Hydrodynamic analysis of flagellated bacteria swimming in corners of rectangular channels

Henry Shum and Eamonn A. Gaffney

Phys. Rev. E 92, 063016 – Published 16 December 2015

Abstract

The influence of nearby solid surfaces on the motility of bacteria is of fundamental importance as these interactions govern the ability of the microorganisms to explore their environment and form sessile colonies. Reducing biofouling in medical implants and controlling the transport of bacterial cells in a microfluidic device are two applications that could benefit from a detailed understanding of swimming in microchannels. In this study, we investigate the self-propelled motion of a model bacterium, driven by rotating a single helical flagellum, in such an environment. In particular, we focus on the corner region of a large channel modeled as two perpendicular sections of no-slip planes joined with a rounded corner. We numerically solve the equations of Stokes flow using the boundary element method to obtain the swimming velocities at different positions and orientations relative to the channel corner. From these velocities, we construct many trajectories to ascertain the general behavior of the swimmers. Considering only hydrodynamic interactions between the bacterium and the channel walls, we show that some swimmers can become trapped near the corner while moving, on average, along the axis of the channel. This result suggests that such bacteria may be found at much higher densities in corners than in other parts of the channel. Another implication is that these corner accumulating bacteria may travel quickly through channels since they are guided directly along the corner and do not turn back or swim transversely across the channel.

Received 14 October 2015
Revised 15 November 2015

DOI:https://doi.org/10.1103/PhysRevE.92.063016

Authors & Affiliations

Henry Shum^*

Department of Chemical & Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA

Eamonn A. Gaffney^†

Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

^*phs13@pitt.edu
^†gaffney@maths.ox.ac.uk

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
Details of the model bacterium and its configuration in the corner of a large channel. (a) Schematic of model bacterium with labels for orientation vectors and parameters specifying shape. With $\bar{a} = {(A_{1} A_{2}^{2})}^{1 / 3}$ the volumetric radius of the cell body, we use the following parameters for the boundary accumulating swimmer: flagellum length, $L = 7.5 \bar{a}$ ; flagellar wavelength, $λ = 1.8 \bar{a}$ ; the radius of the helix formed by the flagellum, $a = λ / 2 π$ ; the flagellar radius, $a^{T} = 0.05 \bar{a}$ ; and the envelope parameter, dictating how rapidly the helical radius decays on approaching the cell body, $k_{E} = 2 π / λ$ . The boundary escaping swimmer has a shorter flagellum, $L = 5 \bar{a}$ . Figure reproduced from Shum et al. [12]. (b) A depiction of the swimmer near a channel corner. The velocity of the swimmer, averaged over a revolution of the flagellum, is determined by its position and orientation relative to the walls. Up to symmetry, this configuration is specified by $y^{B}, z^{B}$ , and $e_{1}^{B}$ .
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Figure 2
Subdivision of the space near the corner of a large channel. The vertical and horizontal walls at this edge are shown in thick, solid lines. Dashed lines indicate the division of the fluid space into regions 1–4 and other regions that we do not explicitly analyze. The dynamics in the marked half space and free space regions can be inferred from previous work. The space between the walls and the enumerated regions is largely inaccessible because the finite volume occupied by the swimmer intersects or closely approaches the walls when the junction position $x^{B}$ is in this space. The model bacteria are displayed to scale in region 4, oriented parallel to the channel $x$ axis (left) and facing towards the vertical wall (right). The swimmers on the top, with shorter flagella, are boundary escapers, and the swimmers on the bottom are boundary accumulators.
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Figure 3
Representative trajectories near a channel corner for (a) boundary accumulating and (b) boundary escaping swimmers. The channel walls are drawn schematically as dark gray strips connected by a rounded corner. For each of the four regions of the corner, delimited by thin, black borders, we show examples of common outcomes for random trajectories entering the region. Starting locations for trajectories are indicated by crosses. Within each region, dashed (green) curves are used for trajectories that start on the left region boundary, dot-dashed (orange) curves start on the bottom, solid (blue) curves start on the right, and dotted (red) curves start on the top. Arrows indicate direction of motion along trajectories prior to exiting from the region, while the absence of an arrow from a trajectory entails that the bacterium becomes trapped within the respective region. For clarity, the four channel corner regions are drawn slightly separated (by light gray strips) and the axis scales are magnified by a factor of four below the separations at $y / \bar{a} = 2$ and $z / \bar{a} = 2$ , respectively.
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Figure 4
Trajectories of a boundary accumulating bacterium relaxing to a periodic orbit in region 3 of the channel corner. (a) $y − z$ projections of the trajectories. Three initial conditions, marked by crosses, are used to compute trajectories of the 4D ODE system (8). All three trajectories converge to the same figure-8 orbit, plotted with a thick, gray curve. The direction of motion around the orbit is indicated by sequentially numbered arrows. Also shown (solid, black curve) is the trajectory starting from the uppermost initial condition, computed using second-order time stepping with velocities obtained directly by the BEM without phase averaging. The vertical scale is expanded by a factor of five relative to the horizontal scale to magnify the small variations in $z$ . (b) $η − ζ$ projections of the trajectories plotted in the same manner as in (a). Positive values of $η$ correspond to the bacterium facing the left and positive values of $ζ$ signify that the swimmer points downward. The numbered points around the periodic orbit correspond to the numbering in (a). (c) The top-down $(x − y)$ view of the trajectories shown in (a) and (b). The $x$ and $y$ axes are set to the same scale and the leftward moving model bacterium is illustrated to scale at around $x / \bar{a} = - 200$ .
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Figure 5
Trajectories of (a) boundary accumulating and (b) boundary escaping bacteria simulated with the interpolation technique in the combined channel corner incorporating regions 1–4. Five trajectories are shown for each swimmer, starting from the same set of initial positions, marked by crosses. In all cases, the initial orientation was fixed with $η_{0} = ζ_{0} = 0.2$ so that the swimmer was pointing obliquely toward the walls. In (a), one trajectory terminates at the arrow closest to the corner due to the swimmer colliding with the lower wall. All other trajectories either escape from the corner region or converge to the corner accumulating orbit.
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