E. coli bacterium tumbling in bulk and close to surfaces: A simulation study

Holger Stark; Pierre Martin; Tapan Chandra Adhyapak

REVIEW 3 major objections 4 minor 72 references

E. coli bacterium tumbling in bulk and close to surfaces: A simulation study

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Near a bounding surface, E. coli tumble angles shrink and turn forward-biased, explaining reported suppression of tumbling.

desk verdict Near-wall tumbling simulation with a genuinely new in-plane forward-bias result; solid bulk validation, but authors must clarify which flagellum reverses and address the Re=0.4 gap. read the letter →

arxiv 2504.20893 v1 pith:WLS66WFN submitted 2025-04-29 cond-mat.soft physics.bio-ph

Pierre Martin , Tapan Chandra Adhyapak , Holger Stark This is my paper · ORCID

classification cond-mat.softphysics.bio-ph

keywords E.colitumblingflagellarpolymorphismKirchhoffrodmultiparticlecollisiondynamicssurfacehydrodynamicsrun-and-tumbletumbleangledistribution

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 argues that the surface itself reshapes how E. coli tumbles: close to a flat wall, the tumble angle distribution shifts to smaller angles, flagellar dispersion drops, and in-plane reorientation becomes strongly forward-biased, with 25% of near-wall tumbles turning less than 10 degrees. This provides a mechanistic explanation for why earlier experiments reported suppressed tumbling near surfaces: many near-wall tumbles are simply too small to notice. The same model first reproduces the classic bulk tumble-angle distribution, giving confidence that the near-surface changes are caused by the wall rather than by model artifacts. A sympathetic reader would care because it connects single-flagellum mechanics to surface exploration, biofilm formation, and bacterial escape.

What carries the argument

The central object is a discrete flagellum built from an extended Kirchhoff-rod elastic energy, in which each segment's rotational strain vector chooses among four polymorphic ground states (normal, coiled, semi-coiled, curly-I) with a smooth transition cost. The flagella are hydrodynamically coupled to the fluid through multiparticle collision dynamics, which supplies the wall-induced flows and drag. The hook is represented by reduced bending rigidity at the motor attachment. This machinery is what lets a single reversal of one motor produce realistic polymorphic transformation, bundle unbundling, and reorientation statistics, both in bulk and at the wall.

What would settle it

Repeat the near-wall tumble simulations with the same bacterium at Reynolds number 0.4 and at about $10^{-5}$ (for example, by raising viscosity or lowering motor torque). If the mean tumble angle near the wall does not stay well below the bulk value of 61 degrees, or if the fraction of in-plane angles below 10 degrees drops from 25%, the central claim fails. On the experimental side, a high-frame-rate (at least 200 Hz) tracking study of individual E. coli near a glass surface that counts every velocity drop would settle whether such small in-plane tumbles actually occur.

Watch

Extended reading notes

Core claim

In a simulation of E. coli with four flexible, polymorphic flagella in an MPCD fluid, a tumble event is triggered by reversing one motor. In bulk, the tumble-angle distribution has mean 61 degrees and matches the classic experiment (mean 62 degrees). Near a no-slip surface, the mean drops to 41 degrees, the polar angle peaks at 90 degrees (parallel to the wall), and the in-plane angle distribution is forward-biased: 25% of events have in-plane reorientation below 10 degrees. The authors claim these small forward-biased events are the origin of the experimentally reported 'suppressed' tumbling near surfaces, and that they help bacteria stay at the surface while still exploring it.

Load-bearing premise

The near-wall results assume that the simulated fluid at Reynolds number about 0.4 behaves like real E. coli hydrodynamics at Reynolds number about $10^{-5}$; if inertial effects at 0.4 change the wall-induced drag on flagella, the reduced tumble angles and forward bias could be simulation artifacts.

Editorial extensions

If this is right

If the wall shifts tumbling to small forward angles, near-wall E. coli effectively reorient in place, which should increase their residence time at surfaces without requiring fewer tumble events.
Stiffer hooks narrow the tumble-angle distribution and reduce flagellar dispersion; in the model, a 100-fold stiffer hook raises the mean tumble angle and eliminates very large angles.
Because 53% of near-wall tumbles give a polar angle below 90 degrees, tumbling does provide escape attempts, but escape is not simply correlated with flagellar dispersion.
The measured bulk agreement pins the model's tumble mechanics to real E. coli, so the surface comparison is the meaningful new quantity.

