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.

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.

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 bacterium's 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. low Reynolds number | directional motion | chirality T he effectiveness and benefit of motility are largely determined by the dependence of movement behavior on environmental stimuli. For example, chemical stimuli may affect the spreading of tumor cells (1) and allow bacteria to increase uptake by swimming toward larger nutrient concentrations (2, 3), whereas hydrodynamic stimuli can stifle phytoplankton migration (4), allow protists to evade predators (5), and change sperm-egg encounter rates for external fertilizers (6). Microorganisms exhibit a broad range of directed movement responses, called "taxes". Whereas some of these responses, such as chemotaxis (7) and thermotaxis (8), are active and require the ability to sense and respond to the stimulus, others, such as magnetotaxis (9) and gyrotaxis (4), are passive and do not imply sensing, instead resulting purely from external forces.Chemotaxis is the best studied among these directional motions: Bacteria measure chemical concentrations and migrate along gradients (Fig. 1A). For instance, chemotaxis guides Escherichia coli to epithelial cells in the human gastrointestinal tract, favoring infection (10); Rhizobium bacteria to legume root hairs in soil, favoring nitrogen fixation (2); and marine bacteria to organic matter, favoring remineralization (3). Equally as pervasive as chemical gradients in microbial habitats are gradients in ambient fluid velocity or "shear" (Fig. 1B). Although nearly every fluid environment experiences velocity gradientsfrom laminar shear in bodily conduits and soil to turbulent shear in streams and oceans-the effect of velocity gradients on bacterial motility has received negligible attention compared with chemical gradients, partly due to the difficulty of studying motility under controlled flow conditi...

Point-like motile topological defects control the universal dynamics of diverse twodimensional active nematics ranging from shaken granular rods to cellular monolayers. A comparable understanding in higher dimensions has yet to emerge. We report the creation of three-dimensional active nematics by dispersing extensile microtubule bundles in a passive colloidal liquid crystal. Light-sheet microscopy reveals the millimeter-scale structure of active nematics with a single bundle resolution and the temporal evolution of the associated nematic director field. The dominant excitations of three-dimensional active nematics are extended charge-neutral disclination loops that undergo complex dynamics and recombination events. These studies introduce a distinct class of non-equilibrium systems whose turbulent-like dynamics arises from the interplay between internally generated active stresses, the chaotic flows and the topological structure of the constituent defects.

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.Comment: 26 pages, 12 figures, to be published in Reviews of Modern Physic

Evolution from unicellular organisms to larger multicellular ones requires matching their needs to the rate of exchange of molecular nutrients with the environment. This logistic problem poses a severe constraint on development. For organisms whose body plan is a spherical shell, such as the volvocine green algae, the current (molecules per second) of needed nutrients grows quadratically with radius, whereas the rate at which diffusion alone exchanges molecules grows linearly, leading to a bottleneck radius beyond which the diffusive current cannot meet metabolic demands. By using Volvox carteri, we examine the role that advection of fluid by the coordinated beating of surface-mounted flagella plays in enhancing nutrient uptake and show that it generates a boundary layer of concentration of the diffusing solute. That concentration gradient produces an exchange rate that is quadratic in the radius, as required, thus circumventing the bottleneck and facilitating evolutionary transitions to multicellularity and germ-soma differentiation in the volvocalean green algae.advection ͉ multicellularity ͉ Volvox

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 forcefree 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-rateindependent 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. motility | propulsion | rheology S mall motile organisms often swim in complex fluids. Mammalian spermatozoa beat their flagella to move through cervical fluid (1). The Lyme disease spirochete Borrelia burgdorferi flexes and rotates its body to move through the extracellular matrix of our skin (2). The nematode Caenorhabditis elegans undulates its body to move through soil saturated with water (3). While there is an extensive framework for understanding the mechanics of swimming of small organisms in purely viscous Newtonian liquids such as water, our understanding of the basic principles of swimming in non-Newtonian fluids is still in its infancy (4). The behavior of complex fluids is varied, and there are many possible nonNewtonian effects a swimmer could encounter, including elastic response of the fluid, shear-dependent viscosity, adhesion to suspended particles or fibers, or the permeability of a porous medium. In this article we focus on swimming in an elastic liquid, and our goal is to determine how the speed of a model bacterial swimmer is changed by elastic effects.It is known that helically shaped bacteria such as Leptospira or B. burgdorferi swim more rapidly in solutions with methylcellulose than in nonviscoelastic solutions of the same viscosity (2, 5). On the other hand, C. elegans, which moves using planar undulations of its body, swims more slowly in a viscoelastic fluid than in a vi...

Motivated by 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.The physical environment of the cell places severe constraints on mechanisms for motility. For example, viscous effects dominate inertial effects in water at the scale of a few microns. Therefore, swimming cells use viscous resistance to move, since mechanisms that rely on imparting momentum to the surrounding fluid, such as waving a rigid oar, do not work [1,2]. The fundamental principles of swimming in the low-Reynolds number regime of small-scale, slow flows have been established for many years [2,3,4,5], yet continue to be an area of active research. However, when a sperm cell moves through the viscoelastic mucus of the female mammalian reproductive tract, the theory of swimming in a purely viscous fluid is inapplicable. Observations of sperm show that they are strongly affected by differences between viscoelastic and viscous fluids. In particular, the shape of the flagellar beating pattern as well as swimming trajectories and velocities depend on the properties of the medium [6,7,8].The interplay of medium properties and flagellar motility or transport also arises in other situations, such as spirochetes swimming in a gel [9], and cilia beating in mucus to clear foreign particles in the human airway [10]. Motivated by these phenomena, we develop a theory for swimming filaments in a viscoelastic medium. We begin by analyzing the swimming of an infinite filament with a prescribed beating pattern in a fluid described by the Upper Convected Maxwell (UCM) model [11]. We deduce the hydrodynamic force per unit length acting on the filament and the swimming velocity to leading order in the deformation of the filament. Our results extend the findings of Lauga [12], who considered a variety of fading memory models for the case of a prescribed beat pattern on a planar sheet. We further apply our results to a simple model flagellum with active internal forces, and find that changes in flagellum shapes play a crucial role in distinguishing the effects of viscoelastic media.Newtonian fluids are characterized by a simple constitutive relation, in which stress is proportional to strain rate. Non-newtonian fluids cannot be characterized by a simple universal constitutive relation, and exhibit a range of phenomena such as elasticity, shear thinning, and yield stress behavior. We choose to focus our attention on fluids with fading memory, in which the stress relaxes over time to the viscous stress. We consider small amplitude deflections of an infinite filam...