Response of monoflagellate pullers to a shearing flow: A simulation study of microswimmer guidance

Microscale swimming may be intuited to be dominated by background flows, sweeping away any untethered bodies with the prevalent flow direction. However, it has been observed that many microswimmers utilize ambient flows as guidance cues, in some cases resulting in net motion upstream, contrary to the dominant background fluid direction and our accompanying intuition. Thus the hydrodynamic response of small-scale motile organisms requires careful analysis of the complex interaction between swimmer and environment. Here we investigate the effects of a Newtonian shear flow on monoflagellated swimmers with specified body symmetry, representing, for instance, the Leishmania mexicana promastigote, a parasitic hydrodynamic puller that inhabits the microenvironment of a sandfly vector midgut and is the cause of a major and neglected human tropical disease. We observe that a lack of symmetry-breaking cellular geometry results in the periodic tumbling of swimmers in the bulk, with the rotations exhibiting a linear response to changes in shearing rate enabling analytic approximation. In order to draw comparisons with the better-studied pushers, we additionally consider virtual Leishmania promastigotes in a confined but typical geometry, that of a no-slip planar solid boundary, and note that in general stable guided taxis is not exhibited amongst the range of observed behaviors. However, a repulsive boundary gives rise to significant continued taxis in the presence of shearing flow, a phenomenon that may be of particular pertinence to the infective lifecycle stage of such swimmers subject to the assumption of a Newtonian medium. We finally propose a viable and general in vitro method of controlling microswimmer boundary accumulation using temporally evolving background shear flows, based on the analysis of phase-averaged dynamics and verified in silico.

Figure 1

Computational representation of a monoflagellate. We model monoflagellated swimmers with an axisymmetric ellipsoidal body and an attached flagellum, as shown here for a virtual promastigote. The attachment location is denoted ${\mathit{x}}_0\left(t\right)$ in the laboratory reference frame ${\widehat{x}}_1{\widehat{x}}_2{\widehat{x}}_3$, and forms the origin of the swimmer-fixed reference frame $x_1{x}_2{x}_3$. Planar boundaries will typically be specified by ${\widehat{x}}_1=\text{const}$, with accompanying swimmer separation $h$ and orientation $\theta$ shown.

Figure 2

Describing planar flagellate motion in the bulk. Shown for a virtual promastigote, we define $\theta$ to be the clockwise angle between the swimmer-fixed major axis, $x_1$, and the principal axis of the shear flow, ${\widehat{x}}_1$, with the flow specified in dimensional form as ${\mathit{u}}_b={\gamma }_d({\widehat{x}}_2,0,0)$, expressed in and relative to the laboratory frame. The monoflagellate's beating plane lies in the plane of the background shearing flow (see Appendix for results concerning the more general case).

Figure 3

Free-space angular evolution of virtual monoflagellates, as predicted by phase averaging, in comparison to passive ellipsoids. (a) Temporal evolution of the angular displacement of a virtual promastigote in the bulk over one period, for background shear flow of shearing rates between 0.2 and 1 ${\mathrm{s}}^{-1}$. Here the period of rotation is seen to increase as shearing rate decreases. (b) The Jeffery's orbit of an ellipsoidal particle in shear flow with the same period as the virtual promastigote is shown for ${\gamma }_d=1{\mathrm{s}}^{-1}$ (blue, solid), showing remarkably similar dynamics to the corresponding promastigote orbit (red, dashed). The residual plot is inset, displaying error on the order of ${10}^{-2}$. (c) The dimensional period of promastigote rotation as a function of dimensional shearing rate (black dots). The periods of the promastigote rotation exhibit the same dependence on shearing rate as Jeffery's orbits, as given by the smooth curve. Hence the interaction between the timescales of the background flow and flagellar beating are seemingly of little consequence in the phase-averaged system.

Figure 4

Period of rotation for virtual puller monoflagellates in shear flow of rate ${\gamma }_d=1{\mathrm{s}}^{-1}$, for a range of body lengthscales. Body lengthscale is shown as a scaling factor applied to the typical promastigote body parameters introduced in Sec. 2a, where a scaling of 1 corresponds to the typical Leishmania mexicana lengthscale. The maximum period of rotation is highlighted by the dashed vertical line and occurs for a scaling $\sim 1$, which corresponds approximately to a typical promastigote.

Figure 5

Swimmer trajectories computed from phase-averaged analysis, initialized at the origin of the laboratory frame in a background shear flow and shown after 2 $\mathrm{s}$. (a) Trajectories with initial orientation sampled from a uniform distribution. The density of endpoints provides a naive estimate of the probability of initially randomly oriented swimmers being in a given region after a specified time. (b) Trajectories where initial orientation is sampled according to the distribution of orientations during a Jeffery's orbit. A differing distribution of endpoints to (a) can be clearly seen, giving a markedly different but physically realistic quantification of how the location evolves for a swimmer with unknown orientation. Here the background shear flow is prescribed in the laboratory frame as ${\mathit{u}}_b={\gamma }_d({\widehat{x}}_2,0,0)$, where results are shown for ${\gamma }_d=1{\mathrm{s}}^{-1}$ and are qualitatively robust to changes in both shear rate and simulation length.

