The hydrodynamics of swimming microorganisms

The hydrodynamics of swimming microorganisms

Abstract

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 microns and below). The focus is on the fundamental flow physics phenomena occurring in this inertia-less realm, and the emphasis is on the simple physical picture. We review 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: early calculations of the speed of a swimmer with prescribed stroke, and the application of resistive-force theory and slender-body theory to flagellar locomotion. After reviewing 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.

pacs:
47.63.-b, 47.63.Gd, 87.17.Jj, 87.18.Ed, 47.63.mf, 47.61.-k, 47.15.G-
1

1 Introduction

Our world is filled with swimming microorganisms: The spermatozoon that fuse with the ovum during fertilization, the bacteria that inhabit our guts, the protozoa in our ponds, and the algae in the ocean.

The reasons microorganisms move are familiar. Bacteria such as Escherichia coli detect gradients in nutrients and move to regions of higher concentration [1]. The spermatozoa of many organisms swim to the ovum, sometimes in challenging environments such as tidal pools in the case of sea urchins or cervical mucus in the case of humans [2]. Paramecium cells swim to evade predator rotifers.

What is perhaps less familiar is the fact that the physics governing swimming at the micron scale is different from the physics of swimming at the macroscopic scale. The world of microorganisms is the world of low “Reynolds number,” a world where inertia plays little role and viscous damping is paramount. As we describe below, the Reynolds number is defined as , where is the fluid density, is the viscosity, and and are a characteristic velocity and length scale of the flow, respectively. Swimming strategies employed by larger organisms that operate at high Reynolds number, such as fish, birds, or insects [3, 4, 5, 6, 7, 8], do not work at the small scale. For example, any attempt to move by imparting momentum to the fluid, as is done in paddling, will be foiled by the large viscous damping. Therefore microorganisms have evolved propulsion strategies that successfully overcome and exploit drag. The aim of this review is to explain the fundamental physics upon which these strategies rest.

The study of the physics of locomotion at low Reynolds number has a long history. In 1930, Ludwig [9] pointed out that a microorganism that waves rigid arms like oars is incapable of net motion. Over the years there have been many classic reviews, from the general perspective of animal locomotion [10], from the perspective of fluid dynamics at low Reynolds number [11, 12, 13, 14, 3, 15, 16], and from the perspective of the biophysics and biology of cell motility [17, 18, 19, 20, 21, 1]. Nevertheless, the number of publications in the field has grown substantially in the past few years. This growth has been spurred in part by new experimental techniques for studying cell motility. Traditionally, motile cells have been passively observed and tracked using light microscopy. This approach has led to crucial insights such as the nature of the chemotaxis strategy of E. coli [1]. Advances in visualization techniques, such as the fluorescent staining of flagella [22] in living, swimming bacteria, continue to elucidate the mechanics of motility. A powerful new contribution is the ability to measure forces at the scale of single organisms and single motors. For example, it is now possible to measure the force required to hold a swimming spermatozoon [23, 24, 25], algae [26] or bacterium [27] in an optical trap. Atomic force microscopy also allows direct measurement of the force exerted by cilia [28]. Thus the relation between force and the motion of the flagellum can be directly assessed. These measurements of force allow new approaches to biological questions, such the heterogeneity of motor behaviour in genetically identical bacteria. Measurements of force together with quantitative observation of cell motion motivate the development of detailed hydrodynamic theories that can constrain or rule out models of cell motion.

The goal of this review is to describe the theoretical framework for locomotion at low Reynolds number. Our focus is on analytical results, but our aim is to emphasize physical intuition. In §2, we give some examples of how microorganisms swim. After a brief general review of low-Reynolds number hydrodynamics (§3), we outline the fundamental properties of locomotion without inerti (§4). We then discuss the classic contributions of Taylor [29], Hancock [30] and Gray [31], who all but started the field more than 50 years ago (§5); we also outline many of the subsequent works that followed. We proceed by introducing the different ways to physically actuate a flagella-based swimmer (§6). We then move on to introduce topics of active research. These areas include the role of hydrodynamic interactions, such as the interactions between two swimmers, or between a wall and a swimmer (§7); locomotion in non-Newtonian fluids such as the mucus of the female mammalian reproductive tract (§8); and the design of artificial swimmers and the optimization of locomotion strategies in an environment at low Reynolds number (§9). Our coverage of these topics is motivated by intellectual curiosity and the desire to understand the fundamental physics of swimming; the relevance of swimming in biological processes such as reproduction or bacterial infection; and the practical desire to build artificial swimmers, pumps, and transporters in microfluidic systems.

Our review is necessarily limited to a small cross-section of current research. There are many closely related aspects of “life at low Reynolds number” that we do not address, such as nutrient uptake or quorum sensing; instead we focus on flow physics. Our hope is to capture some of the current excitement in this research area, which lies at the intersection of physics, mechanics, biology, and applied mathematics, and is driven by clever experiments that shed a new light on the hidden world of microorganisms. Given the interdisciplinary nature of the subject, we have tried to make the review a self-contained starting point for the interested student or scientist.

2 Overview of mechanisms of swimming motility

Figure 1: Sketches of microscopic swimmers, to scale. (a) E. coli. (b) C. crescentus. (c) R. sphaeroides, with flagellar filament in the coiled state. (d) Spiroplasma, with a single kink separating regions of right-handed and left-handed coiling. (e) Human spermatozoon. (f) Mouse spermatozoon. (g) Chlamydomonas. (h) A smallish Paramecium.

In this section we motivate our review with a short overview of mechanisms for swimming motility. We define a “swimmer” to be a creature or object that moves by changing its body shape in a periodic way. To keep the scope of the article manageable, we do not consider other mechanisms that could reasonably be termed “swimming,” such as the polymerization of the actin of a host cell by pathogens of the genus Listeria [32], or the gas-vesicle mediated buoyancy of aquatic micoorganisms such as Cyanobacteria [33].

Many microscopic swimmers use one or more appendages for propulsion. The appendage could be a relatively stiff helix that is rotated by a motor embedded in the cell wall, as in the case of E. coli (Fig. 1a), or it could be a flexible filament undergoing whip-like motions due to the action of molecular motors distributed along the length of the filament, as in the sperm of many species [21] (Figs. 1e and f). For example, the organelle of motility in E. coli and Salmonella typhimurium is the bacterial flagellum, consisting of a rotary motor, a helical filament, and a hook which connects the motor to the filament  [34, 35, 36]. The filament has a diameter of  nm, and traces out a helix with contour length m. In the absence of external forces and moments, the helix is left-handed with a pitch m and a helical diameter [22]. There are usually several flagella per cell. When the motor turns counter-clockwise (when viewed from outside the cell body), the filaments wrap into a bundle that pushes the cell along at speeds of 25–35 m/s (see §7.3.2) [37]. When one or more of the motors reverse, the corresponding filaments leave the bundle and undergo “polymorphic” transformations in which the handedness of the helix changes; these polymorphic transformations can change the swimming direction of the cell [22].

There are many variations on these basic elements among swimming bacteria. For example, Caulobacter crescentus has a single right-handed helical filament (Fig. 1b), driven by a rotary motor that can turn in either direction. The motor preferentially turns clockwise, turning the filament in the sense to push the body forward [38]. During counterclockwise rotation the filament pulls the body instead of pushing. The motor of the bacterium Rhodobacter sphaeroides turns in only one direction but stops from time to time [39]. The flagellar filament forms a compact coil when the motor is stopped (Fig. 1c), and extends into a helical shape when the motor turns. Several archaea also use rotating flagella to swim, although far less is known about the archaea compared to bacteria. Archaea such as the various species of Halobacterium swim more slowly than bacteria, with typical speeds of 2–3 m/s [40]. Although archaeal flagella also have a structure comprised of motor, hook, and filament, molecular analysis of the constituent proteins shows that archaeal and bacterial flagella are unrelated (see [41] and references therein).

