The hydrodynamics of swimming microorganisms
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IOP PUBLISHING REPORTS ON PROGRESS IN PHYSICS
Rep. Prog. Phys. 72 (2009) 096601 (36pp) doi:10.1088/0034-4885/72/9/096601
The hydrodynamics of swimming
microorganisms
Eric Lauga
1
and Thomas R Powers
2
1
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla,
CA 92093-0411, USA
2
Division of Engineering, Brown University, Providence, RI 02912-9104, USA
E-mail: elauga@ucsd.edu and Thomas
Powers@brown.edu
Received 17 December 2008, in final form 5 May 2009
Published 25 August 2009
Online at
stacks.iop.org/RoPP/72/096601
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
micrometers and below. At this scale, inertia is unimportant and the Reynolds number is small.
Our emphasis is on the simple physical picture and fundamental flow physics phenomena in
this regime. We first give a brief overview of the mechanisms for swimming motility, and of
the basic properties of flows at low Reynolds number, paying special attention to aspects most
relevant for swimming such as resistance matrices for solid bodies, flow singularities and
kinematic requirements for net translation. Then we review classical theoretical work on cell
motility, in particular early calculations of swimming kinematics with prescribed stroke and
the application of resistive force theory and slender-body theory to flagellar locomotion. After
examining the physical means by which flagella are actuated, we outline areas of active
research, including hydrodynamic interactions, biological locomotion in complex fluids, the
design of small-scale artificial swimmers and the optimization of locomotion strategies.
(Some figures in this article are in colour only in the electronic version)
This article was invited by Christoph Schmidt.
Contents
1. Introduction 2
2. Overview of mechanisms of swimming motility 2
3. Flows at low Reynolds number 4
3.1. General properties 4
3.2. Motion of solid bodies 4
3.3. Flow singularities 5
4. Life at low Reynolds number 6
4.1. Reinterpreting the Reynolds number 6
4.2. The swimming problem 6
4.3. Drag-based thrust 7
4.4. The scallop theorem 7
5. Historical studies, and further developments 9
5.1. Taylor’s swimming sheet 9
5.2. Local drag theory for slender rods 11
5.3. Slender-body theory 14
6. Physical actuation 15
6.1. Boundary actuation 15
6.2. Distributed actuation 16
7. Hydrodynamic interactions 18
7.1. Interactions between cells 18
7.2. Interactions between cells and boundaries 20
7.3. Interactions between flagella 22
8. Swimming in complex fluids 24
9. Artificial swimmers and optimization 26
9.1. Designing artificial swimmers 26
9.2. Exploiting low-Re locomotion 27
9.3. Optimization 27
10. Conclusion and outlook 29
Acknowledgments 29
References 30
0034-4885/09/096601+36$90.00 1 © 2009 IOP Publishing Ltd Printed in the UK
Rep. Prog. Phys. 72 (2009) 096601 E Lauga andTRPowers
1. Introduction
Our world is filled with swimming microorganisms: the
spermatozoa that fuse with the ovum during fertilization, the
bacteria that inhabit our guts, the protozoa in our ponds,
the algae in the ocean; these are but a few examples of a wide
biological spectrum. 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 micrometer 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 Re
is defined as Re = ρUL/η, where ρ is the fluid density, η
is the viscosity and U and L are characteristic velocity and
length scales of the flow, respectively. Swimming strategies
employed by larger organisms that operate at high Reynolds
number, such as fish, birds or insects [3–8], are ineffective 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 [3, 11–16], and from the perspective
of the biophysics and biology of cell motility [1, 17–21].
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]. These techniques continue to improve
and find applicability to a wide range of microorganisms [22].
Other recent advances in visualization techniques, such as
the fluorescent staining of flagella [23] in living, swimming
bacteria, also 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 [24–26], alga [27] or bacterium [28] in an optical
trap. Atomic force microscopy also allows direct measurement
of the force exerted by cilia [29]. 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 behavior
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 section 2, we give some examples
of how microorganisms swim. After a brief general review
of low Reynolds number hydrodynamics (section 3), we
outline the fundamental properties of locomotion without
inertia (section 4). We then discuss the classic contributions
of Taylor [30], Hancock [31] and Gray [32], who all but
started the field more than 50 years ago (section 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 (section 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
(section 7); locomotion in non-Newtonian fluids such as the
mucus of the female mammalian reproductive tract (section 8)
and the design of artificial swimmers and the optimization
of locomotion strategies in an environment at low Reynolds
number (section 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
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 deforming its body in
a periodic way. To keep the scope of the paper 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 [33], or the gas-
vesicle mediated buoyancy of aquatic micoorganisms such as
Cyanobacteria [34].
