AMER. ZOOL., 36:608-618 (1996)
A New Look at Locomotion in Microorganisms: Rotating and Translating1
HUGH C. CRENSHAW
Department' of Zoology, Box 90325, Duke University,
Durham, North Carolina 27708-0325
SYNOPSIS. The locomotion and orientation of free-swimming microorganisms have been widely studied for more than a century. With few
exceptions, only the two-dimensional translational velocity of the organism in question is ever reported, yet a complete description of motion
requires a three-dimensional description of both the translational and rotational velocities. Recent theoretical work, considering both the translational and rotational velocities, has demonstrated that a previously unrecognized orientation mechanism exists—helical klinotaxis. Efforts to test
the predictions of this theory are described, concluding that helical klinotaxis is probably utilized by a diverse assemblage of free-swimming
microorganisms.
croorganisms. This is particularly true of
Organisms move through their environ- free-swimming microorganisms. Their moments, searching for food, mates, appropri- tion is truly three-dimensional (3D) in that
ate temperature, correct pH, sunlight, and preferred axes or planes of motion are not
many other factors important to life. The obvious. If you examine pond water, you
probability of a successful search is greatly will see rotifers, ciliates, flagellates, and
increased if the organism is able to orient bacteria moving in all directions in the conto appropriate stimuli. Orientation, there- tainer.
If a microorganism orients to a stimulus,
fore, has been widely studied to understand
both the physiological mechanisms under- frequently that stimulus also is 3D. A
lying orientation and the ecological conse- chemical that attracts mates diffuses in
quences of the resulting distributions of or- three dimensions, or the fluid motion that
carries the chemical through space is 3D.
ganisms.
Even if the stimulus is primarily one-diThe orientation of organisms has been
studied almost exclusively in two spatial di- mensional, the 3D motion of the organism
mensions, treating the organism's world as modulates the stimulus. For example, cona flat plane. This is largely acceptable for sider the motion of a phototactic alga that
terrestrial and benthic animals because mo- has an eyespot, creating a directional photion in the third dimension (usually vertical toreceptor (a photoreceptor that senses not
or parallel with the action of gravity) is a only the intensity of a beam of light at one
small component of their motion. This is point in space but also that beam's direcless acceptable for animals in a three-di- tion). As the alga moves in three dimenmensional environment (water or air), but sions, its photoreceptor points in all direcit can be a reasonable simplification for an- tions. In a collimated beam of light {i.e., all
of the light rays point in one direction, so
imals with straight, or rectilinear, motion.
This simplification, however, can obscure the stimulus is primarily one-dimensional)
important aspects of the orientation of mi- the motion of the photoreceptor produces a
signal that is the result not only of the
beam's direction and intensity, but also the
' From the Symposium Aquatic Locomotion: New organism's, and thus the photoreceptor's,
Approaches to Invertebrate and Vertebrate Biome- three-dimensional motion (see Foster and
chanics presented at the Annual Meeting of the Society
for Integrative and Comparative Biology, 27-30 De- Smyth, 1980).
cember 1995, at Washington, D.C.
Studying three-dimensional motion,
INTRODUCTION
608
ROTATING AND TRANSLATING IN MICROORGANISMS
however, is difficult, largely because there
are no conventional techniques for tracking
microorganisms in three dimensions. Standard optical systems (such as video cameras
and microscopes) create two-dimensional
images. Additionally, there are no widely
used techniques for analyzing 3D positional
data in biology, and there are no widely recognized statistics, similar to circular statistics, for analyzing orientation in three dimensions.
Studying the orientation of microorganisms has its own unique problems. Microorganisms move at low Reynolds number,
so nearby walls, such as the walls of an
observation vessel, introduce viscous wall
effects (Vogel, 1994) that can alter the organism's behavior. For example, sea urchin
spermatozoa that ordinarily swim along a
helical trajectory, become hydrodynamically trapped by a microscope slide such that
they swim in fiat spirals apposed to the
glass slide (Gray, 1955). The use of a large
observation vessel places the walls at a distance from organisms that are observed at
the center of the vessel, but here thermal
convection currents can have speeds that
are much larger than those of the organism,
confounding one's attempts to track the organism.