There are also bacteria that swim with no external flagellar filaments. The flagella of spirochetes lie in the thin periplasmic space between the inner and outer cell membranes [42]. The flagellar motors are embedded in the cell wall at both poles of the elongated body of the spirochete, and the flagellar filaments emerge from the motor and wrap around the body. Depending on the species, there may be one or many filaments emerging from each end of the body. In some cases, such as the Lyme disease spirochete Borrelia burgdorferi, the body of the spirochete is observed to deform as it swims, and it is thought that the rotation of the periplasmic flagella causes this deformation which in turn leads to propulsion [43, 44]. The deformation can be helical or planar. These bacteria swim faster in gel-like viscous environments than bacteria with external flagella [45, 46]. Other spirochetes, such as Treponema primitia, do not change shape at all as they swim, and it is thought that motility develops due to rotation of the outer membrane and cytoplasmic membrane in opposite senses [43, 47]. Finally, we mention the case of Spiroplasma, helically shaped bacteria with no flagella (Fig. 1d). These cells swim via the propagation of pairs of kinks along the length of the body [48]. Instead of periplasmic flagella, the kinks are thought to be generated by contraction of the cytoskeleton [49, 50, 51].

Eukaryotic flagella and cilia are much larger than bacterial flagella, with a typical diameter of 200 nm, and with an intricate internal structure [21]. The most common structure has nine microtubule doublets spaced around the circumference and running along the length of a flagellum or cilium, with two microtubules along the center. Molecular motors (dynein) between the doublets slide them back and forth, leading to bending deformations that propagate along the flagellum. There is a vast diversity in the beat pattern and length of eukaryotic flagella and cilia. For example, the sperm of many organisms consists of a head containing the genetic material propelled by a filament with a planar or even helical beat pattern, depending on the species [52]. The length of the flagellum is m in some Hymenoptera [53], m for hippos, m for humans [2] (Fig. 1e), m for mice (Fig. 1f), and can be 1 mm [54] or even several cm long in some fruit flies [55] (although in the last case the flagella are rolled up into pellets and offered to the female via a “pea-shooter” effect [55]).

Many organisms have multiple flagella. Chlamydomonas reinhardtii is an algae with two flagella that can exhibit both ciliary and flagellar beat patterns (Fig. 1g). In the ciliary case, each flagellum has an asymmetric beat pattern [21]. In the power stroke, each flagellum extends and bends at the base, sweeping back like the arms of a person doing the breaststroke. On the recovery stroke, the flagellum folds, leading as we shall see below to less drag. When exposed to bright light, the alga swims in reverse, with its two flagella extended and propagating bending waves away from the cell body as in the case of sperm cells described above [56]. Paramecium is another classic example of a ciliated microorganism. Its surface is covered by thousands of cilia that beat in a coordinated manner [57], propelling the cell at speeds of m/s (Fig. 1h). Arrays of beating cilia are also found lining the airway where they sweep mucus and foreign particles up toward the nasal passage [58].

3 Flows at low Reynolds number

3.1 General properties

We first briefly discuss the general properties of flow at low Reynolds numbers. For more detail we refer to the classic monographs by Happel and Brenner [59], Kim and Karilla [60], and Leal [61]; a nice introduction is also offered by Hinch [62], as well as a more formal treatment by Pozrikidis [63].

To solve for the force distribution on a organism, we need to solve for the flow field and pressure in the surrounding fluid. For an incompressible Newtonian fluid with density and viscosity , the flow satisfies the Navier-Stokes equations

(1)

with boundary conditions appropriate to the problem at hand. The Navier-Stokes equations are a pointwise statement of momentum conservation. Once and are known, the stress tensor is given by , and the force and torque acting on the body are found by integrating along its surface

(2)

The Reynolds number is a dimensionless quantity which qualitatively captures the characteristics of the flow regime obtained by solving Eq. (1), and it has several different physical interpretations. Consider a steady flow with typical velocity around a body of size . The Reynolds number is classically defined as the ratio of the typical inertial terms in the Navier-Stokes equation, , to the viscous forces per unit volume, . Thus, . A low-Reynolds number flow is one for which viscous forces dominate in the fluid.

A second interpretation can be given as the ratio of time scales. The typical time scale for a local velocity perturbation to be transported convectively by the flow along the body is , whereas the typical time scale for this perturbation to diffuse away from the body due to viscosity is . We see therefore that , and a low Reynolds number flow is one for which fluid transport is dominated by viscous diffusion.

We can also interpret as a ratio of forces on the body. A typical viscous stress on a bluff body is given by , leading to a typical viscous force on the body of the form . A typical inertial stress is given by a Bernoulli-like dynamic pressure, , and therefore an inertial force . We see that the Reynolds number is given by , and therefore in a low-Reynolds number flow the forces come primarily from viscous drag.

A fourth interpretation, more subtle, was offered by Purcell [14]. He noted that, for a given fluid, has units of force, and that any body acted upon by the force will experience a Reynolds number of unity, independent of its size. Indeed, it is easy to see that and , and therefore a body with a Reynolds number of one will have . A body moving at low Reynolds number experiences therefore forces smaller than ( nN for water).

What are the Reynolds numbers for swimming microorganisms [3]? In water ( kg/m, Pas), a swimming bacterium such as E. coli with m/s and m has a Reynolds number . A human spermatozoon with m/s and m moves with . The larger ciliates, such as Paramecium, have mm/s and m, and therefore  [13]. At these low Reynolds numbers, it is appropriate to study the limit , for which the Navier-Stokes equations (1) simplify to the Stokes equations

(3)

Since swimming flows are typically unsteady, we implicitly assume the typical frequency is small enough so that the “frequency Reynolds number” is also small. Note that Eq. (3) is linear and independent of time, a fact with important consequences for locomotion, as we discuss below.

Before closing this subsection, we point out an important property of Stokes flows called the reciprocal theorem. It is a principle of virtual work which takes a particularly nice form thanks to the linearity of Eq. (3). Consider a volume of fluid , bounded by a surface with outward normal , in which you have two solutions to Eq. (3), and . If the stress fields of the two flows are and , then the reciprocal theorem states that the mixed virtual works are equal:

(4)

3.2 Motion of solid bodies

When a solid body submerged in a viscous fluid is subject to a external force smaller than , it will move with a low Reynolds number. What is its trajectory? Since Eq. (3) is linear, the relation between kinetics and kinematics is linear. Specifically, if the solid body is subject to an external force , and an external torque , it will move with velocity and rotation rate satisfying

(5)

or the inverse relation

(6)

The matrix in Eq. (5) is the “resistance” matrix of the body, and the matrix of Eq. (6) is the “mobility” matrix. The reciprocal theorem (4) forces these matrices to be symmetric [59]. Dimensionally, since low-Re stresses scale as , the sub-matrices scale as , , , and similarly , , . For most problems, the details of the geometry of the body make these matrices impossible to calculate analytically. The simplest example is that of a solid sphere of radius , for which we have isotropic translational and rotational drag, , and ; the cross-couplings and vanish by symmetry.

Three important features of Eqs. (5-6) needed to be emphasized for their implications for locomotion. The first important property is drag anisotropy: The matrices , , , and need not be isotropic (proportional to ). As we discuss in §4, drag anisotropy is a crucial ingredient without which biological locomotion could not occur at low Reynolds number. For a simple illustration, consider a slender prolate spheroid of major axis and minor axis with . If denotes the direction along the major axis of the spheroid, we have , with and .

Secondly, there exist geometries for which the matrices and are non-zero: chiral bodies, which lack a mirror symmetry plane. In that case, there is the possibility of driving translational motion through angular forcing—this strategy is employed by bacteria with rotating helical flagella (see §6).

Thirdly, these matrices are important as they allow calculate the diffusion constants of solid bodies. The fluctuation-dissipation theorem states that, in thermal equilibrium at temperature , the translational diffusion constant of a solid body is given by the Stokes-Einstein relationship , where is Boltzmann’s constant, while the rotational diffusion constant is given by . The typical time scale for a body to move by diffusion along its own size is , while is the typical time scale for the reorientation of the cell by thermal forces alone. For a non-motile E. coli bacterium at room temperature, we have 0.1 m/s in water; while the time scale for thermal reorientation of the cell axis, , is a few minutes.