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 [35] (figure 1(a)), or it could be a flexible filament
undergoing whip-like motions due to the action of molecular
2
Rep. Prog. Phys. 72 (2009) 096601 E Lauga andTRPowers
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.
motors distributed along the length of the filament, as in the
sperm of many species [21] (figures 1(e) and (f)). For example,
the organelle of motility in E. coli and Salmonella typhimurium
is the bacterial flagellum, consisting of a rotary motor [36], a
helical filament, and a hook which connects the motor to the
filament [1, 20, 37]. The filament has a diameter of ≈20 nm and
traces out a helix with contour length ≈10 µm. In the absence
of external forces and moments, the helix is left-handed with
a pitch ≈2 .5 µm and a helical diameter ≈0.5 µm[23]. 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 µms
−1
(see section 7.3.2)[38]. 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 [23].
There are many variations on these basic elements among
swimming bacteria. For example, Caulobacter crescentus
has a single right-handed helical filament (figure 1(b)), 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 [39]. During counter-
clockwise 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 [40].
The flagellar filament forms a compact coil when the motor
is stopped (figure 1(c)), 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 with bacteria. Archaea such as the various species of
Halobacterium swim more slowly than bacteria, with typical
speeds of 2–3 µms
−1
[41]. 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 [42] 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 [43]. 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 [44, 45].
The deformation can be helical or planar. These bacteria
swim faster in gel-like viscous environments than bacteria
with external flagella [46, 47]. 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 [44, 48]. Finally, we mention the case of Spiroplasma,
helically shaped bacteria with no flagella (figure 1(d)). These
cells swim via the propagation of pairs of kinks along the
length of the body [49]. Instead of periplasmic flagella,
the kinks are thought to be generated by contraction of the
cytoskeleton [ 50–52].
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 [53]. The length of the
flagellum is 12 µm in some Hymenoptera [54], ≈20 µm for
hippos, ≈40 µm for humans [2] (figure 1(e)), ≈80 µm for mice
(figure 1(f)), and can be 1 mm [55] or even several centimeters
long in some fruit flies [56] (although in the last case the flagella
are rolled up into pellets and offered to the female via a ‘pea-
shooter’ effect [ 56]).
Many organisms have multiple flagella. Chlamydomonas
reinhardtii is an alga with two flagella that can exhibit both
ciliary and flagellar beat patterns (figure 1(g)). 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 [57]. Paramecium is another classic example of a
ciliated microorganism. Its surface is covered by thousands
3
Rep. Prog. Phys. 72 (2009) 096601 E Lauga andTRPowers
of cilia that beat in a coordinated manner [58], propelling the
cell at speeds of ≈500 µms
−1
(figure 1(h)). Arrays of beating
cilia are also found lining the airway where they sweep mucus
and foreign particles up toward the nasal passage [59].
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 [60], Kim and Karrila [61]
and Leal [62]; an introduction is also offered by Hinch [63] and
Pozrikidis [64].
To solve for the force distribution on an organism, we need
to solve for the flow field
u and pressure p in the surrounding
fluid. For an incompressible Newtonian fluid with density ρ
and viscosity η, the flow satisfies the Navier–Stokes equations
ρ
∂
∂t
+
u ·∇
u =−∇p + η∇
2
u, ∇·u = 0, (1)
together with the boundary conditions appropriate to the
problem at hand. In the case of swimming of a deformable
body, the no-slip boundary condition states that the velocity of
the fluid at the boundary is equal to the velocity of the material
points on the body surface. The Navier–Stokes equations are a
pointwise statement of momentum conservation. Once
u and
p are known, the stress tensor is given by
σ =−p1 + η[∇u +
(∇
u)
T
](1 is the identity tensor), and the hydrodynamic force
F and torque L acting on the body are found by integrating
over its surface S,
F (t) =
S
σ · n dS, L(t) =
S
x × (σ · n) dS,
(2)
where
x denotes positions on the surface S and n the unit
normal to S into the fluid (in this paper torques are measured
with respect to some arbitrary origin).
The Reynolds number is a dimensionless quantity which
qualitatively captures the characteristics of the flow regime
obtained by solving equation (1), and it has several different
physical interpretations. Consider a steady flow with typical
velocity U around a body of size L. The Reynolds number
Re is classically defined as the ratio of the typical inertial
terms in the Navier–Stokes equation, ∼ρ
u ·∇u, to the viscous
forces per unit volume, ∼η∇
2
u. Thus, Re = ρLU/η.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 t
adv
∼ L/U, whereas the typical time scale for this
perturbation to diffuse away from the body due to viscosity
is t
diff
∼ ρL
2
/η. We see therefore that Re = t
diff
/t
adv
, and a
low Reynolds number flow is one for which fluid transport is
dominated by viscous diffusion.