The work of Howard Berg and collaborators, who were largely responsible for describing the chemo-orientation of bacteria
(Berg and Brown, 1972; Berg, 1978; see
Berg, 1975 for review), provides an excellent example of how analysis of the 3D motion of a microorganism can reveal unseen
aspects of orientation behavior. Berg (1971,
1978) developed a microscope that could
track bacteria in three dimensions to test the
idea that orientation could be achieved by
random motion that is biased by the direction of a stimulus field. This form of orientation is called a "kinesis" or, more generally, a "biased random walk."
Berg had to test the assumption of the
theory as well as its prediction. The assumption was that when an organism
changed direction, the new direction had no
relationship to the direction of the stimulus—it was chosen randomly. The prediction was that net motion in the direction of
the stimulus could be achieved if the fre-
609
FIG. 1. The relationship between rotational velocity,
o), translational velocity, V, and a helical trajectory. A
helical trajectory results whenever an organism moves
with constant rotational and translational velocity such
that these two vectors are neither parallel nor perpendicular. V is always tangential to the trajectory, and u>
is parallel to the axis of the helix. (Redrawn from
Crenshaw, 1990.)
quency of changes in direction was influenced by stimulus intensity. The subtlety
here is that, in a linear chemical concentration gradient, the prediction could be tested
with two-dimensional tracking, but the assumption could not. A change in direction
for bacteria involves a stop and tumble behavior, which can easily be detected with
two-dimensional tracking. However, testing
the assumption required knowledge of the
direction of motion in three dimensions before and after the turn.
My lab has striven both to develop techniques for tracking organisms in three dimensions and to use these techniques to
study the orientation behaviors of microorganisms, especially behaviors that do not
involve a biased random walk. The remainder of this paper will be divided into three
parts. The first briefly describes a newly
identified orientation behavior (Crenshaw,
1990, 19936). The second describes the
techniques we have developed for 3D tracking and analysis. The third describes our
use of these techniques to test whether helical klinotaxis is exhibited by microorganisms.
HELICAL MOTION AND ORIENTATION
Most microorganisms swim along a helical trajectory—their path traces the
threads of a screw (Fig. 1). Helical swimmers include larger bacteria, most freeswimming protists, many of the motile
spores of fungi and of plants, many micrometazoa (including larval invertebrates),
and many of the spermatozoa of animals.
610
HUGH C. CRENSHAW
a/p -•
UJ2
FIG. 2. The motion of a rigid body, (a) A coordinate
system can be fixed to the body of the organism, using
the organism's anterior/posterior (a/p), dorsal ventral
(d/v), and left/right (1/r) axes, (b) The 3D translational
velocity (V) has three components relative to the body
axes, (c) The 3D rotational velocity (a>) has three components relative to the body axes, (d) Motion with constant translational velocity (V) pointing anteriorly. (Redrawn from Crenshaw, 1990.)
There are few explanations of why helical motion is so common. Jennings (1901)
explains that the rotation of an organism
swimming along a helix is like the rifling
of a bullet—it keeps an otherwise meandering object moving along a nearly
straight trajectory. Purcell (1977) explains
that helical motion is the default trajectory
for an organism moving at low Reynolds
number. Motion at low Reynolds number
must be driven by cyclic, asymmetric deformations of the body, and each deformation causes the body to translate and rotate.
This repeated translation and rotation produces a helical trajectory (a point that is
more completely described in the next paragraph). Crenshaw (1989a, b, 1990, 1993a,
b) and Crenshaw and Edelstein-Keshet
(1993) explain that helical motion permits
an organism to orient directly to an external
stimulus, without a biased random walk.
This mechanism has been termed "helical
klinotaxis."