3.3 Flow singularities

Since the Stokes’ equations, Eq. (3), are linear, traditional mathematical methods to solve for flow and pressure fields can rely on linear superposition. The Green’s function to Stokes flow with a Dirac-delta forcing of the form is given by

(7)
(8)

The tensor is known as the Oseen tensor, and the fundamental solution, Eq. (7), is termed a stokeslet [30]. Physically, it represents the flow field due to a point force, , acting on the fluid at the position as a singularity. The velocity field is seen to decay in space as , a result which can also be obtained by dimensional analysis. Indeed, for a three-dimensional force acting on the fluid, and by linearity of Stokes’ flow, the flow velocity has to take the form , where is the angle between the direction of and , and where is the magnitude of the force. Dimensional analysis leads to with a decay.

An important property of the stokeslet solution for locomotion is directional anisotropy. Indeed, we see from Eq. (7) that if we evaluate the velocity in the direction parallel to the applied force, we obtain that , whereas the velocity in the direction perpendicular to the force is given by . For the same applied force, the flow field in the parallel direction is therefore twice that in the perpendicular direction (). Alternatively, to obtain the same velocity, one would need to apply a force in the perpendicular direction twice as large as in the parallel direction (). Such anisotropy, which is reminiscent of the anisotropy in the mobility matrix for long slender bodies (§3.2; see also §5.2) is at the origin of the drag-based propulsion method employed by swimming microorganisms (see §4.3).

From the fundamental solution above, Eq. (7), the complete set of singularities for viscous flow can be obtained by differentiation [64]. One derivative leads to force-dipoles, with flow fields decaying as . Two derivatives leads to source-dipole (potential flow also known as a doublet), and force-quadrupoles, with velocity decaying in space as . Higher-order singularities are easily obtained by subsequent differentiation.

A well-chosen distribution of such singularities can then be used to solve exactly Stokes’ equation in a variety of geometry. For example, the Stokes flow past a sphere is a combination of a stokeslet and a source-dipole at the center of the sphere [65]. For spheroids, the method was pioneered by Chwang and Wu [64], and we refer to Refs. [60, 66] for a textbook treatment. A linear superposition of singularities is also at the basis of the boundary integral method to computationally solve for Stokes flows using solely velocity and stress information at the boundary (see Refs. [63, 66]).

When a flow field is described by a number of different flow singularities, the singularity with the slowest spatial decay is the one that dominates in the far field. Since a cell swimming in a viscous fluid at low-Reynolds numbers is force- and torque-free (Eq. 9 below), the flow singularities that describe point-forces (stokeslets) and point-torques (antisymmetric force-dipole, or rotlets) cannot be included in the far-field description. As a result, the flow field far from a swimming cell is in general well represented by a symmetric force-dipole, or stresslet [67]. Such far-field behavior has important consequences on cell-cell hydrodynamic interactions as detailed in §7.1.

4 Life at low Reynolds number

We now consider the general problem of self-propelled motion at low Reynolds number. We call a body a “swimmer” if by deforming its surface it is able to sustain movement through fluid in the absence of external (non-hydrodynamic) forces and torques. Note that the “body” includes appendages such as the cilia covering a Paramecium or the helical flagella of E. coli.

4.1 Reinterpreting the Reynolds number

We first offer an alternative interpretation of the Reynolds number in the context of swimming motion. Let us consider a swimmer of mass and size swimming with velocity through a viscous fluid of density and viscosity . Suppose the swimmer suddenly stops deforming its body; it will then decelerate according to Newton’s law . What is the typical length scale over which the swimmer will coast due to the inertia of its movement? For motion at high Reynolds number, as in the case of a human doing the breaststroke, the typical drag is , leading to a deceleration . The swimmer coasts a length . If the swimmer has a density , we see that the dimensionless coasting distance is given by the ratio of densities, . A human swimmer in water can cruise for a couple of meters. In contrast, for motion at low Reynolds number, the drag force has the viscous scaling, , and the swimmer can coast a distance , where is the density of the swimmer. For a swimming bacterium such as E. coli, this argument leads to  nm [14]. For , The Reynolds number can therefore be interpreted as a nondimensional cruising distance.

A consequence of this analysis is that in a world of low-Reynolds number, the response of the fluid to the motion of boundaries is instantaneous. This conclusion was anticipated by our second interpretation of the Reynolds number (§3), where we saw that in the limit of very low , velocity perturbations diffuse rapidly relative to the rate at which fluid particles are carried along by the flow. To summarize, the rate at which the momentum of a low- swimmer is changing is completely negligible when compared to the typical magnitude of the forces from the surrounding viscous fluid. As a result, Newton’s law becomes a simple statement of instantaneous balance between external and fluid forces and torques

(9)

In most cases, there is no external forces, and . Situations where is non-zero include the locomotion of nose-heavy or bottom-heavy cells [68]; in all other cases we will assume .

4.2 The swimming problem

Mathematically, the swimming problem is stated as follows. Consider a body submerged in a viscous fluid. In a reference frame fixed with respect to some arbitrary reference point in its body, the swimmer deforms its surface in a prescribed time-varying fashion given by a velocity field on its surface, . The velocity field is the “swimming gait”. A swimmer is a deformable body by definition, but it may be viewed at every instant as a solid body with with unknown velocity and rotation rate . The instantaneous velocity on the swimmer’s surface is therefore given by , which provides the boundary conditions needed to solve Eq. (3). The unknown values of and are determined by satisfying Eq. (9)

The mathematical difficulty of solving the swimming problem arises from having to solve for the Stokes flow with unknown boundary condition; in that regard, low- swimming is reminiscent of an eigenvalue problem. A great simplification was derived by Stone and Samuel [69], who applied the reciprocal theorem, Eq. (4), to the swimming problem. Recall that the reciprocal theorem involves two different flow problems for the same body. Let and the velocity and stress fields we seek in the swimming problem. For the second flow problem, suppose and are the velocity and stress fields for the dual problem of instantaneous solid body motion of the swimmer with velocity and rotation rate . This problem correspond to subjecting the shape, , to an external force, , and torque, . Applying the reciprocal theorem (4), we obtain [69]

(10)

Equation (10) shows explicitly how the swimming velocity and rotation rate may be found in terms of the gait , given the solution to the dual problem of the flow induced by the motion of the rigid body with instantaneous shape , subject to force and torque . Note that since and are arbitrary, Eq. (10) provides enough equations to solve for all components of the swimming kinematics. Note also that for squirming motion, where the shape of the swimmer surface remains constant (), Eq. (10) simplifies further. For a spherical squirmer of radius   [69], we have the explicit formulas [69]

(11)

4.3 Drag-based thrust

Figure 2: Physics of drag-based thrust: The drag anisotropy for slender filaments provides a means to generate forces perpendicular to the direction of the local actuation (see text for notation).

Most biological swimmers exploit the motion of slender appendages (“flagella”) for locomotion. This limit of slender bodies allows us to provide a physical, intuitive way to understand the origin of locomotion through drag; the specifics of biological and artificial flagellar actuation will be discussed in §6.

The fundamental property allowing for drag-based thrust of slender filaments is their drag anisotropy, as introduced in §3.2. Indeed, consider a thin filament immersed in a viscous fluid which is motionless but for flows induced by the deformations of the filament. The shape of the filament is described by its tangent vector at distance along the filament, and its instantaneous deformation is described by the velocity field , where is time. For asymptotically slender filaments, (see §5), as in the case of prolate spheroids, the local viscous drag force per unit length opposing the motion of the filament is

(12)

where and are the projections of the local velocity on the directions parallel and perpendicular to the filament, i.e. and ; and are the corresponding drag coefficients (typically ).

The origin of drag-based thrust relies on the following two physical ideas: (a) The existence of drag anisotropy means that propulsive forces can be created at a right angle with respect to the local direction of motion of the filament, and (b) a filament can deform in time-periodic way and yet create non-zero time-averaged propulsion. To illustrate these ideas, consider the beating filament depicted in Fig. 2. Any short segment of the filament may be regarded as straight and moving with velocity at an angle with the local tangent (Fig. 2, inset). This velocity resolves into components and , leading to drag per unit length components and . For isotropic drag, , and the force on the filament has the same direction as the velocity of the filament; however, if , the drag per unit length on the filament includes a component which is perpendicular to the direction of the velocity,

(13)

In addition, in order to generate a net propulsion from a time-periodic movement, we see from Eq. (13) that both the filament velocity and its orientation angle need to vary periodically in time. For example, actuation with a given and , followed by the change and , leads to a a propulsive force with a constant sign; in contrast, actuation in which only changes periodically leads to zero average force.