Reading between the lines

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

If the forward-bias is real, then the standard run-and-tumble picture near surfaces should be revised: runs may not be longer, but each tumble changes direction less, so effective exploration is dominated by many small turns.
A testable extension: track near-wall tumbles with high-speed microscopy and count all velocity drops; the model predicts a population of tumbles with in-plane angle below 10 degrees that previous assays may have discarded.
Because the simulation runs at Reynolds number about 0.4, roughly 10^4 times the real value, the near-wall forward bias should be re-checked at lower Reynolds numbers before it is used to interpret experiments.
The same polymorphic flagellum model could be applied to other peritrichous bacteria to see whether surface-biased tumbling is generic or specific to E. coli.

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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

Colleague —

The paper deserves a serious look. Its real new result is the near-wall tumble statistics: compared to bulk (mean 61°), tumbling near a wall shifts the tumble-angle distribution down (mean 41°) and produces a strong in-plane forward bias, with 25% of events below 10°. That is a concrete, mechanistic way to reconcile ref. 39's report that tumbling is suppressed at surfaces with ref. 41's claim that it drives escape. The result is an emergent output of the simulation, not a fit, so it has predictive weight.

What the paper does well: the model is the group's established Kirchhoff-rod/MPCD framework, extended to tumbling near a surface; they validate against Berg-Brown in bulk and get a decent match for a fixed tumble time. The hook-stiffness sweep and the dispersion measure (smallest eigenvalue of the bundle's moment of inertia) are thoughtful and clearly explained. Also to their credit, they explicitly flag the four-flagella symmetry as a simplification and say the influence on tumbling needs further work—that is honest.

Soft spots, in rough order of importance. First, the Reynolds number: Re≈0.4 is four orders of magnitude above real E. coli, and the near-wall results depend on wall-induced hydrodynamics. They report that the swimming speed and bundle rotation are off from experiments (though the ratio v/ωr is right), and they don't test convergence toward Stokes flow. That is a real gap, though arguably a known MPCD limitation. Second, the paper is silent on which flagellum reverses in the near-wall runs. Section 2.2 says the symmetric design lets them avoid randomizing the reversing flagellum; that is valid in bulk, but the wall breaks the symmetry. If all near-wall tumble events used the same body-fixed flagellum, the P(γ), P(φ), and the 25% figure are conditional on motor position, not the marginal distribution an experiment would sample. The long trajectory has only eight near-wall events and can't average over that either. This is not fatal—the effect might survive averaging—but the authors need to state the protocol and ideally run all four motors. Third, the phenomenological ground-state energies are hand-tuned to produce curly-I; the paper leans on earlier work for that, which is fine but leaves the model sensitive to parameters that aren't independently fixed. Fourth, no error bars on the quoted fractions, and data are only 'available upon request.'

Who benefits: anyone working on bacterial motility, surface residence, or flagellar polymorphism; the near-wall reorientation result is worth discussing even with the caveats. I'd accept it for peer review and ask the authors to clarify the motor-identity protocol, add a low-Re check if feasible, and provide error bars or raw data.

Referee Report

3 major / 4 minor

Summary. The paper develops a mesoscale MPCD model of E. coli with four flexible flagella modeled by an extended Kirchhoff-rod theory that includes polymorphic transformations, and uses it to study single tumble events in bulk and near a flat no-slip wall. In bulk, for a fixed tumble time of 30 bundle periods the simulated tumble-angle distribution (mean 61°, standard deviation 34°) is compared with Berg and Brown's experimental values (mean 62°, standard deviation 26°); increasing the hook bending rigidity narrows the distribution and reduces flagellar dispersion. Near a wall, independent tumble simulations yield a mean tumble angle of 41° (versus 61° in bulk), an in-plane reorientation distribution in which 25% of events have an in-plane angle below 10°, and a polar-angle distribution peaking near 90°. The authors propose that these small in-plane reorientations explain the experimentally reported suppression of tumbling and prolonged run times near surfaces.

