Bundled slender-body theory for elongated geometries in swimming bacteria

Size and shape of a living microorganism have been recognized as important factors for its movement through a viscous fluid. Understanding how subtle variations in cellular geometry affect the hydrodynamic forces requires solving three-dimensional (3D) Stokes equations, e.g., by resolving an object's 2D surface in a boundary integral method. A reduction of these computational costs involves using available symmetries to simplify the boundary geometries, such as representing a slender body by its body centerline, known as the slender-body theory. Here, we extend the range of the aspect ratio that can be treated by a standard slender-body theory, by representing the body with a bundle of thin filaments that each still approximately satisfies the slender-body criteria. We show that this bundled slender-body theory can be used to determine the dependency of hydrodynamic forces on varying geometric factors of a moving object. As a direct application of this method, we study the optimized kinematics of a monotrichous bacterium that has a curved cell body and swims by rotating a helical flagellum. We show that the curvature in its cell body can play a nontrivial role in the swimming motility, depending on the chirality emerging from the cell-flagellum alignment.

Figure 1

Schematic presentation of the slender-body theory (SBT). The zoomed-in view shows a circular cross section of a slender body with a radius $a$, displaced by an arc length $s$ along its body centerline. Unit vectors $(\widehat{\mathbf{i}},\widehat{\mathbf{j}},\widehat{\mathbf{k}})$ define a local coordinate system, with the tangential direction of the body centerline along $\widehat{\mathbf{i}}$ and the translational direction lying in the “$\widehat{\mathbf{i}}\text{−}\widehat{\mathbf{j}}$” plane. In the SBT, a linear integral of the singular force $\mathbf{f}\left(s\right)$, known as the stokeslet, along the entire body centerline contributes to the flow velocity located at $\mathbf{x}$ through a Green's function.

Figure 2

(a),(b) Distribution of hydrodynamic forces $\mathbf{f}\left(s\right)$ along the body centerline of a prolate spheroid (with a slenderness $a/b=2a/L=0.1$ and a pitch angle ${\theta }_0=\pi /4$) that translates (a) and rotates (b) along its helical axis. The results from the slender body theory (solid lines) and the boundary element method (dashed lines) are both shown as a comparison. The insets show the geometry and motion of the spheroid. (c),(d) The same computation for a thicker spheroid (with a slenderness $a/b=0.2$).

Figure 3

Schematic drawing of a bSBT representation of an elongated body with a finite aspect ratio. (a) A prolate spheroid (with a slenderness $a/b=0.25$) is modeled as three fine filaments that can be accessed by the SBT. A view of the middle cross section of this body is shown in (b).

Figure 4

Translational and rotational drag on a translating and rotating spheroid with its slenderness $a/b<1$. (a),(b) The force and torque along the longitudinal (blue) and transverse (red) directions, as obtained from the SBT (solid lines). (c),(d) The force and torque along the longitudinal (blue) and transverse (red) directions, as obtained from the bSBT (solid lines) for $N_s=3$ subunits. (e),(f) The force and torque along the longitudinal (blue) and transverse (red) directions, as obtained from the bSBT (solid lines) for $N_s=6$ subunits. In (a)–(f), the analytic solutions are shown (in dashed lines) as comparisons.

Figure 5

Resistance matrices of a helical filament at various aspect ratio $a/b$, as computed by the SBT (a) and the bSBT with $N_s=6$ subunits (b). The lines in different colors show the $\xi$ (solid blue), $\sigma$ (solid red), $\kappa$ (dashed green), and ${\kappa }^{\prime }$ (solid green) elements in the computed matrices [Eq. (17)]. The “exact” solutions as obtained through the BEM are shown in symbols “$\circ$” as comparisons.

