Viscotaxis: Microswimmer Navigation in Viscosity Gradients

The survival of many microorganisms, like Leptospira or Spiroplasma bacteria, can depend on their ability to navigate towards regions of favorable viscosity. While this ability, called viscotaxis, has been observed in several bacterial experiments, the underlying mechanism remains unclear. We provide a framework to study viscotaxis of biological or synthetic self-propelled swimmers in slowly varying viscosity fields and show that suitable body shapes create viscotaxis based on a systematic asymmetry of viscous forces acting on a microswimmer. Our results shed new light on viscotaxis in Spiroplasma and Leptospira and suggest that dynamic body shape changes exhibited by both types of microorganisms may have an unrecognized functionality: to prevent them from drifting to low viscosity regions where they swim poorly. The present theory classifies microswimmers regarding their ability to show viscotaxis and can be used to design synthetic viscotactic swimmers, e.g., for delivering drugs to a target region distinguished by viscosity.

Figure 1

The sedimentation of (a) a sphere, (c) a dumbbell, and (d) more complex objects in a linear viscosity profile $\eta (y)={\eta }_0+\lambda y$ (illustrated by the shading of the background color) universally follows (b) a $Y(t)\propto \sqrt{t}$ law at late times, where $Y$ is the y component of the hydrodynamic center of mass [the long blue arrow in (d)]. (b) shows the sedimentation dynamics based on numerical integration of Eqs. (1) and (2) of a single sphere [the two (red) curves which merge corresponding to two different initial $Y$ values], a symmetric dumbbell (blue, top curve at late times), and an irregular triangle (green, lower curve) (using arbitrary parameters). The black dashed line shows the $y(t)=\sqrt{t}$ line for comparison.

Figure 2

Viscotaxis in linear swimmers. (a) Uniaxial swimmers do not show viscotaxis and move in the direction of their initial orientation [typical trajectory obtained by solving Eqs. (1) and (2) numerically in arbitrary units shown in (c)]. (b) Nonuniaxial swimmers generically show viscotaxis which is always positive for linear swimmers [typical trajectory in (d)]. (e) Spiroplasma and (f) Leptospira bacteria changing their body shape from roughly uniaxial to nonuniaxial shapes (snapshots from Refs. [35, 39]). (g) Mechanism of viscotaxis. The viscous drag acting on body parts at high viscosity (sphere 1) dominates the drag acting on spheres at low viscosity (sphere 2), which turns the swimmer up the gradient.

Figure 3

(a) Viscotaxis in chiral swimmers can be (b) positive [typical trajectory obtained by solving Eqs. (1) and (2) numerically in arbitrary units] or (c) negative. Negative viscotaxis emerges from the fact that (c),(d) back-driven swimmers move faster down the gradient than upwards and systematically overshoot down the gradient while circling (see the text for details).