Significance. If the near-wall statistics are robust, this is a useful mechanistic hypothesis for why tumbling near surfaces appears suppressed: tumbles still occur but reorient the cell mostly within the surface plane and often with very small heading change. The model's strengths are its explicit treatment of flagellar polymorphism, flexible hooks, and hydrodynamic coupling, and the bulk tumble-angle statistics are based on 932–1032 independent events and agree well with a classic experiment. The near-wall predictions, however, rest on two load-bearing assumptions that are not yet verified: the fluid dynamics is computed at Reynolds number about 0.4 rather than the Stokes regime of real E. coli, and the near-wall tumble statistics may be conditional on a single flagellar motor position because of the assumed fourfold axisymmetry. These issues, rather than the bulk phenomenology, determine whether the quantitative claims (41° mean angle, 25% forward bias) can be transferred to real bacteria.

major comments (3)

[Section 2.4 and Section 4.2] Section 2.4 reports Re≈0.4 for the MPCD fluid, roughly four orders of magnitude above the value for E. coli. The near-surface effects in Fig. 8 — reduced tumble angle, forward-biased in-plane reorientation, and reduced flagellar dispersion — are attributed to wall-induced hydrodynamic interactions and enhanced drag. At Re=0.4 inertial contributions are not negligible, so the quantitative distributions may not represent the Stokes regime that governs real E. coli. Please either perform a convergence study at lower Re (for example by increasing the viscosity or reducing the lattice constant) and show that P(γ), P(φ), and ⟨I1⟩/I_bundle^1 are unchanged, or provide a quantitative argument that the wall-induced flows relevant to tumbling are Stokes-like despite the nominal Reynolds number.
[Section 2.2 and Section 4.2] Section 2.2 states that the bacterium is 'fully axisymmetric' so that the authors can avoid 'randomly choosing the specific position of the reversely rotating flagellum.' This is a deliberate modeling choice, but it makes the bulk and near-wall cases different: in bulk the four flagella are equivalent, while near a wall one flagellum is closest to the surface and the others are not. Section 4.2 says only that 'tumbling is initiated by reversing the rotation of one flagellum' and does not report whether all four motor positions were sampled, whether the initial roll angle about the body axis was randomized, or whether the same flagellum was used throughout. If a single body-fixed flagellum was used, the 41° mean and the 25% below-10° fraction in Fig. 8 are conditional distributions, not the marginal distributions that experiments observe where motor reversals are stochastic across flagella. The eight near-wall events in Fig. 7 are too few to average over motor identity. Please report the sampling protocol and, if only one motor was used, provide the marginal P(γ), P(φ), and P(θ) over the four motors or justify that the conditional statistics are representative.
[Section 4.2] Unlike the bulk case, where the text reports N=932 and N=1032 for the two tumble-time protocols, Section 4.2 does not state the number of independent near-wall tumble events used for Fig. 8, and no error bars are shown. The headline numbers — mean γ=41°, 25% of events with φ<10°, and 53% escape-oriented events — need statistical support. Please report N and uncertainties (for example, bootstrap confidence intervals) for all three distributions.

minor comments (4)

[Section 2.1] Equation (3) includes a term A/2 ξ²(∂sΩ)² that is said to allow smooth transitions between polymorphic regions of size ξ, but the numerical value of ξ is never given; please state it in Section 2.4.
[Section 3.3, Fig. 5] The text says that for hook rigidities of 100Ah and 400Ah the tumble-angle distributions 'become more narrow,' but the reported mean angles of 87° and 69° are above the reference mean of 61°; please clarify whether 'narrow' refers to the variance and report the standard deviations for these distributions.
[Section 5 and Data availability] The model relies on several phenomenological parameters (ground-state energies δ(n), hook rigidity Ah, motor reversal protocol) and the paper does not assess sensitivity of the near-wall results to these choices; a brief sensitivity discussion would strengthen the conclusions.
[Data availability] The data availability statement says data are available upon request from one of the authors; for reproducibility, please consider depositing the simulation code and parameter files in a public repository.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the near-wall tumble statistics are emergent simulation outputs benchmarked against independent experiments.