It is important to realize that this result relies on two ideas associated with two different length scales. One is purely local, and states that with the appropriate geometry and actuation, a force can be created in the direction perpendicular the motion of the filament. This conclusion relies explicitly on the properties of Stokes flows, and in a world with isotropic viscous friction (), locomotion would not be possible [31, 70]. The second idea is a global constraint that says that the periodic actuation of the filament needs to be sufficiently subtle to generate non-zero forces on average; this property is generally known as Purcell’s scallop theorem [14].

4.4 The scallop theorem

As pointed out above, the Stokes equation—Eq. (3)—is linear and independent of time. These properties lead to kinematic reversibility, an important and well-known symmetry property associated with the motion of any body at zero Reynolds number [59, 61]. Consider the motion of a solid body with an instantaneous prescribed velocity, , and rotation rate, , together with the flow field surrounding it. If we apply the scaling and , then by linearity, the entire flow and pressure field transform as and . Consequently, the instantaneous flow streamlines remain identical, and the fluid stresses undergo a simple linear scaling, resulting in the symmetry and for the force and torque acting on the body. In particular, when , this means that an instantaneous reversing of the forcing does not modify the flow patterns, but only the direction in which they are occurring.

When applied to low-Reynolds number locomotion, the linearity and time-independence of Stokes equation of motion lead to two important properties [14]. The first one is that of rate independence: If a body undergoes surface deformation, the distance travelled by the swimmer between two different surface configurations does not depend on the rate at which the surface deformation occurs but only on its geometry (i.e. the sequence of shapes the swimmer is going through between these two configurations).

A mathematical proof of this statement can be outlined as follows. We consider for simplicity swimmers with no rotational motion (an extension to the case is straightforward). Consider a body deforming its surface between two different configurations identified by time and . We denote by the positions of points on the surface of the swimmer. From Eq. (10), we know that the instantaneous speed of locomotion is given by a general integral of the form

(14)

where we have used . The net motion of the swimmer between and is therefore given by

(15)

Now consider the same succession of swimmer shapes, but occurring at a different rate. We describe it by a mapping , with for , such that the shape is the same as the shape for all times. We now have

(16)

where

(17)

using the chain rule. We see therefore that , and therefore . The net distance traveled by the swimmer does not depend on the rate at which it is being deformed, but only on the geometrical sequence of shape. One consequence of this property is that many aspects of low-Reynolds number locomotion can be addressed using a purely geometrical point of view [71, 72, 73, 74, 75].

The second important property of swimming without inertia is the so-called scallop theorem: If the sequence of shapes displayed by a swimmer deforming in a time-periodic fashion is identical when viewed after a time-reversal transformation (a class of surface deformation termed “reciprocal deformation”), then the swimmer cannot move on average. This second property puts a strong geometrical constraint on the type of swimming motion which will be effective at low Reynolds numbers.

An outline of the proof can be offered as follows (again, we consider purely translational motion for simplicity). Let us consider a swimmer deforming its body between times and , and a sequence of shape described by . We assume that so that we are looking at the swimming motion over one period of surface deformation. The net distance traveled by the swimmer is given by Eqs. (1415). We now consider the motion between and obtained by time-reversal symmetry of the first motion; we describe it by a temporal mapping , with and , defined such that the shape is the same as the shape (t). In that case, using similar arguments at those used to demonstrate the first property above, we see that

(18)

and reversing the sequence of shape leads therefore to the opposite distance traveled. However, since the body deformation is reciprocal, the sequence of shape between and is the same as between and , and therefore the distance traveled should be the same independently of the direction of time: . By combining the two results, we see therefore that : Reciprocal motion cannot be used for locomotion at low Reynolds numbers. Note here that in order to demonstrate this result, we do not need to assume anything about the geometry of the fluid surrounding the swimmer, so the scallop theorem remains valid near solid walls, and more generally in confined environments.

In his original article, Purcell illustrated this result by using the example of a scallop, a mollusk that opens and closes its shell in a time period fashion. A low-Reynolds number scallop undergoes a reciprocal deformation, and therefore cannot swim in the absence of inertia (independent of the rate of opening and closing)2. Another example of a reciprocal deformation is a dumbbell, made of two solid spheres separated by time-periodic distance. More generally, bodies with a single degree of freedom deform in a reciprocal fashion, and cannot move on average.

Successful swimmers must display therefore non-reciprocal body kinematics. In his original paper, Purcell proposed a simple example of non-reciprocal body deformation, a two-hinged body composed of three rigid links rotating out-of-phase with each other, now refereed to as Purcell’s swimmer [70]. Another elementary example is a trimer, made of three rigid spheres whose separation distances vary in a time-periodic fashion with phase differences [76]. More examples are discussed in §9. Note that, mathematically, the presence of non-reciprocal kinematics is a necessary but not sufficient condition to obtain propulsion. A simple counterexample is a two swimmers which are mirror-images of each other and arranged head-to-head; although the kinematics of the two bodies taken together is non-reciprocal, the mirror symmetry forbids net motion of their center of mass.

For biological bodies deforming in a continuous fashion, the prototypical non-reciprocal deformation is a wave. Consider a continuous filament of length deforming with small amplitude (i.e. for which ); in that case, the propulsive force generated along the filament, Eq. (13), is given by

(19)

If the filament deforms as a planar wave traveling in the - direction, , the force is given by and propulsion is seen to occur in the direction opposite to that of the wave (). Mathematically, a wave-like deformation allows the product to keep a constant sign between and , and therefore all portions of the filament contribute to generating propulsion. In general, all kinds of three-dimensional wave-like deformations lead to propulsion, in particular helical waves of flexible filaments [77].

Finally, it is worth emphasizing that the scallop theorem is strictly valid in the limit where all the relevant Reynolds numbers in the swimming problem are set to zero. Much recent work has been devoted to the breakdown of the theorem with inertia, and the transition from the Stokesian realm to the Eulerian realm is found to be either continuous or discontinuous depending on the spatial symmetries in the problem considered [78, 79, 80, 81, 82, 83, 84].

5 Historical studies, and further developments

In this section we turn to the first calculations of the swimming velocities of model microorganisms. We consider two simple limits: (1) propulsion by small amplitude deformations of the surface of the swimmer, and (2) propulsion by the motion of a slender filament. Although these limits are highly idealized, our calculations capture essential physical aspects of swimming that are present in more realistic situations.

5.1 Taylor’s swimming sheet

In 1951, G. I. Taylor asked how a microorganism could propel itself using viscous forces alone, rather than imparting momentum to the surrounding fluid as fish do [29]. To answer this question, he calculated the flow induced by propagating transverse waves of small amplitude on a sheet immersed in a viscous fluid. In this subsection, we review Taylor’s calculation [29]. The sheet is analogous to the beating flagellum of a spermatazoon, but since the flow is two-dimensional, the problem of calculating the induced flow is greatly simplified. The height of the sheet over the plane is

(20)

where the -direction is parallel to the direction of propagation of the wave, is the amplitude, is the wavenumber, and is the frequency of the oscillation. Note that we work in the reference frame in which the material points of the sheet move up and down, with no -component of motion. The problem is further simplified by the assumption that the amplitude is small compared to the wavelength . Note that the motion of Eq. (20) implies that the sheet is extensible. If the sheet is intensible, then the material points of the sheet make narrow figure eights instead of moving up and down; nevertheless the extensible and inextensible sheets have the same swimming velocity to leading order in .

To find the flow induced by the traveling-wave deformation, solve the Stokes equations with no-slip boundary conditions at the sheet,

(21)

with an unknown but uniform and steady flow far from the sheet,

(22)

Since we work in the rest frame of the sheet, is the swimming velocity of the sheet in the laboratory frame, in which the fluid is at rest at . Although it turns out in this problem that the leading order swimming speed is steady in time, other situations lead to unsteady swimming speeds. In all cases we are free to use non-inertial frames—even rotating frames—without introducing fictitious forces, since inertia may be disregarded at zero Reynolds number.