An explanation of helical klinotaxis first
requires a description of how organisms
FIG. 3. Trajectories resulting from motion with constant V and ID. (a) If V and o> are parallel, the trajectory
is a straight line along which the organism rotates its
body, (b) If V and a> are perpendicular, the trajectory
is a circle. For any other angle between V and u, the
trajectory is a helix (see Fig. 1). (Redrawn from Crenshaw, 1990.)
swim along a helical trajectory. This is easiest if the organism is treated as a rigid
body, in which the motions of parts of the
organism, relative to each other, are not
considered. Motion of a rigid body in three
dimensions is completely described by the
body's translational velocity, V, and rotational velocity, to (Figs. 1 and 2)2. If the
body has constant V and w, then one of
three trajectories will result:
(1) If w is parallel to V, then the trajectory
will be a straight line, and the organism's
body will rotate on an axis that is parallel
to the direction of motion (Fig. 3a).
(2) If co is perpendicular to V, then the trajectory will be a circle (Fig. 3b).
(3) For any other orientation of to relative
to V, the trajectory will be a helix (Fig. 1).
The rest of this description is easier if V
and to are described relative to axes fixed
to the body of the organism, e.g., the an2
For those more familiar with the term "angular
velocity," angular velocity and rotational velocity are
synonymous. In this case, rotational velocity is defined
by the "right-hand rule": Holding one's hand with
thumb stretched outward and fingers curved, the thumb
gives the axis of rotation and the fingers give the direction of positive rotation.
ROTATING AND TRANSLATING IN MICROORGANISMS
terior/posterior, dorsal/ventral, and left/right
axes of the organisms' body (Fig. 2a). V
and co can then be described by three components, one for each body axis (Fig. 2b,
c). We can further simplify things if V is
constant, always pointing anteriorly (i.e.,
the dorsal/ventral and left/right components
of V are zero) (Fig. 2d).
Under these conditions, the rotational velocity, to, is parallel to the axis of the helix,
which is the organism's net direction of motion (Fig. 1). Thus, an organism that swims
along a helical trajectory actually moves in
the direction of co, not of V. See Crenshaw
(1989a, b, 1993a) for a more complete discussion.
Orientation to a stimulus, therefore, is accomplished by alignment of co, and thus the
axis of the helix, with the stimulus. This
point has long been understood for phototaxis to a beam of light, having been described qualitatively by Jennings (1904)
and Buder (1917) and more recently, and
more rigorously, by Foster and Smyth
(1980).
It has not been understood, however, how
alignment of co is accomplished. Crenshaw
(1993ft) demonstrates that alignment occurs
if the components of co are simple functions
of stimulus intensity, with the stimulus being a chemical concentration gradient or a
beam of light3. This is best explained with
an example. Let space be described by the
coordinate system, XYZ. Consider a concentration gradient of some chemical attractant such that the concentration increases in
the positive-X direction (Fig. 4a). If an organism moves with co pointing in the
positive-Z direction, then the axis of the helix is perpendicular to the gradient, and the
organism's helical motion will carry it up
and down the gradient (Fig. 4a). In Figure
4b, the organism begins to move down the
gradient. If the anterior/posterior compo3
Note that I must give precedence here to Herbert
Jennings (1904) and Charles Brokaw (1958). Jennings
appears to be the first to describe this behavior (see
pp. 54-57 of his monograph), but he was unable to
explain the kinematics of the motion, and his description has not received much attention. Charles Brokaw
used different mathematical techniques to describe
many aspects of this mechanism in his doctoral thesis
(Brokaw, 1958).