full rationale

The paper's central claims—the bulk tumble-angle distribution, the hook-stiffness dependence, and the near-surface shift to smaller tumble angles with in-plane forward bias—are outputs of an MPCD + Kirchhoff-rod simulation, not fits to the target statistics. The only explicitly calibrated quantity, the ground-state energy offsets δ(n), is chosen to reproduce the curly-I polymorphic state during reverse motor rotation, a qualitative input about flagellar conformation; it does not by construction determine the reported mean tumble angle (61° bulk, 41° near wall) or the 25% in-plane forward-bias fraction. The bulk P(γ) is benchmarked against the independent Berg & Brown measurements, and the near-surface trends are compared with separate experimental studies (Molaei et al.; Junot et al.; Lemelle et al.). The self-citations (Vogel & Stark, Adhyapak & Stark, Zantop & Stark) are methodological references for the extended Kirchhoff-rod model, ground-state vectors, and MPCD implementation; they are not invoked as uniqueness theorems or as substitutes for the new simulation evidence. The in-plane angle φ is defined from the simulated final orientation and is a derived statistic, not an input parameter. No equation is visibly identical to an input, and no fitted parameter is renamed as a prediction. Any concerns about Reynolds-number mismatch or conditional motor choice near the wall are correctness/robustness issues, not circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the extended Kirchhoff-rod/MPCD model from the authors' earlier work, with hand-tuned polymorphic ground-state energies and a constant hook rigidity. No new physical entities are introduced. The main burdens are the Reynolds number mismatch, the simplified hook representation, and the axisymmetric four-flagella geometry.

free parameters (4)

Polymorphic ground state energies delta(n) = {0.0, 25, 10, 0.0} kBT
Chosen by hand (Section 2.4) as pure phenomenological parameters so that a reversely rotating flagellum transitions to the curly-I state, as observed in experiments. These energies directly affect the speed and extent of the polymorphic transformation and hence the tumble angle distribution.
Hook bending rigidity Ah = 2.5 kBT a0
Taken from prior work (refs. 21, 68), assumed constant in the reference case even though experiments (refs. 32, 33) report dynamic variation under rotation. The paper later varies Ah by factors 10 to 400, so the central surface results use the single reference value.
Motor torque reversal time = 3.5 tau_b
The torque is linearly switched from -Tm to Tm over 3.5(tau_b) (Section 3.1). This timescale is chosen for smoothness and affects the initial phase of the tumble, though its influence on the final angle distribution is not systematically tested.
Gamma distribution shape parameter k = 2
For the variable tumble time scenario, the Gamma shape parameter is set to k=2 with theta=15(tau_b) to give a mean of 30(tau_b) (Section 3.1). Motivated by ref. 41, but the choice of k=2 is a modeling input.

assumptions (4)

domain assumption The extended Kirchhoff rod free energy with a min over polymorphic ground states (eqn. 3) quantitatively captures polymorphic transformations during tumbling.
The model relies on the authors' earlier extended Kirchhoff rod framework (refs. 20, 21, 55). The ground-state vectors are fixed inputs from ref. 55. There is no independent validation in this paper that this energy functional reproduces the dynamics of polymorphic transitions.
domain assumption The flagellar hook can be represented by a short Kirchhoff rod segment with reduced bending and twisting rigidity, rotating about a fixed motor axis tilted at 55 degrees.
The hook is not explicitly resolved (Section 2.2). The universal-joint property is replaced by a reduced-rigidity elastic segment with a fixed motor axis. The finite stiffness of this segment is known (refs. 32, 33) to vary with rotation, which the model ignores in the reference case.
domain assumption The symmetrically attached four-flagella geometry is representative of E. coli tumbling behavior.
The choice of four flagella symmetrically anchored at the rear (Section 2.2) is made to reduce the parameter space. Real E. coli have variable flagellar number and arrangement, and the authors acknowledge in the conclusions that the influence on tumbling needs further investigation.
domain assumption MPCD at the chosen collision rule and density reproduces the relevant low-Reynolds-number hydrodynamics of swimming E. coli.
The simulations run at Re about 0.4 (Section 2.4), while real E. coli swim at Re about 10^-5. The assumption that the flow field, surface attraction, and flagellar drag are faithfully reproduced despite this Reynolds number mismatch is not tested.

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Cite this review

Pith. "Pith review of E. coli bacterium tumbling in bulk and close to surfaces: A simulation study." pith.science (2026). https://pith.science/paper/WLS66WFN

@misc{pith2026250420893,
  author       = {Pith},
  title        = {Pith review of: E. coli bacterium tumbling in bulk and close to surfaces: A simulation study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLS66WFN}},
  note         = {Machine review of arXiv:2504.20893}
}

read the original abstract

Motility is fundamental to the survival and proliferation of microorganisms. The E. coli bacterium propels itself using a bundle of rotating helical flagella. If one flagellum reverses its rotational direction, it leaves the bundle, performs a polymorphic transformation, and the bacterium tumbles. The E. coli bacterium is hydrodynamically attracted to surfaces. This prolongs its residence time, while tumbling facilitates surface detachment. We develop a model of E. coli that uses an extended Kirchhoff rod theory to implement flagellar flexibility as well as different polymorphic conformations and perform hydrodynamic simulations with the method of multiparticle collision dynamics (MPCD). To establish a reference case, we determine the distribution of tumble angles in the bulk fluid, which shows good agreement with experiments for a fixed tumble time. Increasing the hook stiffness, narrows the tumble angle distribution and reduces the flagellar dispersion during tumbling. Close to a bounding surface, the tumble angle distribution is shifted to smaller angles, while flagellar dispersion is reduced. Reorientation within the plane favors the forward direction, which might be an explanation for prolonged run times observed in experiments

