(15) and is a standard Stokes solution expansion; removing those citations would not alter the derivation. The only flagged limitation is Sec. 2.3: "The balance of normal tractions is not needed to determine the flow at given fl(t), but it can be used to calculate the active tractions responsible for the given deformations, if necessary." This means the race ranking is a statement about prescribed shape kinematics, not a demonstration that the optimal strokes are physically drivable by particular active mechanisms. That is a scope limitation, not circularity: the stated model class is precisely 'given deformations,' and the ranking is derived, not assumed. No step reduces Eq. (27) to its inputs by construction, and no load-bearing claim rests on a self-citation chain. Hence score 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No fitted parameters appear in the derivation: stroke amplitudes and phases are input degrees of freedom that are scanned to locate maxima, and lambda is a physical viscosity contrast. The main assumptions are the Stokes/quasi-static approximation, the choice to prescribe deformations rather than derive them from active forces, the restriction to radial axially symmetric strokes, the minimal interface models, and the neglect of internal dissipation in the solid. None of these introduces a new entity or a constant tuned to reproduce the claimed ranking.

assumptions (8)

domain assumption Quasi-steady Stokes flow with negligible inertia; product of Reynolds and Strouhal numbers is small (Sec. 2, before Eq. 1).
Standard for low-Reynolds-number microswimmers; justifies neglecting unsteady inertia in the outer and inner fluids.
ad hoc to paper Deformations are prescribed inputs, not coupled to active force generation (Sec. 2: 'we consider the time-dependent deformations as given').
The central comparison is kinematic; the realizability of the optimal strokes by physically plausible active tractions is not verified.
ad hoc to paper Restriction to radial, axially symmetric, achiral deformations f(theta, t) about a fixed center of deformation (Eqs. 2-3).
Excludes tangential, non-axisymmetric, and rotational deformations; the race ranking is proven only within this class.
standard math Second-order perturbation expansion in deformation amplitudes f and time derivatives f_dot, with both treated as same-order small quantities (Sec. 3).
Valid for small-amplitude, slow strokes; terms of order epsilon cubed and higher are neglected.
domain assumption Vesicle membrane is locally inextensible with no bending, shear, or membrane viscosity (Sec. 2.2).
The surface-area constraint, Eq. (13), is the only membrane physics; richer membrane rheology is ignored.
domain assumption Droplet interface is a sharp surface with constant surface tension and no Marangoni stresses (Sec. 2.3).
Standard minimal droplet model; tangential stress continuity, Eq. (11), is used, but normal traction balance is not enforced.
domain assumption Solid has no internal dissipation; outer-flow dissipation is a lower bound on total dissipation (Sec. 2.1 and Sec. 4).
The solid Lighthill efficiency is an upper bound; including internal dissipation would only lower it.
domain assumption Interior and ambient fluids have equal mass density and viscosity contrast lambda (Sec. 2).
Neutral buoyancy is assumed; lambda is a physical input parameter rather than a fitted constant.

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Cite this review

Pith. "Pith review of Amoeboid propulsion of active solid bodies, vesicles and droplets: a comparison." pith.science (2026). https://pith.science/paper/OLZENRTR

@misc{pith2026250208420,
  author       = {Pith},
  title        = {Pith review of: Amoeboid propulsion of active solid bodies, vesicles and droplets: a comparison},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLZENRTR}},
  note         = {Machine review of arXiv:2502.08420}
}

abstract

We present a unified discussion of three types of near-spherical amoeboid microswimmers, driven by periodic, axially symmetric, achiral deformations (swim strokes): a solid deformable body, a vesicle with incompressible fluid membrane, and a droplet. Minimal models are used, which characterize the swimmer type only by boundary conditions. We calculate the swimming velocities, the dissipated power and the Lighthill efficiencies within a second order perturbation expansion in the small deformation amplitudes. %Our approach uses spherical harmonics to represent surface deformations and a system of general solutions of the Stokes equation based on vector spherical harmonics. For solid bodies, we reproduce older results by Lighthill and Blake, for vesicles and for droplets we add new results. The unified approach allows for a detailed comparison between the three types of microswimmers. We present such comparisons for swim strokes made up of spherical harmonics of adjacent orders $l$ and $l+1$, as well as for a manifold of swim strokes, made up of spherical harmonics up to order $l=4$, which respect volume- and surface-incompressibility. This manifold is two-dimensional, which allows to present swimming velocities and efficiencies in a compact graphical form. In a race in which each swimmer can choose the stroke that maximizes its speed, the droplet always comes in first, the vesicle comes in second, while the particle finishes third. However, if the three swimmers perform the same stroke, other order of rankings become possible. The maximum of the total efficiency of a droplet is greater than that of a vesicle if the internal dissipation is small. The efficiency of the solid body turns out to be typically two orders of magnitude smaller than that of vesicles and droplets. Optimizing the Lighthill efficiency and optimizing the swimming velocity result in different optimal swim strokes

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Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

1 E. K. Paluch, I. M. Aspalter and M. Sixt, Annu. Rev. Cell Dev. Biol., 2016, 32,

work page 2016
[3]

Abercrombie, J

14 | 1–15 + P V S O B M / B N F < Z F B S > < W P M > 3 M. Abercrombie, J. E. Heaysman and S. M. Pegrum, Exp. Cell Res., 1970, 62,

work page 1970
[6]

Kree and A

21 R. Kree and A. Zippelius, Eur. Phys. J. E, 2022, 45,

work page 2022
[15]

22 R. Kree, L. Rückert and A. Zippelius, Phys. Rev. Fluid, 2021, 6, 034201. 23 A. R. Sprenger, V. A. Shaik, A. M. Ardekani, M. Lisicki, A. J. T. M. Mathijssen, F. Guzmán-Lastra, A. M. M. Hart- mut Löwen1 and A. Daddi-Moussa-Ider, Eur. Phys. J. E, 2020, 43,

work page 2021
[58]

Taylor, Proc

24 G. Taylor, Proc. R. Soc. Lond., 1951, A209,

work page 1951
[94]

29 B. U. Felderhof and R. B. Jones, Physica, 1994, A202,

work page 1994
[109]

6 J. R. Blake, J. Fluid Mech., 1971, 46,

work page 1971
[119]

Shapere and F

30 A. Shapere and F. Wilczek, J. Fluid Mech., 1989, 198,

work page 1989

Show all 18 references

[124]

Mateos-Maroto, F

16 A. Mateos-Maroto, F. Ortega, R. G. Rubio, J.-F. Berret and F. Martínez-Pedrero, Particle, 2019, 36, 1900239. 17 C. Abaurrea-Velasco, T. Auth and G. Gompper, New J. Phys , 2019, 21, 123024. 18 H. R. Vutukuri, M. Hoore, C. Abaurrea-Velasco, L. van Buren, A. Dutto, T. Auth, D....

2019
[199]

Ebata, Y

7 H. Ebata, Y. Nishigami, H. Fujiwara, S. Kidoaki and M. Ichikawa, Amoeboid movement utilizes the shape coupled bifurcation of an active droplet to boost ballistic motion , 2024, https://arxiv.org/abs/2402.03759. 8 N. P. Barry and M. S. Bretscher, PNAS, 2010, 107, 11376. 9 A. ...

2024 arXiv
[389]

Bergert, A

4 M. Bergert, A. Erzberger, R. A. Desai, I. M. Aspalter, A. C. Oates, G. Charras, G. Salbreux and E. K. Paluch,Nat. cell Biol., 2015, 17,

2015
[447]

Lauga and T

25 E. Lauga and T. Powers, Rep. Prog. Phys., 2009, 72, 096601. 26 O. S. Pak and E. Lauga, J Eng Math, 2014, 88, 1–28. 27 A. Shapere and F. Wilczek, J. Fluid Mech., 1989, 198,

2009
[469]

2 H. G. Othmer, Wiley Interdiscip Rev Comput Mol Sci., 2019, 9,

2019
[524]

5 M. J. Lighthill, Comm. Pure Appl. Math., 1952, 5,

1952
[557]

28 B. U. Felderhof and R. B. Jones, Physica, 1994, A202,

1994
[587]

31 B. U. Felderhof and R. B. Jones, Phys. Rev. , 2014, E90, 023008. 32 Q. Wang and H. G. Othmer, J. Math. Biol., 2018, 76,

2014
[1699]

Kree and A

33 R. Kree and A. Zippelius, Phys. of Fluids, 2023, 35, 042002. 34 A. Farutin, S. Rafai, D. K. Dysthe, A. P. Duperrey , P. Peyla and C. Misbah, Phys. Rev. Lett., 2013, 111, 228102. 35 M. Morozov and S. Michelin, Journal of Fluid Mechanics , 2018, 860, 711–738. 36 N. Yoshinaga,...

2023
[1893]

40 G. I. Taylor, Proc. R. Soc. Lond. A, 1934, 146, 501–523. 41 U. Seifert, Eur. Phys. J B, 1999, 8, 405–415. 42 J. M. Rallison, J. Fluid Mech, 1980, 98, 625–633. 43 S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Dover Publ.,

1934

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

1 E. K. Paluch, I. M. Aspalter and M. Sixt, Annu. Rev. Cell Dev. Biol., 2016, 32,

work page 2016

[2] [3]

Abercrombie, J

14 | 1–15 + P V S O B M / B N F < Z F B S > < W P M > 3 M. Abercrombie, J. E. Heaysman and S. M. Pegrum, Exp. Cell Res., 1970, 62,

work page 1970

[3] [6]

Kree and A

21 R. Kree and A. Zippelius, Eur. Phys. J. E, 2022, 45,

work page 2022

[4] [15]

22 R. Kree, L. Rückert and A. Zippelius, Phys. Rev. Fluid, 2021, 6, 034201. 23 A. R. Sprenger, V. A. Shaik, A. M. Ardekani, M. Lisicki, A. J. T. M. Mathijssen, F. Guzmán-Lastra, A. M. M. Hart- mut Löwen1 and A. Daddi-Moussa-Ider, Eur. Phys. J. E, 2020, 43,

work page 2021

[5] [58]

Taylor, Proc

24 G. Taylor, Proc. R. Soc. Lond., 1951, A209,

work page 1951

[6] [94]

29 B. U. Felderhof and R. B. Jones, Physica, 1994, A202,

work page 1994

[7] [109]

6 J. R. Blake, J. Fluid Mech., 1971, 46,

work page 1971

[8] [119]

Shapere and F

30 A. Shapere and F. Wilczek, J. Fluid Mech., 1989, 198,

work page 1989