Although is unknown, Taylor found that no additional conditions are required to determine ; instead, there is a unique value of consistent with the solution to the Stokes equations and the no-slip boundary condition (21). It is also important to note that although the Stokes equations are linear, the swimming speed is not a linear function of the amplitude , since enters the no-slip boundary condition both on the right-hand side of Eq. (21) and implicitly on the left-hand side through Eq. (20). In fact, symmetry implies that the swimming speed must be an even function of . Replacing by amounts to translating the wave (20) by half a wavelength. But any translation of the wave cannot change the swimming speed; therefore, is even in .

Taylor solved the swimming problem by expanding the boundary condition (21) in , and solving the Stokes equations order by order. We will consider the leading term only, which as just argued is quadratic in . Since the swimming velocity is a vector, it must be proportional to the only other vector in the problem, the wavevector . For example, if we were to consider the superposition of two traveling waves on the sheet, propagating in different directions, we would expect the swimming direction to be along the vector sum of the corresponding wavevectors. Dimensional analysis determines the remaining dependence of on the parameters of the problem: . Taylor’s calculation yields the proportionality constant, with sign:

(23)

Note that dimensional considerations require the swimming speed to be independent of viscosity. This result holds due to our somewhat unrealistic assumption that the waveform (20) is prescribed, independent of the load. However, the rate that the sheet does work on the fluid does depend on viscosity. The net force per wavelength exerted by the sheet on the fluid vanishes, but by integrating the local force per area () against the local velocity (), Taylor found

(24)

Note that only the first-order solution for the flow is required to calculate . In §6 we consider more realistic models that account for the internal mechanisms that generate the deformation of the swimmer. Such models can predict a viscosity-dependence in the swimming speed, since the shape of the beating filament may depend on viscosity [85, 86, 87]. And in §8 we show how the speed of a swimmer in a complex fluid can depend on material parameters, even for the swimming problem with prescribed waveform.

According to Eq. (23), the swimmer moves in the direction opposite to the traveling wave. It is instructive to also consider the case of a longitudinal wave, in which the material points in the frame of the sheet undergo displacement , yielding the no-slip boundary condition

(25)

For a longitudinal wave, the swimming velocity is in the same direction as the traveling wave.

Figure 3: Physical interpretation of the swimming direction for Taylor’s swimming sheet. The vertical displacements (black vertical arrows and black line) along the sheet as the wave of deformation is traveling to the right (blue lines and rightward arrows) leads to creation of flow vorticity of alternating signs (curved red arrows), which induces leftward motion of the sheet (straight red leftward arrows).

The direction of swimming of Taylor’s sheet can be understood on the basis of the following simple arguments. Let us consider a sheet deforming as a sine-wave propagating to the right (Fig. 3). Consider the vertical displacements occurring along the sheet as a result of the propagating wave. During a small interval of time, the original wave (Fig. 3, thick blue line) has moved to the right (Fig. 3, thin blue line), resulting in vertical displacements given by a profile which is out of phase with the shape of the sheet. Indeed, wherever the sheet has a negative slope, the material points go up as the wave progresses to the right, whereas everywhere the sheet has a positive slope, the material points go down as a result of the wave. The resulting distribution of vertical velocity along the sheet is illustrated in Fig. 3 by the black line and vertical arrows. This velocity profile forces the surrounding fluid, and we see that the fluid acquires vorticity of alternating sign along the sheet (illustrated by the curved red arrows in Fig. 3). The vorticity is seen to be positive near the sheet crests, whereas it is negative near the sheet valleys. The longitudinal flow velocities associated with this vorticity distribution allow us to understand the swimming direction. In the case of positive vorticity, the velocity of the induced vortical flow is to the left at the wave crest, which is the current position of the sheet. In the case of negative vorticity, the induced flow velocity is to the left in the valleys, which is also where the sheet is currently located. As a consequence, the longitudinal flow field induced by the transverse motion of the sheet leads to flow contribution which are to the left in all cases, and the sheet is seen to swim to the left (straight red leftward arrows). Note that if the sheet were not free to move, then it would create an external flow field that cancels the sheet-induced flow, and the net flow direction would therefore be to the right—the sheet acts as a pump.

There are many generalizations to Taylor’s 1951 calculation. Taylor himself considered the more realistic geometry of an infinite cylinder with a propagating transverse wave [88]. In this case, there is a new length scale , the radius of the cylinder, and the calculation is organized as a power series in rather than . In the limit , the swimming velocity has the same form as the planar sheet [88]. With a cylinder, we can study truly three-dimensional deformations of a filament, such as helical waves. A helical wave can be represented by the superposition of two linearly polarized transverse waves, with perpendicular polarizations and a phase difference of . If these waves have the same amplitude, speed, and wavelength, then the swimming velocity is twice the velocity for a single wave. Although the hydrodynamic force per unit wavelength acting on the waving filament vanishes, there is a nonvanishing net hydrodynamic torque per unit wavelength, which is ultimately balanced by the counter-rotation of the head of the organism [88].

The Taylor sheet calculation may also be extended to finite objects. For example, to model the locomotion of ciliates such as Opalina and Paramecium, Lighthill introduced the “envelope model,” in which the tips of the beating cilia that cover the cell body are represented by propagating surface waves [89, 90, 91]. Perhaps the simplest version of the envelop model is the two-dimensional problem of an undulating circle in the plane, which may equivalently be viewed as an infinite cylinder with undulations traveling along the circumferential direction [91]. Unlike the problem of a rigid cylinder towed through liquid at zero Reynolds number, the undulating cylinder does not suffer from the Stokes paradox [65, 92], since the total force on the cylinder is zero. And unlike the Taylor sheet problem, where the swimming speed emerges self-consistently, the condition of vanishing force is required to determine the swimming speed of the undulating cylinder. The problematic solutions that lead to the Stokes paradox are the same ones that lead to a net force, as well as a diverging kinetic energy, and are therefore eliminated in the swimming problem [91].

In three dimensions, the swimming speed is also determined by the condition of vanishing total force [89, 90], but since the solution to the problem of towing a sphere with an external force is well-behaved, we may also consider solutions with nonzero force. These solutions must be well-behaved if we are to apply reciprocal theorem of §3, which gives perhaps the shortest route to calculating the swimming speed [69, 93]. We can also gain additional insight into why the swimming speed for a prescribed deformation of the surface is independent of viscosity. Using the linearity of Stokes flow, at any instant we may decompose the flow field generated by the swimmer into a “drag flow” and a “thrust flow,”  [3]. The drag flow is the flow induced by freezing the shape of the swimmer and towing it at velocity with a force , to be determined. The thrust flow is the flow induced by the swimmer’s motion at that instant when it is prevented from moving by an anchoring force , which is determined by the shape and and rate of change of shape of the swimmer at that instant. Superposing the two flows, and adjusting to cancel yields the swimming speed . Since the linearity of Stokes flow implies that both and depend linearly on viscosity, the swimming velocity does not depend on viscosity. Note that the same conclusion follows from an examination of the reciprocal theorem formula, Eq. (10).

Finally, in the sheet calculation, it is straightforward to include the effects of inertia and show that if flow separation is disregarded, the swimming speed decreases with Reynolds number, with an asymptotic value at high Reynolds number of half the value of Taylor’s result (23[94, 95]. At zero Reynolds number, the effect of a nearby rigid wall is to increase the swimming speed as the gap between the swimmer and the wall decreases, for prescribed waveform [94]. However, if the swimmer operates at constant power, the swimming speed decreases as the gap size decreases [94].

5.2 Local drag theory for slender rods

All the calculations of the previous subsection are valid when the amplitude of the deflection of the swimmer is small. These calculations are valuable since they allow us to identify qualitative trends in the dependence of the swimming velocity on geometric and, as we shall see in §8, material parameters. However, since real flagella undergo large-amplitude deformations, we cannot expect models based on small-amplitude deformations to give accurate results. Fortunately, we may develop an alternative approximation that is valid for large deformations by exploiting the fact that real flagella are long and thin. The idea is to model the flow induced by a deforming flagellum by replacing the flagellum with a line of singular solutions to Stokes flow of appropriate strength. In this subsection and the following subsection we develop these ideas, first in the simplest context of local drag theory, also known as resistive force theory, and then using the more accurate slender-body theory.