Figure 6

The shear-guided swimming path of a monoflagellate in a specified time-dependent shear. The phase-averaged approximation (red, dashed) is observed to closely resemble the long-time simulation of the swimmer (blue, solid), beginning from the origin of the laboratory frame with $x_1$ parallel to ${\widehat{x}}_1$. The locations of background flow reversal, corresponding to a specific time and switching the shearing rate from ${\gamma }_d=1{\mathrm{s}}^{-1}$ to ${\gamma }_d=-1{\mathrm{s}}^{-1}$, are highlighted on the paths (black, filled). The prescribed path is shown as a black dotted curve, seen to coincide with the phase-averaged approximation. Inset is the initial section of the trajectory. Quantitative agreement is improved with increased refinement of the phase-averaged and long-time simulations, but computation of the latter is prohibitively expensive. Overall good agreement over such a timescale (60 s) validates the phase-averaged approximation and the resulting method of path prediction.

Figure 7

Phase planes describing the phase-averaged dynamics of a large-bodied puller in shearing flow of rate ${\gamma }_d=1{\mathrm{s}}^{-1}$ near a no-slip planar boundary. The black dashed line separates off the region of the phase plane where configurations intersect with the boundary. (a) Approximate stable and unstable manifolds of the saddle in $0<\theta <\pi /2$ are shown as green (left) and red (right) curves, respectively. A sample trajectory exemplifying boundary collision is shown as a black curve, with the start and endpoints shown as a red circle and a blue diamond, respectively. The separatrix, partitioning the phase plane into collision or bulk-bound configurations, is shown as a dot-dashed blue curve. (b) The phase plane corresponding to a reversal of shear direction from (a). The separatrix corresponding to the original flow direction (blue, dot-dashed) is shown superimposed upon the phase plane, a mirror of the reversed separatrix (red, dashed). The cross-hatched region lying between the two curves includes those configurations whose characteristic long-time behavior will be changed by a shear reversal. The smoothed path of a sample long-time simulation in the reversed flow is shown (red, solid), which crosses the original separatrix from beneath. Following the crossing, if the flow direction is reversed, the dynamics become those represented in (a), changing the long-time behavior of the swimmer, with the trajectory following the direction of the large black arrow.

Figure 8

Motion of virtual promastigotes in a shear flow near a repulsive planar boundary. (a) With the $h$ nullcline displayed in green (approximately vertical) and the separatrix displayed as a blue dot-dashed curve, multiple trajectories are shown across the phase plane, with start and endpoints of full simulations being shown as red circles and blue diamonds, respectively. Shown in black are smoothed traces of motion, resultant of simulating the full dynamics, in addition to their phase-averaged continuations, demonstrating a repeated washout tumbling for configurations beginning near the saddle point, and quasiperiodic behavior for those around $\theta =3\pi /2$. (b) Results of a full simulation of motion, with the first and last frames displayed in red (lower) and blue (upper), respectively. (c) A region of the phase plane representing the motion of (b), with the endpoints of the full simulation highlighted. The $h$ nullcline is shown in green (approximately vertical), with the smoothed trajectory shown in black, dashed. At the end of the full simulation the repulsive boundary force becomes negligible, hence the dynamics then follow the phase plane on a collision-bound trajectory (black, solid).

Figure 9

Sample long-time simulations of the motion of a virtual promastigote near a planar wall in shear flow from various nonplanar initial configurations. (a)–(c) Projections of the motion onto the $h\text{−}\theta$ space, allowing direct comparison of trajectories with those of Figs. 7 and 8. The start and endpoints of the motion are shown as red circles and blue diamonds, respectively, with the projected, smoothed path displayed as a black curve and showing remarkable qualitative agreement with those presented in Figs. 7 and 8. Sample paths of the planar system are shown as red dashed curves for comparison. Here (a) and (b) correspond to the repulsive boundary of Fig. 8, whilst (c) is analogous to the passive boundary of Fig. 7. The trajectory shown in (c) is truncated at 30 s as the swimmer collides with the boundary. (d)–(f) The smoothed evolution of the swimmer Tait-Bryan angles corresponding to the trajectories of (a)–(c). Periodic nonplanar motion is exemplified in (d), whilst the angles corresponding to out-of-plane rotation in (e) and (f) can be seen to evolve in general on a slower timescale than the in-plane angle $\theta$, in agreement with the general observation of Ishimoto [10]. Initial conditions are (a) $(\psi ,\varphi ,\theta )=(-1.40,0.02,-1.31)$, (b) $(\psi ,\varphi ,\theta )=(-1.57,0,1.57)$, and (c) $(\psi ,\varphi ,\theta )=(-2.10,-0.05,1.58)$, with angles given in radians and $h=5.5\mu \mathrm{m}$ in all cases. Discontinuities in (e) and (f) arise as the range of $\theta$ is $[-\pi ,\pi )$.

Figure 10

Sample long-time simulations of the motion of a virtual promastigote in the bulk in a background shear flow. The smoothed evolution of the planar Tait-Bryan angle $\theta$ is shown for a swimmer oriented within the plane of the shear flow (red, solid) and for a swimmer with a nonplanar initial configuration (black, dashed), demonstrating good agreement over a long timescale. Initially $(\psi ,\varphi ,\theta )=(-0.79,0,0)$ for the nonplanar swimmer, with all angles being given in radians. The discontinuity in the plot simply arises as the range of $\theta$ is $[-\pi ,\pi )$.