611
FIG. 4. Orientation to a chemical concentration gradient by helical klinotaxis. (a) Initially an organism
swims in a chemical concentration gradient with the
concentration increasing in the positive-X direction,
(b) The organism begins responding such that ioB is
proportional to chemical concentration. As the organism swims down the gradient, a>j becomes smaller, and
(o becomes more nearly perpendicular to V, which is
pointing down the gradient. The result is that the axis
of the helix rotates into the gradient, (c) As the organism begins to swim up the gradient, <D, becomes larger,
and u> becomes more nearly parallel to V, which is now
pointing up the gradient. The result is that the axis of
the helix again rotates into the gradient. (Redrawn
from Crenshaw, 19936.)
nent of co is proportional to the chemical
concentration, then this component becomes smaller as the organism moves down
the gradient. This causes co, and thus the
axis of the helix, to become more nearly
perpendicular to V. This causes co to align
with the concentration gradient because V
is pointed down the gradient. (Note that the
corollary to this statement is that whenever
co changes direction relative to the body of
the organism, the axis of the helix changes
direction in space (Crenshaw and EdelsteinKeshet, 1993).) As the organism completes
one turn of the helix and begins to swim
back up the gradient (Fig. 4c), the anterior/
posterior component of co becomes larger,
and co becomes more nearly parallel to V.
This causes co to further align with the concentration gradient because V is now pointed up the gradient. With repeated turns of
the helix, the axis of the helical trajectory
will completely align with the concentration
gradient.
Orientation to a beam of light can occur
by a similar mechanism. Most phototactic
algae have directional photoreceptors. As
Foster and Smyth (1980) explain, rotation
of the cell body causes the signal produced
by the photoreceptor to oscillate, much like
612
HUGH C. CRENSHAW
motion up and down a chemical concentration gradient causes the stimulus intensity
to oscillate, and, like above, orientation will
occur if the components of the rotational
velocity are simple functions of the signal
produced by the directional photoreceptor
(Crenshaw, 1993/?).
Computer simulations demonstrate that
helical klinotaxis is a remarkably rigorous
orientation behavior, working for a wide variety of stimuli and permitting either positive or negative orientation (Crenshaw,
3D TRACKING AND ANALYSIS OF
TRAJECTORIES
As with Berg and Brown's (1972) analysis of bacterial chemotaxis, tests of the
theory of helical klinotaxis require knowledge of the 3D trajectory of an organism as
it responds to a stimulus. Using a technology different than that used in Berg's 3D
tracking microscope, I developed a microscope that permits 3D tracking of an organism in the center of a large observation vessel, removing viscous wall effects (Fig. 5)
(Crenshaw, 1991). This 3D tracking microscope uses two video cameras, oriented at
right angles, to sample the organism's XYZ
coordinates 60 times per second with submicron precision.
This 3D tracking microscope permits the
simultaneous tracking of several objects, allowing correction of motion due to convection currents. An immotile particle near the
moving microorganism is tracked along
with the microorganism. The motion of the
immotile object, which is due entirely to the
convection current, is then subtracted from
the motion of the microorganism, yielding
the motion of the microorganism relative to
the water.
V and to can be determined from the 3D
trajectory, if some simplifying assumptions
are made (Crenshaw, 1993<a): V has constant direction with respect to the body of
the organism, and co has only two components, one parallel to V and one perpendicular to V, co and co±, respectively. V is then
given by the velocity of the trajectory, and
co and <a± are then given by the trajectory's
torsion and curvature, respectively. More
precisely, co^ equals the curvature times the
laser
strobe
camera lens/condenser
observation vessel
darkfield annulus
/
camera lens/condenser
computer-^
video camera
.split-frame generator
videotape recorder
FIG. 5. The 3D tracking microscope. Two microscopes are aligned such that their optical axes are perpendicular to one another and perpendicular to gravity.
The image from each microscope is viewed by a video
camera, and the signals from the two cameras are combined by a split frame generator such that the view of
one camera is on the left-hand side of the resulting
video image and the view of the other camera is on
the right-hand side of the video image. A small point
in the center of the vessel is focused onto the center
of the view of both cameras. These two, perpendicular
views define a volume in the center of the vessel in
which microorganisms are tracked. No attempt is made
to move the vessel or the microscopes. Rather, microorganisms are allowed to freely enter and leave the
scanned volume. The videotape is analyzed field-byfield, collecting the XYZ coordinates of the microorganism at the field rate of video (60 times per second
with NTSC video). (Redrawn from Crenshaw, 1991.)
organism's speed, and co, equals the torsion
times the organism's speed. The velocity,
curvature, and torsion of the trajectory are
calculated directly from the XYZ coordinates over time. These calculations are very
sensitive to error, or noise, in the measured
trajectory. Nevertheless, I do not smooth
the trajectory, nor do I fit curves, removing
the possibility that artefact is introduced by
smoothing or fitting.