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Reference graph

Works this paper leans on

72 extracted references · 70 canonical work pages

[1]

K. M. Ottemann and J. F. Miller, Mol. Microbiol., 1997, 24, 1109–1117

work page 1997
[2]

L. A. Pratt and R. Kolter, Mol. Microbiol., 1998, 30, 285–293

work page 1998
[3]

Gest, Notes Rec

H. Gest, Notes Rec. R. Soc. Lond., 2004, 58, 187–201

work page 2004
[4]

Wadhwa and H

N. Wadhwa and H. C. Berg, Nat. Rev. Microbiol., 2022, 20, 161–173

work page 2022
[5]

Lauga, Annu

E. Lauga, Annu. Rev. Fluid Mech., 2016, 48, 105–130

work page 2016
[6]

Ishikawa, Annu

T. Ishikawa, Annu. Rev. Fluid Mech., 2024, 56, 119–145

work page 2024
[7]

Bray , Cell Movements: From Molecules to Motility, Garland Science, 2nd edn., 2000

D. Bray , Cell Movements: From Molecules to Motility, Garland Science, 2nd edn., 2000

work page 2000
[8]

Watnick and R

P. Watnick and R. Kolter,J. Bacteriol., 2000, 182, 2675–2679

work page 2000

Show all 72 references

[9]

H. C. Berg and L. Turner, Biophys. J., 1993, 65, 2201–2216

1993
[10]

H. C. Berg, The rotary motor of bacterial flagella, 2003. 10

2003
[11]

Turner, W

L. Turner, W. S. Ryu and H. C. Berg, J. Bacteriol., 2000, 182, 2793–2801

2000
[12]

H. C. Berg, E. coli in Motion, 1st edn., 2004

2004
[13]

N. C. Darnton, L. Turner, S. Rojevsky and H. C. Berg, J. Bac- teriol., 2007, 189, 1756–1764

2007
[14]

H. C. Berg and D. A. Brown, Nature, 1972, 239, 500–504

1972
[15]

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Kor- ber and H. M. Lappin-Scott, Annu. Rev. Microbiol., 1995, 49, 711–745

1995
[16]

Van Houdt and C

R. Van Houdt and C. W. Michiels, Res. Microbiol., 2005, 156, 626–633

2005
[17]

A. P. Berke, L. Turner, H. C. Berg and E. Lauga, Phys. Rev. Lett., 2008, 101, 038102

2008
[18]

G. Li, J. Bensson, L. Nisimova, D. Munger, P. Mahautmr, J. X. Tang, M. R. Maxey and Y. V. Brun, Phys. Rev. E , 2011, 84, 041932

2011
[19]

Drescher, J

K. Drescher, J. Dunkel, L. H. Cisneros, S. Ganguly and R. E. Goldstein, Proc. Natl. Acad. Sci. U. S. A. , 2011, 108, 10940– 10945

2011
[20]

Vogel and H

R. Vogel and H. Stark, Phys. Rev. Lett., 2013, 110, 158104

2013
[21]

T. C. Adhyapak and H. Stark, Soft Matter , 2016, 12, 5621–5629

2016
[22]

C. R. Calladine, Nature, 1975, 255, 121–124

1975
[23]

T. C. Adhyapak and H. Stark, Phys. Rev. E, 2015, 92, 052701

2015
[24]

W. Lee, Y. Kim, B. E. Griffith and S. Lim, Phys. Rev. E, 2018, 98, 052405

2018
[25]

E. E. Riley , D. Das and E. Lauga,Sci. Rep., 2018, 8, 10728

2018
[26]

K. Son, J. S. Guasto and R. Stocker, Nat. Phys., 2013, 9, 494– 498

2013
[27]

Hirano, S

T. Hirano, S. Yamaguchi, K. Oosawa and S. Aizawa, J. Bacte- riol., 1994, 176, 5439–5449

1994
[28]