We can also interpret Re as a ratio of forces on the
body. A typical viscous stress on a bluff body is given by
σ
viscous
∼ ηU/L, leading to a typical viscous force on the body
of the form f
viscous
∼ ηUL. A typical inertial stress is given by
a Bernoulli-like dynamic pressure, σ
inertial
∼ ρU
2
, leading to
an inertial force f
inertial
∼ ρU
2
L
2
. We see that the Reynolds
number is given by Re = f
inertial
/f
viscous
, 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,
F = η
2
/ρ
has units of force, and that any body acted upon by the force
F will experience a Reynolds number of unity, independent
of its size. Indeed, it is easy to see that Re = f
viscous
/F and
Re = (f
inertial
/F )
1/2
, and therefore a body with a Reynolds
number of one will have f
inertial
= f
viscous
= F . A body
moving at low Reynolds number therefore experiences forces
smaller than
F , where F ≈ 1 nN for water.
What are the Reynolds numbers for swimming
microorganisms [3]? In water (ρ ≈ 10
3
kg m
−3
, η ≈
10
−3
Pa s), a swimming bacterium such as E. coli with U ≈
10 µms
−1
and L ≈ 1–10 µm has a Reynolds number Re ≈
10
−5
–10
−4
. A human spermatozoon with U ≈ 200 µms
−1
and L ≈ 50 µm moves with Re ≈ 10
−2
. The larger ciliates,
such as Paramecium,haveU ≈ 1mms
−1
and L ≈ 100 µm,
and therefore Re ≈ 0.1[13]. At these low Reynolds numbers,
it is appropriate to study the limit Re = 0, for which the
Navier–Stokes equations (1) simplify to the Stokes equations
−∇p + η∇
2
u = 0, ∇·u = 0. (3)
Since swimming flows are typically unsteady, we implicitly
assume the typical frequency ω is small enough so that the
‘frequency Reynolds number’ ρLω
2
/η is also small. Note
that equation (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 [60]. It
is a principle of virtual work which takes a particularly nice
form thanks to the linearity of equation (3). Consider a volume
of fluid V , bounded by a surface S with outward normal
n,in
which two solutions to equation (3) exist,
u
1
and u
2
, satisfying
the same boundary conditions at infinity. If the stress fields of
the two flows are
σ
1
and σ
2
, then the reciprocal theorem states
that the mixed virtual works are equal:
S
u
1
· σ
2
· n dS =
S
u
2
· σ
1
· n dS. (4)
3.2. Motion of solid bodies
When a solid body submerged in a viscous fluid is subject to an
external force smaller than
F , it will move with a low Reynolds
number. What determines its trajectory? Since equation (3)is
linear, the relation between kinetics and kinematics is linear.
Specifically, if the solid body is subject to an external force
F , and an external torque L, it will move with velocity U and
rotation rate
Ω satisfying
F
L
=
AB
B
T
C
·
U
Ω
, (5)
4
![Figure 19. Enhanced diffusion near a carpet of swimming cells [321]. Trajectory of fluorescent particles over a 10 s interval when located above a flat surface coated with attached bacteria (two-dimensional view from above the surface). (a) The particles are located 80µm above the surface, and move by Brownian motion. (b) When the particles are located 3µm above the surface, their effective diffusion is significantly larger than Brownian motion [321]. Picture reproduced from [321] courtesy of Kenny Breuer and by permission from the Biophysical Society, copyright 2004 (doi:10.1016/S0006-3495(04)74253-8).](/figures/figure-19-enhanced-diffusion-near-a-carpet-of-swimming-cells-mucn1zb8.webp)
![Figure 18. Locomotion powered by chemical reactions [313]. Polystyrene colloidal spheres in a solution of water and hydrogen peroxide ([H2O2]) with increasing concentration. Top: control experiment (inert spheres). Bottom: experiment where the spheres are half-coated by platinum, a catalyst for the reduction of [H2O2] into oxygen and water, which leads to directed swimming of the spheres (coupled to Brownian rotational diffusion). Picture reproduced from [313] courtesy of Ramin Golestanian and by permission from the American Physical Society, copyright 2007 (doi:10.1103/PhysRevLett.99.048102).](/figures/figure-18-locomotion-powered-by-chemical-reactions-313-12vm186e.webp)