Figures 6 and 7a present the trajectories
of two spermatozoa of the sea urchin, Arbacia punctulata (techniques for handling
of sperm are described in the next section).
The trajectory in Figure 6 is straight, and
speed, co, and cox for this trajectory are all
nearly constant (data not shown). The tra-
613
ROTATING AND TRANSLATING IN MICROORGANISMS
z
-•Y
-•Y
FIG. 6. Helical trajectory of a spermatozoan of A.
punctulata. This spermatozoan is unstimulated, and its
trajectory is straight. Two, orthogonal plane projections of the 3D trajectory are presented. The trajectory
begins at the dot (•). (Redrawn from Crenshaw, 1990.)
jectory in Figure 7a has two sections where
the axis of the helix changes direction.
These sections of the trajectory are marked
by decreases in o>± (Fig. 8b); o>n (Fig. 7c)
and speed (Fig. 7d) remain relatively constant. These two trajectories demonstrate
that changes in the components of to cause
the axis of the helix to change direction, as
predicted by Crenshaw and Edelstein-Keshet (1993).
CHEMOTAXIS BY HELICAL KLINOTAXIS
Spermatozoa of the sea urchin, Arbacia
punctulata, orient to a concentration gradient of an attractant released by the egg of
the species (Ward et al., 1985). The attractant is a polypeptide, named "resact" (Suzuki et al., 1984), and the events from receptor binding to initial second messenger
events are fairly well understood (Ramarao
and Garbers, 1985; Cook et al., 1994). Nevertheless, the means by which orientation is
achieved has not been described (Ward et
al., 1985; Miller, 1985; Cook et al., 1994).
I have been tracking these sperm to determine if they exhibit helical klinotaxis.
Sperm are obtained from male urchins
stimulated with mild electrical currents
(Lutz and Inoue, 1986). The ejaculate is
stored on ice and used within 1 hour of collection. Ejaculate is diluted into artificial
sea water to a concentration of about 1 X
12
8-
I
3
0
12-
1
3°
4-
0
300200-
Ia 1000
1
2
3
4
Time (s)
FIG. 7. Helical trajectory of a spermatozoan of A.
punctulata in which the axis of the helix bends, (a) 3D
trajectory in which two, orthogonal plane projections
of the 3D trajectory are presented. The trajectory begins at the dot (•). Two changes of direction are evident, (b) co± decreases two times, corresponding to the
two changes of direction of the trajectory, (c) o>, remains fairly constant. The high noise is typical of this
parameter, (d) Speed remains fairly constant. (7a is redrawn from Crenshaw, 1990.)
105 spermml"1 within the observation vessel of the 3D tracking microscope. A concentration gradient of resact (described below) is presented to the sperm beginning 30
sec after dilution.
HUGH C. CRENSHAW
614
a
3"
(s/s
1
a
7,0-
3°
10-
o-
1
l
. i/M
» M iJlMA
r V.
FIG. 8. Trajectory of a spermatozoan of A. punctulata
that is exposed to a chemical concentration gradient of
resact. The chemical concentration increases in the
positive-Z direction. The arrowheads provide registry
between the trajectory and the measured parameters,
(a) 3D trajectory in which two, orthogonal plane projections of the 3D trajectory are presented. The trajectory begins at the dot (•) and ends when the sperm
contacts the dialysis membrane. o)± (b), to (c), and
speed (d) all change at the point where the trajectory
aligns with the gradient. See text for complete explanation.