T. Kato, F. Makino, T. Miyata, P. Horváth and K. Namba,Nat. Commun., 2019, 10, 5295

2019
[29]

M. T. Brown, B. C. Steel, C. Silvestrin, D. A. Wilkinson, N. J. Delalez, C. N. Lumb, B. Obara, J. P. Armitage and R. M. Berry , J. Bacteriol., 2012, 194, 3495–3501

2012
[30]

Bianchi, F

S. Bianchi, F. Saglimbeni, G. Frangipane, M. C. Cannarsa and R. Di Leonardo, PRX Life, 2023, 1, 013016

2023
[31]

Spöring, V

I. Spöring, V. A. Martinez, C. Hotz, J. Schwarz-Linek, K. L. Grady , J. M. Nava-Sedeño, T. Vissers, H. M. Singer, M. Rohde, C. Bourquin, H. Hatzikirou, W. C. K. Poon, Y. S. Dufour and M. Erhardt, PLoS Biol., 2018, 16, 1–19

2018
[32]

A. L. Nord, A. Biquet-Bisquert, M. Abkarian, T. Pigaglio, F. Seduk, A. Magalon and F. Pedaci, Nat. Commun., 2022, 13, 2925

2022
[33]

Zhang, C

X. Zhang, C. Zhang, R. Zhang and J. Yuan, Phys. Rev. Lett., 2023, 130, 138401

2023
[34]

Schaar, A

K. Schaar, A. Zöttl and H. Stark, Phys. Rev. Lett., 2015, 115, 038101

2015
[35]

S. M. Mousavi, G. Gompper and R. G. Winkler, Soft Matter, 2020, 16, 4866–4875

2020
[36]

Lauga, W

E. Lauga, W. R. DiLuzio, G. M. Whitesides and H. A. Stone, Biophys. J., 2006, 90, 400–412

2006
[37]

Perez Ipiña, S

E. Perez Ipiña, S. Otte, R. Pontier-Bres, D. Czerucka and F. Pe- ruani, Nat. Phys., 2019, 15, 610–615

2019
[38]

Bianchi, F

S. Bianchi, F. Saglimbeni and R. Di Leonardo, Phys. Rev. X, 2017, 7, 011010

2017
[39]

Molaei, M

M. Molaei, M. Barry , R. Stocker and J. Sheng,Phys. Rev. Lett., 2014, 113, 068103

2014
[40]

Lemelle, T

L. Lemelle, T. Cajgfinger, C. C. Nguyen, A. Dominjon, C. Place, E. Chatre, R. Barbier, J.-F. Palierne and C. Vaillant, Biophys. J., 2020, 118, 2400–2410

2020
[41]

Junot, T

G. Junot, T. Darnige, A. Lindner, V. A. Martinez, J. Arlt, A. Dawson, W. C. K. Poon, H. Auradou and E. Clément,Phys. Rev. Lett., 2022, 128, 248101

2022
[42]

Giacché, T

D. Giacché, T. Ishikawa and T. Yamaguchi,Phys. Rev. E, 2010, 82, 056309

2010
[43]

J. Hu, A. Wysocki, R. G. Winkler and G. Gompper, Sci. Rep., 2015, 5, 9586

2015
[44]

M. Kong, Y. Wu, G. Li and R. G. Larson, Soft Matter, 2015, 11, 1572–1581

2015
[45]

Wu-Zhang, P

B. Wu-Zhang, P. Zhang, R. Baillou, A. Lindner, E. Clément, G. Gompper and D. A. Fedosov,bioRxiv, 2025

2025
[46]

Dvoriashyna and E

M. Dvoriashyna and E. Lauga, PLoS ONE, 2021, 16, 1–26

2021
[47]

Zöttl and H

A. Zöttl and H. Stark, Eur. Phys. J. E, 2018, 41, 61

2018
[48]

J. Hu, M. Yang, G. Gompper and R. G. Winkler, Soft Matter, 2015, 11, 7867–7876

2015
[49]

S. B. Babu and H. Stark, New J. Phys., 2012, 14, 085012

2012
[50]

A. W. Zantop and H. Stark, Soft Matter , 2020, 16, 6400–6412

2020
[51]

Eisenstecken, J

T. Eisenstecken, J. Hu and R. G. Winkler, Soft Matter, 2016, 12, 8316–8326

2016
[52]

