Thomas R. Powers

The hydrodynamics of swimming microorganisms

Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming, tens of micrometers and below. At this scale, inertia is unimportant and the Reynolds number is small. Our emphasis is on the simple physical picture and fundamental flow physics phenomena in this regime. We first give a brief overview of the mechanisms for swimming motility, and of the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming such as resistance matrices for solid bodies, flow singularities and kinematic requirements for net translation. Then we review classical theoretical work on cell motility, in particular early calculations of swimming kinematics with prescribed stroke and the application of resistive force theory and slender-body theory to flagellar locomotion. After examining the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers and the optimization of locomotion strategies. (Some figures in this article are in colour only in the electronic version) This article was invited by Christoph Schmidt.

Fluid-membrane tethers: Minimal surfaces and elastic boundary layers

Thin cylindrical tethers are common lipid bilayer membrane structures, arising in situations ranging from micromanipulation experiments on artificial vesicles to the dynamic structure of the Golgi apparatus. We study the shape and formation of a tether in terms of the classical soap-film problem, which is applied to the case of a membrane disk under tension subject to a point force. A tether forms from the elastic boundary layer near the point of application of the force, for sufficiently large displacement. Analytic results for various aspects of the membrane shape are given.

Force-free swimming of a model helical flagellum in viscoelastic fluids

We precisely measure the force-free swimming speed of a rotating helix in viscous and viscoelastic fluids. The fluids are highly viscous to replicate the low Reynolds number environment of microorganisms. The helix, a macroscopic scale model for the bacterial flagellar filament, is rigid and rotated at a constant rate while simultaneously translated along its axis. By adjusting the translation speed to make the net hydrodynamic force vanish, we measure the force-free swimming speed as a function of helix rotation rate, helix geometry, and fluid properties. We compare our measurements of the force-free swimming speed of a helix in a high-molecular weight silicone oil with predictions for the swimming speed in a Newtonian fluid, calculated using slender-body theories and a boundary-element method. The excellent agreement between theory and experiment in the Newtonian case verifies the high accuracy of our experiments. For the viscoelastic fluid, we use a polymer solution of polyisobutylene dissolved in polybutene. This solution is a Boger fluid, a viscoselastic fluid with a shear-rate-independent viscosity. The elasticity is dominated by a single relaxation time. When the relaxation time is short compared to the rotation period, the viscoelastic swimming speed is close to the viscous swimming speed. As the relaxation time increases, the viscoelastic swimming speed increases relative to the viscous speed, reaching a peak when the relaxation time is comparable to the rotation period. As the relaxation time is further increased, the viscoelastic swimming speed decreases and eventually falls below the viscous swimming speed.

Bacterial rheotaxis

The motility of organisms is often directed in response to environmental stimuli. Rheotaxis is the directed movement resulting from fluid velocity gradients, long studied in fish, aquatic invertebrates, and spermatozoa. Using carefully controlled microfluidic flows, we show that rheotaxis also occurs in bacteria. Excellent quantitative agreement between experiments with Bacillus subtilis and a mathematical model reveals that bacterial rheotaxis is a purely physical phenomenon, in contrast to fish rheotaxis but in the same way as sperm rheotaxis. This previously unrecognized bacterial taxis results from a subtle interplay between velocity gradients and the helical shape of flagella, which together generate a torque that alters a bacteriums swimming direction. Because this torque is independent of the presence of a nearby surface, bacterial rheotaxis is not limited to the immediate neighborhood of liquid–solid interfaces, but also takes place in the bulk fluid. We predict that rheotaxis occurs in a wide range of bacterial habitats, from the natural environment to the human body, and can interfere with chemotaxis, suggesting that the fitness benefit conferred by bacterial motility may be sharply reduced in some hydrodynamic conditions.

A macroscopic scale model of bacterial flagellar bundling

Escherichia coli and other bacteria use rotating helical filaments to swim. Each cell typically has about four filaments, which bundle or disperse depending on the sense of motor rotation. To study the bundling process, we built a macroscopic scale model consisting of stepper motor-driven polymer helices in a tank filled with a high-viscosity silicone oil. The Reynolds number, the ratio of viscous to elastic stresses, and the helix geometry of our experimental model approximately match the corresponding quantities of the full-scale E. coli cells. We analyze digital video images of the rotating helices to show that the initial rate of bundling is proportional to the motor frequency and is independent of the characteristic relaxation time of the filament. We also determine which combinations of helix handedness and sense of motor rotation lead to bundling.

Theory of Swimming Filaments in Viscoelastic Media

Motivated by our desire to understand the biophysical mechanisms underlying the swimming of sperm in the non-Newtonian fluids of the female mammalian reproductive tract, we examine the swimming of filaments in the nonlinear viscoelastic upper convected Maxwell model. We obtain the swimming velocity and hydrodynamic force exerted on an infinitely long cylinder with prescribed beating pattern. We use these results to examine the swimming of a simplified sliding-filament model for a sperm flagellum. Viscoelasticity tends to decrease swimming speed, and changes in the beating patterns due to viscoelasticity can reverse swimming direction.

Twirling and Whirling: Viscous Dynamics of Rotating Elastic Filaments

Motivated by diverse phenomena in cellular biophysics, including bacterial flagellar motion and DNA transcription and replication, we study the overdamped nonlinear dynamics of a rotationally forced filament with twist and bend elasticity. Competition between twist injection, twist diffusion, and writhing instabilities is described by coupled PDEs for twist and bend evolution. Analytical and numerical methods elucidate the twist/bend coupling and reveal two regimes separated by a Hopf bifurcation: (i) diffusion-dominated axial rotation, or twirling, and (ii) steady-state crankshafting motion, or whirling. The consequences of these phenomena for self-propulsion are investigated, and experimental tests proposed.

Dynamics of filaments and membranes in a viscous fluid

Motivated by the motion of biopolymers and membranes in solution, this article presents a formulation of the equations of motion for curves and surfaces in a viscous fluid. We focus on geometrical aspects and simple variational methods for calculating internal stresses and forces, and we derive the full nonlinear equations of motion. In the case of membranes, we pay particular attention to the formulation of the equations of hydrodynamics on a curved, deforming surface. The formalism is illustrated by two simple case studies: (1) the twirling instability of straight elastic rod rotating in a viscous fluid, and (2) the pearling and buckling instabilities of a tubular liposome or polymersome.

Locomotion of Helical Bodies in Viscoelastic Fluids: Enhanced Swimming at Large Helical Amplitudes

The motion of a rotating helical body in a viscoelastic fluid is considered. In the case of force-free swimming, the introduction of viscoelasticity can either enhance or retard the swimming speed and locomotive efficiency, depending on the body geometry, fluid properties, and the body rotation rate. Numerical solutions of the Oldroyd-B equations show how previous theoretical predictions break down with increasing helical radius or with decreasing filament thickness. Helices of large pitch angle show an increase in swimming speed to a local maximum at a Deborah number of order unity. The numerical results show how the small-amplitude theoretical calculations connect smoothly to the large-amplitude experimental measurements.

Separation of Microscale Chiral Objects by Shear Flow

We show that plane parabolic flow in a microfluidic channel causes nonmotile, helically shaped bacteria to drift perpendicular to the shear plane. Net drift results from the preferential alignment of helices with streamlines, with a direction that depends on the chirality of the helix and the sign of the shear rate. The drift is in good agreement with a model based on resistive force theory, and separation is efficient (>80%) and fast (<2 s). We estimate the effect of Brownian rotational diffusion on chiral separation and show how this method can be extended to separate chiral molecules.