A chemical concentration gradient of resact is established as follows. The top of
the observation vessel is closed with a stopper to prevent convection currents within
the vessel driven by evaporation. A stiff
polyethylene tube (6.25 mm outer diameter,
4 mm inner diameter) passes through the
stopper, and the end of the tube inside the
observation vessel is covered with dialysis
membrane (molecular weight cutoff of
about 20,000). A second tube, made of
glass, is filled with resact diluted into artificial sea water to a final concentration of
about 10"6 M. A concentration gradient is
initiated when the resact solution is injected
from the glass tube into the polyethylene
tube. The dialysis tube prevents the mixing
of the resact solution with the sea water inside the vessel and the currents that would
be initiated by this mixing.
The dialysis membrane is positioned in
the observation vessel such that it is just at
the top of the volume scanned by the video
cameras. The inner diameter of the polyethylene tube is much larger than the width of
the scanned volume (typically less than 0.5
mm), so the dialysis membrane creates a
large planar source of resact. Diffusion of
the resact through the dialysis membrane
and into the observation vessel creates a
one-dimensional, vertical concentration
gradient with concentration decreasing
downward.
This concentration gradient is initiated
when the resact solution is injected from the
glass tube into the polyethylene tube. A laser briefly flashes onto the pickup tube of
one video camera to mark the point in time
when the concentration gradient is initiated.
The response of sperm is then recorded as
the concentration gradient is established by
diffusion into the vessel.
The trajectory of a sperm responding to
such a gradient of resact is presented in figure 8a. Initially the sperm swims along a
straight helical trajectory, but at one point
the axis of the trajectory aligns with the
gradient, and the sperm swims up the gradient. When the axis aligns with the gradient, there are large changes in w and w±
(Fig. 8b, c). Speed remains nearly constant
throughout this trajectory (Fig. 8d). As the
sperm swims up the gradient, there is a con-
ROTATING AND TRANSLATING IN MICROORGANISMS
tinuous decrease in u>L and an increase in wuntil the sperm is nearly at the dialysis
membrane. At this point, co± suddenly increases and w decreases.
This trajectory is representative of all
sperm we have observed. That this response
is indeed a response to resact is supported
by controls in which the sperm are tracked
as they swim near a dialysis membrane with
only sea water injected into the polyethylene tubing—no changes in the trajectory
are observed. Furthermore, the distance
from the dialysis membrane at which a response is observed increases with time, as
expected from the evolution of the concentration gradient over time due to diffusion
of resact from the dialysis membrane.
Clearly, the components of w change as
a function of the concentration of resact,
and they change as the sperm orients to a
concentration gradient. At this point in
time, we are still unable to state that the
components of u> change in a manner that
effects orientation. The resolution of our
analyses does not yet allow us to state that
the changes in the components of w are responsible for alignment because we are unable to quantify the chemical concentration
at the position of the sperm in the gradient.
Nevertheless, all results to date are consistent with the hypothesis that the sperm of
A. punctulata orient to gradients of resact
by helical klinotaxis.
615
I have been tracking C. reinhardtii with
the 3D tracking microscope while stimulating the cell with a beam of light. This stimulation beam is a nearly collimated beam of
light from a mercury arc lamp (100 W) that
enters the bottom of the vessel and exits
through the top. IR and UV reflecting filters
are used to eliminate these wavelengths
from the stimulation beam; otherwise, the
spectral distribution of the light is unaltered. The cells are observed with light in the
near IR (\ > 750 nm) to which the cells
are insensitive.4 The stimulation beam is ordinarily blocked by an opaque plate. When
the cells in the observation vessel are to be
stimulated, the plate is lifted out of the
stimulation beam. A fiber optic transmits a
small amount of light from the stimulation
beam to the photo-sensitive surface of one
video camera such that a bright spot appears in the corner of the video image when
the stimulation beam enters the observation
vessel. A bright beam is used to elicit negative orientation.