To introduce local drag theory, we develop an intuitive model for calculating the resistance matrix of a straight rigid rod of length and radius . Our model is not rigorous, but it captures the physical intuition behind the more rigorous theories described below. Suppose the rod is subject to an external force . Suppose further that this force is uniformly distributed over the length of the rod with a constant force per unit length. Our aim is to find an approximate form for the resistance matrix, or equivalently, the mobility matrix, with errors controlled by the small parameter . To this end, we replace the rod with spheres equally spaced along the -axis, with positions . According to our assumption of uniformly distributed force, each sphere is subject to an external force . If there were no hydrodynamic interactions among the spheres, then each sphere would move with velocity, , and the mobility matrix would be isotropic. In fact, the motion of each sphere induces a flow that helps move the other spheres along (Fig. 4).

Figure 4: Model for a rod subject to an external force along the rod (a) or perpendicular to the rod (b). The arrows at the centers of the spheres represent external forces. The flow field of the central sphere only is shown. Each sphere induces a similar flow field that helps push the other spheres along.

To calculate the flow induced by the th sphere, recall from §3 that the flow induced by a moving sphere is the superposition of a stokeslet and a source-dipole. Since we seek the mobility to leading order, we only keep the far-field terms; thus

(26)

Since each sphere moves with the local flow, we identify the velocity of the th sphere with the superposition of the flow induced by the force at and the flows induced by the forces at all the other spheres:

(27)

Since , we replace the sum over by an integral, noting that the density of spheres is , and taking care to exclude a small segment around from the region of integration:

(28)

where the prime on the integral indicates that the region of integration is from to except for the interval . Evaluation of the integral yields

(29)

where we have disregarded end effects by assuming . Keeping only the terms which are leading order in , and using the fact that is constant for a rigid rod, we find

(30)

where is the externally imposed force per unit length. In our model, the only forces acting between any pair of spheres is the hydrodynamic force: There are no internal cohesive forces. Therefore, drag per unit length , and

(31)

where and denote the components perpendicular and parallel to the -axis, respectively, and . Once again, we encounter the anisotropy already mentioned for slender bodies (§3.2) and stokeslets (§3.3) that is necessary for drag-based thrust (§4.3).

In our derivation of Eq. (31), we assumed zero deformation since the filament was straight. Turning now to deformed filaments, suppose that the filament is gently curved, , where , and gives the position of the of the centerline of the filament with arclength coordinate . In the limit of very small curvature, it is reasonable to assume that the viscous force per unit length acting on the curved filament is the same as the viscous force per unit length acting on a straight rod of the same length. Since local drag theory is an expansion in powers of , it is valid for filaments that are “exponentially thin.” That is, to make of order with , we need . Below in §5.3 we introduce slender-body theory, which has the advantage of being accurate for thin () rather than exponentially thin filaments.

Slender-body theory also more accurately captures the hydrodynamic interactions between distant parts of a curved filaments. To see the limitations of the resistive force theory coefficients of Eq. (31), consider a rigid ring of radius and rod diameter , falling under the influence of gravity in a very viscous fluid. Suppose the plane of the ring is horizontal. Compare the sedimentation rate of the ring with that of a horizontal straight rod with length . In both cases, each segment of the object generates a flow which helps push the other segments of the object down. But the segments of the ring are closer to each other, on average, and therefore the ring falls faster. Using the coefficients of local drag theory from Eq. (31) would lead to the same sedimentation rate for both objects. This argument shows the limitations of our local drag theory. One way to improve our theory is to use a smooth distribution of stokeslets and source-dipoles to make a better approximation for the flow induced by the motion of the rod. Applying this approach to a sine wave with wavelength leads to [30, 12]

(32)
(33)

In Ref. [12], Lighthill refined the arguments of [30] and gave more accurate values for and . Despite the limitations of local drag theory, we will see that it is useful for calculating the shapes of beating flagella and the speeds of swimmers.

In the remainder of this section, we describe some of the applications of local drag theory to the problem of swimmers with prescribed stroke. To keep the formulas compact, we work in the limit of small deflections, although local drag theory is equally applicable to thin filaments with large deflections. Consider the problem of a spherical body of radius propelled by a beating filament with a planar sine wave [77]

(34)

As in our discussion of the Taylor sheet, §5.1, we work in the frame of the swimmer. Thus, the problem is to find the flow velocity that yields zero net force and moment on the swimmer. To simplify the discussion, we suppose that external forces and moments are applied to the head to keep it from rotating or moving in the -direction. In real swimmers, there is a transverse component of the velocity and a rotation, which both play an important role in determining the swimmer’s trajectory and the shape of the flagellum [96, 97].

Equation (31) gives the viscous forces per unit length acting on the filament. The total force per unit length has a propulsive component, Eq. (19), arising from the deformation of the filament, and also a drag component, arising from the resistance to translating the swimmer along the direction. Integrating this force per length to find the total -component of force, writing the drag force on the sphere as , and balancing the force on the sphere with the force on the filament yields

(35)

Note that only the perpendicular component of the rod velocity leads to propulsive thrust; the motion of the rod tangential to itself hinders swimming. Inserting the sinusoidal waveform (34) into Eq. (35) and averaging over a period of the oscillation yields

(36)

The form of Eq. (36) is similar to the result for a swimming sheet, Eq. (23); when and , the two expressions are identical. For , the swimming speed is independent of for fixed and , since lengthening the filament increases the drag and propulsive forces by the same amount.

Since the swimmer has finite length, we can define the efficiency as the ratio of the power required to drag the swimmer with a frozen shape though the liquid at speed to the average rate of work done by the swimmer. In other words, the efficiency is the ratio of the rate of useful working to the rate of total working. To leading order in deflection, the efficiency is given for arbitrary small deflection by

(37)

For the sinusoidal traveling wave (34) with ,

(38)

For small deflections, , and the hydrodynamic efficiency is small. Note that the total efficiency is given by the product of the hydrodynamic efficiency and the efficiency of the means of energy transduction.

We now turn to the case of rotating helix, such as the flagellar filament of E. coli. The body of the cell is taken to be a sphere of radius . For simplicity, suppose the radius of the helix is much smaller than the pitch of the helix, or equivalently, that the pitch angle is very small (Fig. 5). Expressions relevant for large amplitudes may be found in Refs. [98, 27]. We also assume that the axis of the helix always lies along the -axis, held by external moments along and if necessary. The helix is driven by a rotary motor embedded in the wall of the body, turning with angular speed relative to the body. The helix rotates with angular speed in the laboratory frame, and the body must counter-rotate with speed to ensure the total component of torque along of the swimmer vanishes. The angular speeds are related by .

Figure 5: Forces on a helical segment pulled through a viscous fluid with speed . Half of a helical pitch is shown. The helix is prevented from rotating by an external moment along .

To calculate the force and moment acting on the helical filament, use Eq. (5), simplified to include only components along -axis:

(39)

where and are the external force and moment, respectively, required to pull the helix with speed and angular rotation rate . To leading order in , and with the assumption that , the resistance coefficients are approximately

(40)
(41)
(42)

These values are deduced by applying Eqs. (31) to special cases of motion. The coefficient may be found by considering the case of a helix pulled at speed but prevented from rotating. Likewise, may be found by examining a helix that rotates about but does not translate along . To find , we may examine the moment required to keep a helix pulled with speed from rotating (Fig. 5). Or we may also examine the force required to keep a helix rotating at angular speed from translating. The equivalence of these two calculations is reflected in the symmetry of the resistance matrix, which ultimately stems from the reciprocal theorem, Eq. (4). Note that the sign of the coupling between rotational and translational motion is determined by the handedness of the helix.

