^{1,a)}and E. A. Gaffney

^{1}

### Abstract

A crucial structure in the motility of flagellated bacteria is the hook, which connects the flagellum filament to the motor in the cell body. Early mathematical models of swimming bacteria assume that the helically shaped flagellum rotates rigidly about its axis, which coincides with the axis of the cell body. Motivated by evidence that the hook is much more flexible than the rest of the flagellum, we develop a new model that allows a naturally straight hook to bend. Hook dynamics are based on the Kirchhoff rod model, which is combined with a boundary element method for solving viscous interactions between the bacterium and the surrounding fluid. For swimming in unbounded fluid, we find good support for using a rigid model since the hook reaches an equilibrium configuration within several revolutions of the motor. However, for effective swimming, there are constraints on the hook stiffness relative to the scale set by the product of the motor torque with the hook length. When the hook is too flexible, its shape cannot be maintained and large deformations and stresses build up. When the hook is too rigid, the flagellum does not align with the cell body axis and the cell “wobbles” with little net forward motion. We also examine the attraction of swimmers to no-slip surfaces and find that the tendency to swim steadily close to a surface can be very sensitive to the combination of the hook rigidity and the precise shape of the cell and flagellum.

We are grateful to Dr. George Wadhams and Professor Judy Armitage for insightful comments concerning the motility of *R. sphaeroides*.

I. INTRODUCTION

II. METHODS

A. Geometrical modeling

B. Equations of motion

C. The Kirchhoff hook model

D. Bacterial dynamics with hook response

E. Steady states of the hook

F. Tracking

III. RESULTS

A. Bend/twist ratio

B. Hook rigidity

C. Hook length

D. Boundary accumulation

IV. DISCUSSION

A. Free space

B. Accumulation at plane boundaries

C. Adjusting the relative hook stiffness

V. CONCLUSION

### Key Topics

- Bacteria
- 22.0
- Torque
- 17.0
- Elasticity
- 16.0
- Rheology and fluid dynamics
- 10.0
- Viscosity
- 10.0

## Figures

Geometrical model of bacterial cell body and flagellum.

Geometrical model of bacterial cell body and flagellum.

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.

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.

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 τ^{M} 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 τ^{M}.

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 τ^{M} 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 τ^{M}.

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

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

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

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

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

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

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).

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).

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.

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.

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.

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.

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 *x*–*y* 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 *x*–*y* 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.

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 *x*–*y* 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 *x*–*y* 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

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

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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