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd edn., 1976

1976
[53]

A. E. H. Love, A treatise on the mathematical theory of elastic- ity, Cambridge university press, 2013

2013
[54]

Vogel and H

R. Vogel and H. Stark, Eur. Phys. J. E: Soft Matter Biol. Phys., 2010, 33, 259–271

2010
[55]

Vogel and H

R. Vogel and H. Stark, Eur. Phys. J. E: Soft Matter Biol. Phys., 2012, 35, 15

2012
[56]

R. E. Goldstein, A. Goriely , G. Huber and C. W. Wolgemuth, Phys. Rev. Lett., 2000, 84, 1631–1634. 11

2000
[57]

P. J. Mears, S. Koirala, C. V. Rao, I. Golding and Y. R. Chemla, eLife, 2014, 3, e01916

2014
[58]

Namba and F

K. Namba and F. Vonderviszt, Q. Rev. Biophys. , 1997, 30, 1–65

1997
[59]

Malevanets and R

A. Malevanets and R. Kapral, J. Chem. Phys. , 1999, 110, 8605–8613

1999
[60]

Malevanets and R

A. Malevanets and R. Kapral, J. Chem. Phys. , 2000, 112, 7260–7269

2000
[61]

Gompper, T

G. Gompper, T. Ihle, D. Kroll and R. Winkler, inMulti-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Ap- proach to the Hydrodynamics of Complex Fluids , ed. C. Holm and K. Kremer, Springer Berlin Heidelberg, Berlin, Heidel- berg, 2009, pp. 1–87

2009
[62]

Noguchi, N

H. Noguchi, N. Kikuchi and G. Gompper, Europhys. Lett. , 2007, 78, 10005

2007
[63]

I. O. Götze, H. Noguchi and G. Gompper, Phys. Rev. E, 2007, 76, 046705

2007
[64]

Ihle and D

T. Ihle and D. M. Kroll, Phys. Rev. E, 2001, 63, 020201

2001
[65]

Noguchi and G

H. Noguchi and G. Gompper, Phys. Rev. E, 2008, 78, 016706

2008
[66]

Ripoll, K

M. Ripoll, K. Mussawisade, R. G. Winkler and G. Gompper, Phys. Rev. E, 2005, 72, 016701

2005
[67]

N. C. Darnton and H. C. Berg, Biophys. J., 2007, 92, 2230– 2236

2007
[68]

A. Sen, R. K. Nandy and A. N. Ghosh, J. Electron Microsc. , 2004, 53, 305–309

2004
[69]

J. K. Dhont, An Introduction to Dynamics of Colloids, Elsevier, Amsterdam, 1996, vol. 2

1996
[70]

Z. Qu, F. Z. Temel, R. Henderikx and K. S. Breuer, Proc. Natl. Acad. Sci. U. S. A., 2018, 115, 1707–1712

2018
[71]

Seyrich, Z

M. Seyrich, Z. Alirezaeizanjani, C. Beta and H. Stark, New J. Phys., 2018, 20, 103033

2018
[72]

Turner, L

L. Turner, L. Ping, M. Neubauer and H. C. Berg, Biophys. J., 2016, 111, 630–639. 12

2016

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

K. M. Ottemann and J. F. Miller, Mol. Microbiol., 1997, 24, 1109–1117

work page 1997

[2] [2]

L. A. Pratt and R. Kolter, Mol. Microbiol., 1998, 30, 285–293

work page 1998

[3] [3]

Gest, Notes Rec

H. Gest, Notes Rec. R. Soc. Lond., 2004, 58, 187–201

work page 2004

[4] [4]

Wadhwa and H

N. Wadhwa and H. C. Berg, Nat. Rev. Microbiol., 2022, 20, 161–173

work page 2022

[5] [5]

Lauga, Annu

E. Lauga, Annu. Rev. Fluid Mech., 2016, 48, 105–130

work page 2016

[6] [6]

Ishikawa, Annu

T. Ishikawa, Annu. Rev. Fluid Mech., 2024, 56, 119–145

work page 2024

[7] [7]

Bray , Cell Movements: From Molecules to Motility, Garland Science, 2nd edn., 2000

D. Bray , Cell Movements: From Molecules to Motility, Garland Science, 2nd edn., 2000

work page 2000

[8] [8]

Watnick and R

P. Watnick and R. Kolter,J. Bacteriol., 2000, 182, 2675–2679

work page 2000