To find the swimming speed and rate of filament rotation, we equate the external forces and moments on the filament to the forces and moments acting on the cell body,

(43)
(44)

where we have introduced a resistance coefficient for the rotation of the sphere. Solving for the three unknowns , , and , we find

(45)

, and . The resistance of the body does not enter since we assume . The speed is linear in since the sign of the speed is given by the handedness of the helix, which is given by the sign of . Note the contrast with the planar wave. If , then the velocity (45) vanishes: Propulsion by means of a rotating helix requires a body. However, a planar wave does not require a body for propulsion [see Eq. (36)]. For the helical filament, the swimming speed decreases with increasing for fixed body size and fixed motor speed , since torque balance forces the filament to rotate more slowly with increasing . Also, since the resistance to rotation of a helix scales as , the helix rotates sufficiently faster as the radius decreases to overcome the reduced thrust force implied by the linear dependence of on , leading to a speed that increases like for decreasing . It is important to note that the motor speed of a swimming bacterium is not directly observable with current techniques; typically the approach just described is used to make a prediction for the relation between observables such as and , which is then compared with measurements [27, 99]

5.3 Slender-body theory

The local drag theory illustrated in the previous section allows an intuitive presentation of the scaling laws for flagella-based locomotion. It turns out however that this theory is quantitatively correct only for exponentially slender filaments. Let be the typical length scale along the flagellum on which its variations in shape occur, such as the wavelength, and the flagellum radius. The local drag theory assumes that . Since real biological flagella have aspect ratios on the order , an improved modeling approach is necessary [12, 100]. The idea, termed slender-body theory and pioneered by Hancock [30], is to take advantage of the slenderness of the filaments and replace the solution for the dynamics of the three-dimensional of the filament surface by that of its centerline using an appropriate distribution of flow singularities. Two different approaches to the method have been proposed.

The first approach consists of solving for the flow as a natural extension of the local theory, and approximating the full solution as a series of logarithmically small terms [101, 102, 103]. Physically, the flow field close to the filament is locally two-dimensional, and therefore diverges logarithmically away from the filament because of Stokes’ paradox of two-dimensional flows [61]. The flow far from the filament is represented by a line distribution of stokeslets of unknown strengths, and diverges logarithmically near the filament due to the singular nature of the stokeslets (the logarithmic divergence is due to the line integration of stokeslet terms). Matching these two diverging asymptotic behaviors allows the determination of the stokeslet strengths order by order as a series of terms of order . The leading-order term in this series, of order , is the local drag theory, and gives a stokeslet distribution proportional to the local velocity. The next order term is in general non-local and provides the stokeslet strength at order as an integral equation on the filament shape and velocity. Terms at higher order can be generated in a systematic fashion [101, 102, 103]. This approach to slender-body hydrodynamics is the logical extension to the local drag theory, and all the terms in the expansion can be obtained analytically which makes it appealing. However, the major drawback to this approach is that each term in the expansion is only smaller than the previous term by a factor , so a large number of terms is necessary in order to provide an accurate model for the flow.

A second approach, asymptotically more accurate but technically more involved, consists in bypassing the logarithmically-converging series by deriving directly the integral equation satisfied by the (unknown) distribution of singularities along the filament. Such approach has been successfully implemented using matched asymptotics [104] or uniform expansions [105], and leads to results accurate at order . This approach is usually preferable since instead of being only logarithmically correct it is algebraically correct. This improved accuracy comes at a price however, and at each instant an integral equation must be solved to compute the force distribution along the filament. An improvement of the method was later proposed by accurately taking into account end effects and a prolate spheroidal cross-section, with an accuracy of order [106].

Figure 6: Lighthill’s singularity construction for the slender-body theory of flagellar hydrodynamics. To compute the flow field at a point along the flagellum, Lighthill picks an intermediate length scale such that and represents the flow by a distribution of stokeslests at a distance further away than from (outer problem) and a distribution of stokeslets and source-dipoles within of (inner problem). The dipole strengths in the inner problem are obtained by requiring that the result be independent of .

In his John von Neumann lecture, Lighthill proposed an alternate method for the derivation of such integral equations. Instead of using asymptotic expansions, he used physical arguments to derive the type and strength of the singularities located along the filament [12, 107]. By analogy with the flow past a sphere, Lighthill first proposed that a line distribution of stokeslets and source-dipoles should be appropriate to represent the flow field induced by the motion of the filament. He then demonstrated that the strength of the dipole distribution should be proportional to the stokeslets strengths using the following argument (see Fig. 6). Consider a location along the filament. By assumption of slenderness, it is possible to find an intermediate length scale along the filament such that . The flow field on the surface of the filament at the position is then the sum of the flow due to the singularities within a distance from (“inner” problem) and those further away than from (“outer” problem). Since , the contribution at from the outer problem is primarily given by a line distribution of stokeslets (the source dipoles decay much faster in space). In the inner problem, Lighthill then showed that it was possible to analytically determine the strength of the dipoles to ensure that the complete solution (sum of the inner and outer problems) was independent of the value of . The dipole strength is found to be proportional to the stokeslet strength, and the resulting value of the velocity of the filament at is given by the integral equation

(46)

In Eq. (46), is the local strength of the (unknown) stokeslet distribution (dimensions of force per unit length), is the Oseen tensor from Eq. (7), is a length scale that appears in the analysis (), and represents the normal component of the stokeslet distribution, i.e. if is local the tangent to the filament. Note that Lighthill’s slender-body analysis is less mathematically rigorous than those presented in Refs. [104, 105], and consequently gives results which are only valid at order [3]. His derivation provides however important physical insight into a subject, the topic of flow singularities, that is usually very mathematical. The resulting integral formulation, Eq. (46), is relatively simple to implement numerically, and can also be used to derive “optimal” resistance coefficients for the local drag theory (see §5.2). His modeling approach has been extended for filament motion near a solid boundary [108, 109, 110, 111], and an alternative approach based on the method of regularized flow singularities [112] has also been devised [113].

6 Physical actuation

In this section we describe common mechanisms for the generation of non-reciprocal swimming strokes. In addition to biological mechanisms such as rotating helices or beating flagella, we also describe simple mechanisms that do not seem to be used by any organism. The non-biological setups are useful to study since they deepen our understanding of the biological mechanisms. For example, the modes of an elastic rod driven by transverse oscillations at one end are useful for understanding the shape of a beating flagellum driven by motors distributed along its entire length. An important theme of this section is the fluid-structure interaction for thin filaments in viscous liquid.

6.1 Boundary actuation

Figure 7: The torque-speed characteristic for the flagellar motor of E. coli for three different temperatures (bold lines). Also shown (thin lines) are the load lines for a cell body with radius of 1 m (left), and a minicell with effective radius of about m (right). Figure reprinted from Ref. [35], courtesy of Howard Berg and by permission from Annual Reviews, copyright 2003.

We begin with the case of boundary actuation, in which an elastic filament is driven by a motor at its base. The rotary motor of the bacterial flagellum is a prime example of such a biological actuating device [34, 35, 114, 115]. The steady-state relation between motor torque and motor speed is shown in Fig. 7 [35]. At low speeds, the motor torque is roughly constant; at higher speeds it decreases linearly with speed, reaching zero torque at about 300 Hz at 23 C. To determine the speed of the motor from the motor torque-speed relation, use torque balance and equate the motor torque with the load torque. By the linearity of Stokes flow, the load torque is linear in rotation speed. In the experiments used to make the graph of Fig. 7, the flagellar filament of E. coli was tethered to a slide, and the rotation of the body was observed. A typical body of 1 m radius has a substantial resistance, leading to the steep load curve on the left of Fig. 7 and a correspondingly low rotation speed. A smaller load, such as that of a minicell, leads to a load curve with smaller slope, and higher rotation speed. The torque-speed characteristic allows us to go beyond the artifice of the previous section where we calculated the swimming speed in terms of the motor speed . Solving Eqs. (39) and (4344) along with yields the swimming speed in terms of the geometrical parameters of the flagellum, the cell body, the drag coefficients, and the properties of the motor.

In the previous section we described how a rotating helix generates propulsion. Since the flagellar filaments of E. coli and S. typhimurium are relatively stiff, a helical shape is necessary to escape the constraints of the scallop theorem as a straight rod rotating about its axis generates no propulsion. Indeed, mutant E. coli with straight flagella do not swim [116]. If the rate of rotation of a straight but flexible rod is high enough for the hydrodynamic torque to twist the rod through about one turn, then the rod will buckle into a gently helical shape that can generate thrust [117, 118, 119]. However, the high twist modulus of the filament and the low rotation rate of the motor make this kind of instability unlikely in the mutant strains with straight flagella. On the other hand, a rotating helix with the dimensions of a flagellar filament experiences much greater hydrodynamic torque since the helical radius (microns) is much greater than the filament radius ( nm). The handedness of the helix also breaks the symmetry of response to the sense of rotation of the motor: Counterclockwise rotation of a left-handed helix in a viscous fluid tends to decrease the pitch of a helix, whereas clockwise rotation tends to increase the pitch [120]. There is no noticeable difference between the axial length of rotating and de-energized flagella for counterclockwise rotation [121]; calculations of the axial extension [120] based on estimates of the bending stiffness of the flagellar filament [122] are consistent with this observation. However, the hydrodynamic torque for clockwise rotation is sufficient to trigger polymorphic transformations, in which a right-handed helical state invades the left-handed state by the propagation of a front [123, 124].

