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The effects of flagellar hook compliance on motility of monotrichous bacteria: A modeling study
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10.1063/1.4721416
/content/aip/journal/pof2/24/6/10.1063/1.4721416
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/6/10.1063/1.4721416

Figures

Image of FIG. 1.
FIG. 1.

Geometrical model of bacterial cell body and flagellum.

Image of FIG. 2.
FIG. 2.

Comparison between (a) the rigid hook model and (b) the current, flexible hook. The alignment angle between the body axis, e B, and the flagellum axis, e T, is denoted ψ. Since the shapes of the flagella differ near the cell body, the rigid model is not simply the limiting case where stiffness is taken to be very large in the flexible hook model.

Image of FIG. 3.
FIG. 3.

Evolution of alignment angle ψ showing transient hook behaviour at different levels of relative stiffness. The steady state alignment angles are plotted as dashed lines (see Fig. 5). (a) k H = 0.5, (b) k H = 0.6, and (c) k H = 1. Time is non-dimensionalized by the timescale , where is the body's volumetric radius, μ is the dynamic viscosity of the fluid, and τM is the magnitude of the motor torque. The timescale corresponds to the period of revolution of a sphere of radius in the viscous fluid under the action of a torque of magnitude τM.

Image of FIG. 4.
FIG. 4.

Exemplar swimmers (a) with a stable hook state and (b) with an unstable steady hook state. The initial configurations (right) are out of equilibrium but a steady state is soon reached in the stable case. The curves between the two instances shown in each case trace the paths of the junction point x B. The trajectories correspond to cases (c) and (a), respectively, from Fig. 3 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4721416.1] [URL: http://dx.doi.org/10.1063/1.4721416.2]10.1063/1.4721416.110.1063/1.4721416.2

Image of FIG. 5.
FIG. 5.

Variations with relative hook stiffness of the steady state (a) alignment angle, (b) stability, and (c) swimming speed. Solid curves indicate data for bacterial shape A while dashed curves correspond to shape B. Swimming speeds in (c) are normalized by U R, the speed computed using the rigid hook model with the equivalent geometrical parameters (enhanced online). [URL: http://dx.doi.org/10.1063/1.4721416.3]10.1063/1.4721416.3

Image of FIG. 6.
FIG. 6.

Variations in steady state alignment angle (solid curve) and swimming speed (dashed curve) with hook length . The quantities are normalized by ψ0 and U 0, respectively, the values for the standard swimmer shape A, which has hook length .

Image of FIG. 7.
FIG. 7.

Time series of (a) the swimmer's height above the wall, h, and (b) the hook alignment angle, ψ, as a bacterium with relative hook stiffness k H = 1 approaches a stable orbit above a plane boundary. The equivalent swimmer using the rigid hook model has a stable accumulation height , indicated by a horizontal dashed line in (a) and the stable alignment angle ψ* ≈ 0.13 in free space is marked by a horizontal dashed line in (b).

Image of FIG. 8.
FIG. 8.

Near-surface swimming behaviour after transience for bacterial shape A with varying relative hook stiffness. (a) Curves marked with dots indicate the minimum and maximum alignment angles attained over a cycle of periodic motion compared with the free space stable angle shown by the unmarked, thick curve. (b) The minimum and maximum swimming heights during stable boundary swimming (curves marked with dots) and the minimum separation distance between the wall and the swimmer (curve marked with crosses). Using the rigid hook model, this swimmer would have a stable swimming height of . The dashed lines on the left hand side mark the relative stiffness value for which the swimmer was found to collide with the wall and no boundary accumulating trajectory could be obtained.

Image of FIG. 9.
FIG. 9.

Near-surface swimming behaviour after transience for bacterial shape B with varying relative hook stiffness. (a) Curves marked with dots indicate the minimum and maximum alignment angles attained over a cycle of periodic motion compared with the free space stable angle shown by the unmarked, thick curve. (b) The minimum and maximum swimming heights during stable boundary swimming (curves marked with dots) and the minimum separation distance between the wall and the swimmer (curve marked with crosses). Using the rigid hook model, this swimmer would have a stable swimming height of . The dashed lines on the left and right hand sides mark relative stiffness values for which the swimmer was found to escape from the wall and no boundary accumulating trajectory could be obtained.

Image of FIG. 10.
FIG. 10.

Trajectories of two geometrically identical swimmers near plane boundaries. Swimmer (i) has relative hook stiffness k H = 0.7 and swimmer (ii) has relative hook stiffness k H = 2. The 3D trajectories are shown in thick curves while projections of the trajectories onto the xy plane are shown in thin curves. The heights of the swimmers above the wall are also indicated at regular time intervals by vertical lines from the xy plane. The bacteria are shown at their respective starting positions. Note the much smaller radius of curvature of swimmer (ii) once the steady circular orbit is reached.

Tables

Generic image for table
Table I.

Key parameters determining geometry of the model bacteria used in this article. The lengthscale is the volumetric radius of the spheroidal cell body.

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/content/aip/journal/pof2/24/6/10.1063/1.4721416
2012-06-08
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The effects of flagellar hook compliance on motility of monotrichous bacteria: A modeling study
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/6/10.1063/1.4721416
10.1063/1.4721416
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