Figure 6

Relative error in the bSBT results. (a) For a half-period helical spheroid with ${\theta }_0=\pi /4$, the relative errors of the bSBT are computed with different number of subunits $N_s$ (insets) for two aspect ratios ($a/b=0.3$ and $a/b=0.5$). Here, the fraction factor $\varphi =0.3$. (b) Same computation as (a) for a different helical symmetry (${\theta }_0=\pi /6$). (c) For same geometry as (a), the relative errors by bSBT are shown at various $\varphi$ (insets) in both the $N_s=3$ and $N_s=6$ cases. (d) Same computation as (c) for a different helical symmetry (${\theta }_0=\pi /6$).

Figure 7

Comparisons of the flow field of bSBT with the “exact” solutions (obtained by the BEM). (a),(b) The flow field due to a prolate spheroid (with the aspect ratio 2:1 and translating at the speed $V$) from the bSBT and the exact solution, respectively. (d),(e) The flow field due to the same prolate spheroid rotating at the angular speed $\omega$ from the bSBT and the exact solution, respectively. (c),(f) The deviation of the bSBT velocity fields from the exact solutions. The no-slip boundary conditions are roughly satisfied (within a 10% discrepancy) at the spheroidal surfaces.

Figure 8

Hydrodynamic model of a monotrichous bacterium. (a) A schematic diagram of a bacterium with a spheroidal cell body and swimming by rotating a single flagellum. The flagellum is aligned along an arbitrary direction relative to the cell axis, determined by a joint angle $\beta$. The relative movement of the flagellum to the counterrotating cell body ${\mathit{\omega }}_f-{\mathit{\omega }}_c$ is prescribed by a rotation along the helical axis (${\widehat{\mathbf{x}}}^{\prime }$) at the motor speed (${\omega }_0$). A 3D trace of the centerline of the cell body with the motor running over 100 revolutions is obtained by the bSBT (b) and the BEM (c). Here, the cell body has a spheroidal shape with its length $L=2\mu \mathrm{m}$ and radius $a=0.25\mu \mathrm{m}$, and the joint angle $\beta =0.8$ rad. For the BEM, the number of mesh points is set as $2\times {10}^4$.

Figure 9

Computed flow field surrounding a swimming bacterium through the bSBT with $L=2\mu \mathrm{m}, a=0.25\mu \mathrm{m}$, and $\beta =0$. (a) An example of the instantaneous flow field in the $x\text{−}y$ plane exhibits nonaxial symmetry along the swimming direction, corresponding to a time-dependent flow structure. (b) A time-averaged flow from (a) shows an axial-symmetric flow structure associated with a force-dipole moment. Here, the colors show the flow speed $u_{x\text{−}y}$ as normalized by the wave speed ${\omega }_0/k$ of the flagellum. The corresponding streamlines are shown in solid curves.

Figure 10

Curved cell body and its alignment relative to the flagellar axis. While indistinguishable from side views (a), the cell body exhibits either the right-handed (b) or the left-handed (c) chiralities, indicated by a blue arrow moving along its body centerline from the joint to the free end. The dashed circles in blue show the radius of the curvature $r_c$ and the 2D planes where the circular arc lies. Here, the helical flagellum is right handed, indicated by a red arrow oriented along the helix from its back to its front.

Figure 11

Motilities of bacteria with curved cell bodies depend on the detailed cell alignments, characterized by the joint angle and the chirality due to the curvature of the body. The left-handed chirality opposite to that of the flagellum, here right handed, leads to higher swimming motilities. A regular spheroid cell body is shown as a comparison.

Figure 12

Cell kinematics and swimming dynamics with identical curvature and joint angle ($\beta =0.8$ rad) but opposite chiralities in cell alignments. The body centerline of a left-handed cell body composes a double-helix-like envelope (a), while that of the right-handed one composes a complex self-intercepting mesh (b). (c) The time-averaged $x\text{−}y$ flow field surrounding the swimmer with left-handed cell alignment [upper and same as (a)] is more localized than that with the right-handed cell alignment [lower and same as (b)]. Here, the flow speed ($u_{x\text{−}y}$) is normalized by the wave speed of the flagellum (${\omega }_0/k$).