Now consider the case of an elastic filament driven by a mechanism that oscillates the base of the filament in the direction normal to the tangent vector of the filament. Although we know of no organism that uses this mechanism to swim, study of this example has proven instructive. In early work, Machin [125] pointed out that the overdamped nature of low Re flow leads to propagating waves of bending with exponential decay of the amplitude along the length of the filament. Since the observed beating patterns of sperm flagella typically have an amplitude that increases with distance from the head, Machin concluded that there must be internal motors distributed along the length of the flagellum that give rise to the observed shape. This problem has served as the basis for many subsequent investigations of the fluid-structure interaction in swimming [125, 126, 127, 128, 129, 130, 97, 131, 87], and has even been applied to the determination of the persistence length of actin filaments [132, 133]. Therefore, we give a brief overview of the most important elements of the problem.

Figure 8: A bent elastic rod. The darkly shaded part of the rod exerts a force and moment on the lightly shaded part through the cross-section at .

Consider a thin rod of length constrained to lie in the -plane, aligned along the -axis in the absence of external loads. We will consider small deflections from the straight state. When the rod is bent into a curved shape, the part of the rod on the outside of the curve is under tension, while the part of the rod on the inside of the curve is under compression (Fig. 8). Therefore, the section of the rod to to right of exerts a moment on the section to the left of , where is the bending modulus and is the curvature of the rod at  [134]. Working to first order in deflection, balance of moments on an element of length of the rod implies

(47)

where is the -component of the force exerted through the cross-section at . Thus, if the rod has a deflection , then an elastic force acts on the element of length at . Balancing this elastic force with the transverse viscous force from resistive force theory yields a hyper-diffusion equation

(48)

The shape of the rod is determined by solving Eq. (48) subject to the appropriate boundary conditions, which are typically zero force and moment at the far end, . At the near end, common choices are oscillatory displacement with clamping , or oscillatory angle with , where and  [125, 127].

The appearance of in Eq. (48) causes the breakdown of kinematic reversibility: Even for a reciprocal actuation such as , Eq. (48) implies that the rod shape is given by propagating waves. Physically, the breakdown of kinematic reversibility occurs because flexibility causes distant parts of the rod to lag the motion of the rod at the base. We saw in §3 that zero-Re flow is effectively quasistatic since the diffusion of velocity perturbations is instantaneous. When the filament is flexible, the time it takes for perturbations in shape to spread along the rod scales as . Since the shape of the rod does not satisfy kinematic reversibility, the flow it induces does not either, and a swimmer could therefore use a waving elastic rod to make net progress.

The wavelength and the decay length of the propagating waves is governed by a penetration length, . Sometimes this length is given in the dimensionless form of the “Sperm number,” . If the rod is waved rapidly, the penetration length is small and propulsion is inefficient since most of the filament has small defection and contributes drag but no thrust. For small frequencies, the rod is effectively rigid , and there is no motion since kinematic reversibility is restored. Thus, we expect the optimum length for propulsion is , since at that length much of the rod can generate thrust to compensate the drag of pulling the filament along  [127, 97]. Note that our discussion of flexibility may be generalized to other situations; for example, the deformation of a flexible wall near a swimmer is not reversible, leading to a breakdown of the scallop theorem even for a swimmer that has a reciprocal stroke [135]. In this case the average swimming velocity decays with a power of the distance from the wall, and therefore this effect is relevant in confined geometries.

An interesting variation on the Machin problem is to rotate a rod which is tilted relative to its rotation axis. If the rod is rigid, then it traces out the surface of a cone. But if it is flexible, then the far end will lag the base, and the rod has a helical shape. As long as the tilt angle is not too small, this shape may be determined without considering effects of twist  [136, 137]. If the driving torque rather than speed is prescribed, there is a transition at a critical torque at which the shape of the rod abruptly changes from gently helical to a shape which is much more tightly wound around axis of rotation, with a corresponding increase in thrust force [138, 139, 140].

Figure 9: Cross-section section of an axoneme from wild type Chlamydomonas, courtesy of Steve King. The diameter is approximately 200 nm. Note that the plane containing the active motors is perpendicular to the plane defined by the beating filament.

6.2 Distributed actuation

We now consider distributed actuation, in which molecular motors are distributed along the length of the filament. Eukaryotic flagella and cilia use distributed actuation. Figure 9 shows a cross-section of the axoneme, the core of a eukaryotic flagellum. As mentioned in the introduction, the axoneme consists of nine microtubule doublets spaced along the circumference of the flagellum, with two microtubules running along the center. In this review we restrict our attention to the case of planar beating, although many sperm flagella exhibit helical beat patterns, and nodal cilia have a twirling, rotational beat pattern [141, 142]. The bending of the eukaryotic flagellum arises from the relative sliding of neighboring microtubule doublets [143, 144, 145, 146]. The sliding is caused by the action of ATP-driven dynein motors, which are spaced every 24 nm along the microtubles [85]. Since the relative sliding of the microtubules at the end near the head is restricted [147], and since each microtubule doublet maintains its approximate radial position due to proteins in the core of the flagellum, the filament must bend when the motors slide microtubule doublets. For example, in Fig. 10, motors have slid the lower doublet to the right relative to the upper doublet.

The simplest approach to understanding how the sliding of the microtubules generates propulsion is to prescribe a density of sliding force and deduce the shape of the flagellum and therefore the swimming velocity from force and moment balance. This approach is taken in Refs. [148, 87], where the effects of viscosity and viscoelasticity are studied. A more complete model would account for how the coordination of the dynein motors arises. Over the years, several different models for this coordination have been suggested. Since sea urchin sperm flagella continue to beat when they have been stripped of their membranes with detergent  [149], it is thought that the motor activity is not coordinated by a chemical signal but instead arises spontaneously via the mechanics of the motors and their interaction [150, 151]. A detailed discussion of molecular motors and the different mechanisms that have been put forth for controlling the beat pattern would take us too far afield [152, 153, 154, 155, 156]. Instead we review regulation by load-dependent motor detachment rate as presented by Riedel-Kruse and collaborators, who showed that this mechanism is consistent with observations of the flagellar shape [147].

Figure 10: Simple model for a planar flagellum actuated by sliding filaments. The dotted line is the center of the flagellum when it is straight. The solid lines represent the two microtubule doublets that the active motors slide relative to each other. The two solid curves and the dotted line all have the same length; the dots divide each curve into quarters as measured along the contour.

The first step is to simplify the problem by considering planar beating and projecting the three-dimensional flagellum of Fig. 9 onto a planar flagellum such as that shown in Fig. 10. Since the lateral spacing between the two filaments is a constant, , the amount that the top filament slides by the bottom is given by

(49)

where is the sliding displacement at the base of the flagellum where , and is the angle the tangent of the flagellum centerline makes with the -axis. In Fig. 10, , and except at either end of the flagellum. We will forbid sliding at the base, but it is straightforward to allow such sliding in our model, and measurements of the shape of bull sperm flagella suggest that  nm [147]. Equation (49) in the case is most easily deduced when the two filaments form the arcs of concentric circles, but it holds for more general curved shapes as well. Also, we will work in the limit of small deflection, but Eq. (49) is valid even for large deflections.

Now consider the forces between the two filaments. Let denote the force per unit length along that the bottom filament exerts on the top filament. This force per unit length could arise from passive resistance to sliding as well as motors. The internal moment acting a cross section has a contribution from bending and an additional piece from :

(50)

Moment balance on an element of the flagellum yields the transverse force acting on a cross section, , which with force balance yields the equation of motion for the filament: