Hydrodynamics-driven phase-locking and collective motility of sessile active dumbbells

Mohd Suhail Rizvi; Urvi Mahendra Bora

REVIEW 4 major objections 6 minor 12 references

Hydrodynamics-driven phase-locking and collective motility of sessile active dumbbells

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Hydrodynamic interactions alone can make a suspension of non-motile, oscillating dumbbells move collectively, with a sharp transition near packing fraction 0.4.

desk verdict Plausible simulation-level mechanism for collective motion in non-motile oscillating dumbbells, but the hydrodynamics-centric claim needs a control without hydrodynamic coupling. read the letter →

arxiv 2501.08065 v1 pith:GFBZIKSH submitted 2025-01-14 cond-mat.soft physics.bio-ph

Urvi Mahendra Bora , Mohd Suhail Rizvi This is my paper · ORCID

classification cond-mat.softphysics.bio-ph

keywords collectivemotilityactivematterhydrodynamicinteractionsphase-lockingsessiledumbbellslimit-cycleoscillationsStokesflowphaseseparation

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

The paper tries to establish that hydrodynamic interactions alone can turn a collection of non-motile, oscillating objects into a collectively moving suspension. It models each cell as a two-bead dumbbell whose myosin-driven contractions produce limit-cycle oscillations in length; by the scallop theorem such a dumbbell cannot swim on its own. In Stokes-flow simulations of 64 dumbbells arranged in a one-dimensional periodic array, the paper finds that above a suspension density of about $\Phi=0.4$ the center of mass switches from diffusive to ballistic motion, the suspension phase-separates into clusters, and the oscillations phase-lock. If true, this provides a mechanism for collective cell motility that requires no intrinsic polarity, adhesion, or alignment rule.

What carries the argument

The central object is the active bead-spring dumbbell: two identical beads connected by a nonlinear spring, with myosin-driven contractility represented by a bound-myosin concentration whose dynamics includes turnover and a conservation term; this system undergoes a Hopf bifurcation to limit-cycle oscillations in length and myosin concentration. The second ingredient is the Oseen tensor, the Green's function of Stokes flow that gives the velocity at one bead caused by a force on another bead. The argument is carried by the phase variable $\psi$ describing each dumbbell's oscillation: pair-level expressions for the cycle-averaged translational velocities show that collective and relative velocities depend on the phase difference $\theta$, giving phase-dependent attraction and repulsion as long as the phases differ. The simulations then show that hydrodynamic interactions feed back into myosin dynamics and produce phase-locking at high density, with finite phase differences among locked dumbbells.

What would settle it

Run the same dumbbell dynamics in two or three dimensions, with larger $N$ and varied spring or myosin parameters, and check whether the center-of-mass exponent $\alpha$ still switches from $\approx1$ to $\approx2$ near $\Phi\approx0.4$ and whether phase-locking still appears; if the ballistic transition disappears or the velocity scaling departs strongly from $\Phi^4$, the central claim fails. An experimental equivalent would be to suspend non-adherent oscillating cells (or synthetic oscillatory colloids) at increasing density in a viscous fluid and look for synchronized shape oscillations and a mean drift.

Watch

Extended reading notes

Core claim

The central claim is that hydrodynamic coupling between sessile active dumbbells produces density-dependent collective dynamics: at low densities the dumbbells oscillate independently and the system's center of mass diffuses, while above $\Phi\approx0.4$ the dumbbells lock their oscillation phases, form hydrodynamic clusters, and the center of mass moves ballistically with velocity scaling roughly as $v_s\sim\Phi^4$. A pair-level analytical argument shows that two dumbbells with a phase difference $\theta$ acquire a cycle-averaged collective velocity and a phase-dependent effective attraction or repulsion, so the same coupling that creates clusters also drives migration once the phases are coordinated. The simulations show that this coordination is absent below $\Phi\approx0.4$, becomes bimodal at intermediate densities (locked or unlocked states), and approaches near-perfect synchronization at $\Phi\approx0.8$–$0.9$. The paper concludes that collective motion of a purely sessile population is possible without polarity or adhesion.

Load-bearing premise

The load-bearing premise is that a strictly one-dimensional periodic array of 64 identical dumbbells with a single set of model parameters represents real suspensions of oscillating cells; if the phase-locking and ballistic motion require this 1D ordering or those specific parameters, the general claim fails.

Editorial extensions

If this is right

Above $\Phi\approx0.4$, the suspension's center of mass crosses from diffusive ($\alpha\approx1$) to ballistic ($\alpha\approx2$) motion, and the collective velocity grows roughly as $\Phi^4$.
Hydrodynamic interactions drive phase separation: at $\Phi\approx0.5$–$0.8$ the homogeneous starting state breaks into several clusters, and near $\Phi\approx0.9$ the dumbbells form a single spanning cluster.
Phase-locking is density-dependent and incomplete: at intermediate densities dumbbells lock at finite phase differences, and at high densities both in-phase and anti-phase locked states are reached depending on initial conditions; the finite phase differences are what generate collective motion.
Translated to cell scales, the collective speed is about $10^{-1}\ \mu$m/min, comparable to slow mesenchymal cell motility, but without polarity or adhesion.

Reading between the lines

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

The pair-level argument implies a testable design rule the paper does not state: a collection of reciprocal oscillators will migrate whenever their phases lock at a nonzero average difference, so tuning density (and thus coupling strength) is a control knob for collective transport without adding polarity.
Because the paper simulates only one-dimensional arrays, whether the transition survives in two or three dimensions is open; if it does, the dumbbell axes should also develop orientational order through hydrodynamic torques, making a 2D or 3D simulation a natural test of the mechanism.
The bimodal mutual-information distributions at intermediate densities suggest bistability between locked and unlocked states; a direct experimental signature would be hysteresis in the phase-correlation as density is swept up and down.
The $\Phi^4$ scaling is measured only for the specific model parameters and the 64-particle system; checking whether the exponent is robust to changes in $N$, $\tau$, and bead size would separate a generic hydrodynamic mechanism from a parameter-dependent coincidence.

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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

Quick take: this is a plausible simulation study showing three density-dependent phenomena in a suspension of oscillating non-motile dumbbells. The new piece is the suspension-level behavior, not the single-agent dynamics. The mechanism claim, however, is under-supported because there is no control run with hydrodynamic coupling switched off.

The good: The model is well grounded in existing bead-spring and Dierkes-type oscillator work. The three phenomena — diffusive-to-ballistic transition, phase separation, and phase-locking — are clearly demonstrated with reasonable diagnostics. The paper is honest about the 1D setup and the need for extensions. The citation pattern looks fair; they cite earlier work on bead-spring swimmers and synchronization.

The soft spots, in order. First, the missing control. Equation (7) includes soft repulsion between beads, which becomes active at Φ≥0.4, exactly where phase-locking emerges. Without a simulation with G=0 (no hydrodynamics), you cannot rule out that steric repulsion alone synchronizes the dumbbells. That would still be a result, but it would not be hydrodynamic phase-locking. The paper never addresses this. Second, the analytical pair argument in Eqs. (15)-(16) is stated without derivation, assumes equal frequencies, and ignores the myosin coupling that the numerics show matters. So it is circular as evidence for phase-locking; the numerics carry that weight. Third, the paper does not give parameter values, initial-condition details, integration scheme, or code, making reproduction difficult. Fourth, the vs~Φ^4 scaling is fit from a few points, and the dimensional translation in the Discussion looks off by an order of magnitude: 10^-2 × 20 µm / 10 s should be ~1 µm/min, not 0.1 µm/min. Minor but sloppy.

None of these are fatal. The central idea is plausible and the qualitative results are probably robust. But the paper needs a revision that adds the hydrodynamic-off control, discloses numerical details, and fixes the dimension error. With that, it could be a solid contribution.

I'd send it to review. The referee should focus on the control and the missing methods. It's not a paradigm shift, but it's a legitimate mechanism worth putting on the table. I wouldn't cite it yet, but I'd bring it to the reading group as a case study in controls for active-matter simulations.

Referee Report

4 major / 6 minor

Summary. This manuscript examines a one-dimensional suspension of bead-spring dumbbells whose internal myosin-like activity produces limit-cycle oscillations in dumbbell length and bound-myosin concentration. In isolation a dumbbell is non-motile because its shape changes are reciprocal; the authors couple the beads to a Stokes fluid via Oseen hydrodynamic interactions and include a soft repulsion between beads. From deterministic simulations of 64 dumbbells with periodic boundary conditions, they report three density-dependent phenomena: a transition from diffusive to ballistic center-of-mass motion near Φ≈0.4 with collective velocity scaling as Φ^4, hydrodynamic phase separation into clusters for intermediate densities, and phase-locking of the myosin and length oscillations at high density. A pair-level analytical argument (Eqs. (11)–(16)) is presented to show that phase differences between dumbbells can produce net translation and phase-dependent attraction/repulsion, and pair simulations quantify phase-locking through mutual information and correlation.

Significance. The reported phenomena, if fully supported, would be a valuable conceptual contribution: they demonstrate a route to collective motility in suspensions of agents that have no intrinsic polarity, no self-propulsion, and no adhesive interactions, with a density threshold and cooperative cluster formation arising from hydrodynamic coupling alone. The paper benefits from systematic density sweeps, complementary order parameters (mutual information and correlation), cluster counting, and a candid discussion of the model's simplifications. However, the manuscript does not yet provide a derivation of the central pair formulas, a parameter table, integration details, code, or the necessary control simulation with hydrodynamics switched off. These omissions leave the quantitative threshold and the attribution of the mechanism to hydrodynamics provisional. I still regard the qualitative scenario as credible and worth publishing after a substantive revision.

major comments (4)

[§2.3, Eq. (7); §3.3] The central attribution of phase-locking and clustering to hydrodynamic interactions is not established because no control simulation is reported with the hydrodynamic coupling switched off (G_αβ=0 in Eq. (7)). At Φ≥0.4 the soft repulsion of Eq. (5) directly couples beads of neighboring dumbbells and can influence dl_i/dt in Eq. (2), providing a steric synchronization mechanism that may reproduce the high-density MI/correlation and cluster signals. Please add simulations without the Oseen coupling (and, if feasible, without repulsion) and show that the phase-locking transition, cluster statistics, and diffusive-to-ballistic transition are absent or strongly modified. Without such a control, Eqs. (2)–(8) do not justify the phrase 'hydrodynamics-driven' in the title and abstract.
[§3.2.1, Eqs. (15)–(16)] The analytical pair result is stated without derivation, and the derivation assumes ω1=ω2 and neglects the effect of inter-dumbbell interactions on myosin binding. Because equal oscillation frequencies are the definition of phase-locking, the argument assumes one of the phenomena it is supposed to explain; it can at most show that, once two dumbbells oscillate at the same frequency with a phase difference θ, their average velocities are nonzero. Please provide the full expansion in an appendix, list all small parameters (including the ordering of ẽ/l̄, a/d, and repulsion range), and reframe the text so the numerical simulations, not Eqs. (15)–(16), carry the burden of demonstrating the emergence of phase-locking.
[§2.3–§2.4, §3.2.2] No simulation parameters are given: k1, k3, l0, a, η, τ, c0, μ, ε, n, N, L, and the initial phase and position distributions are all absent, as are the integration scheme, time step, number of realizations per Φ, and equilibration time. Figure 2C even omits the reference μ used to define T0. Without these values the density threshold Φ≈0.4, the exponent α, and the vs∼Φ^4 scaling cannot be checked or reproduced. Please add a parameter table, a numerical-methods subsection, and a code/data availability statement.
[§3.2.2, Fig. 4] The extraction of α and v_s is under-specified. The text notes that individual realizations at low Φ can be sub- or super-diffusive and that only the average over several simulations gives α≈1, but the fitting window, the number of realizations, and the averaging procedure are not stated. The inset in Fig. 4B spans roughly one decade of Φ; please give the fitted range, the fit uncertainty, and a power-law exponent with error bars. This matters because the diffusive-to-ballistic transition is one of the paper's three headline claims.

minor comments (6)

[§2.4, Fig. 2C] The reference value of μ used to define T0 is missing in the text ('for c0=1.0 and µ = is taken'); please complete and state all parameter values in the figure caption.
[Eqs. (1), (15)–(16)] The symbol k in Eqs. (15)–(16) is not defined; presumably it denotes a combination of elastic parameters, and r should be related to the center-to-center separation d introduced in §3.2.1.
[§3.2.3] The sentence 'an occurrence of clusters of size Nc > nrequires at least n consecutive dumbbell pairs...' contains a typo and does not explain how n is chosen relative to the threshold Δd_c; please rewrite.
[§3.3, Fig. 8] The panel labels such as 'A2' and 'B3' are hard to follow; please label subpanels explicitly with (a), (b), etc., in the figure and refer to them accordingly.
[§4] The claim that the qualitative features will hold in two or three dimensions is explicitly an expectation, not a demonstrated result; please mark it as a conjecture.
[Throughout] Please unify the spelling ('dumbell' vs 'dumbbell') and avoid undefined terms such as 'sessile' when 'non-motile' is meant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three claimed phenomena are obtained from direct numerical simulation of the coupled bead-spring equations, and the analytical pair argument is an explicitly labeled auxiliary interpretation rather than the source of the predictions.

full rationale

All three headline phenomena—density-dependent ballistic transition, phase separation, and phase-locking—are read directly from numerical solutions of the stated equations (2)-(8) for N=64 dumbbells. The MSD exponents, cluster counts, kymographs, and mutual-information/correlation measures are outputs, not inputs. The only analytical step (Sec. 3.2.1) assumes omega1 = omega2 and ignores inter-dumbbell effects on myosin binding; the paper explicitly labels these as simplifications and subsequently relaxes them in the numerical pair study (Sec. 3.3.1), where phase-locking is detected independently via mutual information and correlation. Thus using Eq. (15) later to interpret the phase-locked regime is a consistency argument, not a circular derivation. The vs ~ Phi^4 scaling is presented as a numerical observation, not as a prediction generated from a fitted parameter. Citations to Dierkes et al. supply the single-dumbbell limit-cycle model; citations to Rizvi et al. are examples of bead-spring swimmers and are not load-bearing for the novel claims. No uniqueness theorem or ansatz is imported from the authors' prior work. Potential weaknesses—absence of a hydrodynamic-off control, one-dimensional ordering, and unreported parameter values—are evidentiary limitations rather than definitional circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central simulation rests on the Dierkes oscillator model, Oseen pairwise hydrodynamics, deterministic dynamics, and a 1D periodic geometry; none of these are independently verified here, and the analytical pair argument adds assumptions (equal frequencies, weak amplitudes, ignored myosin coupling) that are not checked in the full numerics.

free parameters (5)

Spring and myosin parameters k1, k3, c0, tau, mu = Not reported
These enter Eqs. (1)-(3) and set the oscillation time period and amplitude, yet no values or ranges are given, so the reported nondimensional velocity scale is not anchored.
Repulsion strength epsilon and exponent n = Not reported
Eq. (5) introduces soft repulsion between beads; the chosen epsilon and n affect whether dumbbells can approach closely at high Phi, which matters for clustering and hydrodynamic forces.
Limit-cycle sinusoidal fit amplitudes (bar l, tilde l, bar c, tilde c, delta) = Not reported numerically
Eqs. (9)-(14) represent the limit cycle as a sinusoid and feed the analytical pair velocities in Eqs. (15)-(16); these amplitudes and phase shifts are taken from simulations, not derived.
System size L (inter-dumbbell spacing at fixed Phi) = Not reported
Phi = (l + 2a)N/L fixes density, but L is never given, so the physical spacing and hydrodynamic coupling strengths are not specified.
Collective velocity scaling exponent = 4 (fitted)
The inset of Fig. 4B reports vs ~ Phi^4; this exponent is inferred from a small number of simulation points without an uncertainty estimate, so it functions as a fitted summary rather than a derived law.

assumptions (6)

standard math A single dumbbell with one reciprocal shape degree of freedom produces no net translation in Stokes flow.
Invoked in Section 3.1; standard consequence of Purcell (1977).
domain assumption Hydrodynamic interactions are pairwise additive and described by the Oseen tensor; near-field lubrication and many-body reflections are neglected.
Used in Eqs. (7)-(8); unchecked at high Phi where beads are close.
domain assumption Thermal fluctuations are negligible compared with actomyosin forces, so stochastic terms can be omitted.
Section 2.1; reasonable for cells, but noise robustness is not tested.
domain assumption A one-dimensional periodic arrangement of dumbbells suspended in 3D fluid is representative of the collective behavior.
Section 2.1 sets the geometry; Section 4 admits that extension to 2D/3D is an expectation, not a demonstrated result.
domain assumption The myosin binding-unbinding equation (2) remains valid for each dumbbell in a hydrodynamically coupled suspension.
The paper applies the single-element oscillator model to all dumbbells and later argues that hydrodynamic interactions feed back into myosin dynamics; this feedback is part of the claimed mechanism.
ad hoc to paper For the pair calculation, separations satisfy d >> l, amplitudes satisfy bar l_i >> tilde l_i, and inter-dumbbell effects on myosin binding are ignored, with omega1 = omega2.
Section 3.2.1; these assumptions exclude the two-way coupling that the numerics identify as essential and assume phase-locking in advance.

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Cite this review

Pith. "Pith review of Hydrodynamics-driven phase-locking and collective motility of sessile active dumbbells." pith.science (2026). https://pith.science/paper/GFBZIKSH

@misc{pith2026250108065,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamics-driven phase-locking and collective motility of sessile active dumbbells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFBZIKSH}},
  note         = {Machine review of arXiv:2501.08065}
}

read the original abstract

Collective motion is a phenomenon observed across length scales in nature, from bacterial swarming and tissue migration to the flocking of animals. The mechanisms underlying this behavior vary significantly depending on the biological system, ranging from hydrodynamic and chemical interactions in bacteria to mechanical forces in epithelial tissues and social alignment in animal groups. While collective motion often arises from the coordinated activity of independently motile agents, this work explores a novel context: the emergence of collective motion in systems of non-motile active agents. Inspired by the oscillatory shape dynamics observed in suspended cells such as neutrophils and fibroblasts, we model active dumbbells exhibiting limit-cycle oscillations in shape as a minimal representation of such systems.Through computational simulations, we demonstrate that hydrodynamic interactions between these dumbbells lead to three key phenomena: a density-dependent transition from sessile to collective motion, hydrodynamics-induced phase separation, and synchronization of oscillatory shape changes. We have explored the role of hydrodynamic interactions on these emergent properties of sessile active dumbbells. These results underscore the critical role of hydrodynamic coupling in enabling and organizing collective behaviors in systems lacking intrinsic motility. This study lays the groundwork for future investigations into the emergent behavior of active matter and its implications for understanding cell motility, tissue dynamics, and the development of bio-inspired materials.

Figures

Figures reproduced from arXiv: 2501.08065 by the authors.

**Figure 2.** Single dumbbell dynamics. Limit cycle oscillations of the dumbells for different values of (A) bead mobilities µ, and (B) reference myosin levels c0. (C) Dependence of time periods of limit cycle oscillations on µ and c0. T0, the time period for c0 = 1.0 and µ = is taken as the characteristic time for the analysis shown in this work. 3.2 Hydrodynamic interactions and collective motion 3.2.1 Pair of active dumbbells:… view at source ↗

**Figure 3.** Collective motion of dumbbells. Mean square displacement (MSD) of the center of mass of active dumbbell suspension for different suspension density values. The exponent α increases with increasing Φ. Insets show the positions of the system’s center of mass as a function of time for different realizations. From the analytical argument presented in section-3.2.1 the collective translational velocity of a pair of dumbb… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: (A) Exponent α (as defined in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Collective swimming velocity. (A) Displacement of representative dumbbells for different values of Φ. (B) Probability distribution of collective Si and relative Di velocities of adjacent dumbbells. Inset in the bottom panel in (A) shows the enlarged version of the same…

**Figure 6.** Figure 6: Hydrodynamic phase separation. (A) Distribution of relative positions of adjacent dumbbells ∆di for one realization of suspension at Φ = 0.7 at different time points. Dotted lines mark ⟨∆di⟩ the average value of ∆di for the particular sample. We have used relative sepa…

**Figure 7.** Figure 7: Density-dependent phase locking. (A) Representative kymographs showing the concentration of bound myosin in the dumbbells for suspension densities Φ = 0.1, 0.3, 0.7, and 0.9. With an increase in suspension density, myosin dynamics in the dumbbells becomes coordinated d…

**Figure 8.** Figure 8: Modes of phase locking and its dependence on suspension density Φ. (A) Mutual Information I(c1; c2) and correlation R(c1, c2) between bound myosin concentrations in two dumbbells at fixed separations. In (A2) the circular markers represent different realizations of the…

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113.148102

doi: 10.1103/PhysRevLett. 113.148102. URL https://link.aps.org/doi/10.1103/PhysRevLett.113.148102. Markus U Ehrengruber, David A Deranleau, and Thomas D Coates. Shape oscillations of human neutrophil leukocytes: characterization and relationship to cell motility. Journal of experimental biology, 199(4): 741–747,

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work page doi:10.1039/c4sm02611j

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21 Zhicheng Zheng, Yuan Tao, Yalun Xiang, Xiaokang Lei, and Xingguang Peng

doi: 10.1038/s42003-020-01300-w. 21 Zhicheng Zheng, Yuan Tao, Yalun Xiang, Xiaokang Lei, and Xingguang Peng. Body orientation change of neighbors leads to scale-free correlation in collective motion. Nature Communications, 15(1):8968,

work page doi:10.1038/s42003-020-01300-w

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doi: 10.1119/1.10903

ISSN 0002-9505. doi: 10.1119/1.10903. URL https://doi.org/10.1119/1.10903. 20 Sriram Ramaswamy. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys., 1 (1):323–345,

doi:10.1119/1.10903

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doi: https: //doi.org/10.1016/0014-4827(85)90263-0

ISSN 0014-4827. doi: https: //doi.org/10.1016/0014-4827(85)90263-0. URL https://www.sciencedirect.com/science/article/ pii/0014482785902630. Angelo C Setaro and Patrick T Underhill. Dumbbell kinetic theory for polymers in a combination of flow and external electric field. Physical Review E, 100(5):052501,

work page arXiv

[6] [2007]

URL https://dx.doi.org/10

doi: 10.1088/1478-3975/4/4/004. URL https://dx.doi.org/10. 1088/1478-3975/4/4/004. Hajime Satoh, Tetsuo Ueda, and Yonosuke Kobatake. Oscillations in cell shape and size during loco- motion and in contractile activities of physarum polycephalum, dictyostelium discoideum, amoeba pro- teus and macrophages. Experimental Cell Research, 156(1):79–90,

work page doi:10.1088/1478-3975/4/4/004

[7] [2010]

Three-bead steering microswimmers

Mohd Suhail Rizvi, Alexander Farutin, and Chaouqi Misbah. Three-bead steering microswimmers. Phys. Rev. E, 97:023102, Feb 2018a. doi: 10.1103/PhysRevE.97.023102. URL https://link.aps.org/doi/10. 1103/PhysRevE.97.023102. Mohd Suhail Rizvi, Alexander Farutin, and Chaouqi Misbah. Size and shape affect swimming of a triangular bead-spring microswimmer. Physic...

doi:10.1103/physreve.97.023102

[8] [2011]

URL https://link.aps.org/doi/ 10.1103/PhysRevLett.106.058104

doi: 10.1103/PhysRevLett.106.058104. URL https://link.aps.org/doi/ 10.1103/PhysRevLett.106.058104. Y. Zeng and B. Liu. Self-propelling and rolling of a sessile-motile aggregate of the bacterium Caulobacter crescentus. Commun Biol, 3:587,

work page doi:10.1103/physrevlett.106.058104