C. reinhardtii is cultured in Sueoka's
high salt medium, bubbled with air, in a 12
hr light/12 hr dark cycle. These culture conditions synchronize the cell cycle of the cultures (Harris, 1989). Cells are used in the
interval 6-10 hours after the lights turn on,
ensuring that all cells are zygotes of nearly
the same size. Cultures are used in logphase growth, with culture densities of approximately 1 X 106 cells per ml. Prior to
PHOTOTAXIS BY HELICAL KLINOTAXIS
an experiment, cells are diluted into fresh
3
Orientation of the flagellate, Chlaymdo- media to a density of about 20 X 10 cells
monas reinhardtii, to light has been widely per ml, given about 30 minutes to adjust to
studied for nearly a century. Nevertheless, the fresh media, and are used within the
we do not understand how this microorgan- next one hour. Cells typically are held in
ism orients to a beam of light (Riiffer and the dark (with IR illumination for obserNultsch, 1991; Witman, 1993). Qualitative- vation) and stimulated for no more than 20
ly, the orientation behavior of C. reinhardtii sec at a time, with at least 60 sec between
is consistent with helical klinotaxis. When successive stimuli, for a total of no more
the microorganism is stimulated by a beam than 10 stimulations.
of light, the axis of the helical trajectory
Figure 9a presents a typical trajectory
aligns with the direction of the beam. If the from photostimulation of C. reinhardtii.
beam is of the correct intensity, the axis This cell initially swims with a trajectory
aligns such that the microorganism swims
toward the source of the light (positive pho4
Red light (\ > 600 nm) is widely used to observe
totaxis). If the beam is too intense, the axis
C. reinhardtii; however, I have observed significant
aligns such that the microorganism swims changes in to when C. reinhardtii is illuminated with
away from the source of light (negative red light. The cell does not orient to red light, but its
motion does change.
phototaxis).
616
HUGH C. CRENSHAW
that is not really helical, but spontaneously
switches to a helical trajectory 2 sec after
we begin tracking.5 The stimulation beam
turns on at 3.8 s. There is a stop response
(the speed drops to nearly zero—Fig. 9d)
when the beam turns on. The cell then exhibits a brief period (3.8 sec to 5.2 sec) of
positive orientation during which the trajectory is a pronounced helix, followed by a
hairpin turn (5.2 sec to 6 sec) as the cell
switches to negative orientation. The radius
and pitch of the helical trajectory change
rapidly during the hairpin turn. During negative orientation, the trajectory is a more
irregular helix.
The transition from positive to negative
orientation is marked by a switch from a
left-hand helical trajectory to a right-hand
helical trajectory. This is reflected in coM as
a switch from u, < 0 to io, > 0 (Fig. 6c).
A right-hand helix is never seen in unstimulated cells. The initial period of positive
orientation does not occur in all cells; a few
cells (< 10%) proceed directly to negative
orientation with no observable period of
positive orientation. The trajectory of these
cells is a right-hand helix after stimulation.
Thus, right-hand helical motion and negative orientation appear correlated.
The change of handedness of the helix
potentially explains the change in the direction of orientation. As Crenshaw (1989&,
1993a, b) explains, when a cell swims
along a helical trajectory, one part of the
cell, such as the eyespot, faces away from
the axis of the helix (Fig. 10a—left-hand
side). A change in the handedness of a helical trajectory causes this part of the cell
to now face toward the axis of the helix
(Fig. 10a—right-hand side). Because the
eyespot determines the directionality of the
photoreceptor (Foster and Smyth, 1980;
Kreimer and Melkonian, 1990), this change
in the handedness of the helix reverses the
phase of the stimulus intensity, relative to
the helical trajectory (Fig. 10b) (Foster and
5
C. reinhardtii does not always swim along a helical trajectory when the cells are illuminated with only
IR light. Most cells swim along meandering trajectories, and I regularly see apparently spontaneous transitions from nearly straight trajectories to helical trajectories and back to straight trajectories.
15
5-\
FIG. 9. Trajectory of C. reinhardtii that is exposed to
a beam of light. The beam points in the positive-Z
direction. The arrowheads indicate when the beam was
turned on. (a) 3D trajectory in which two, orthogonal
plane projections of the 3D trajectory are presented.
The trajectory begins at the dot (•). (b) w± rises following stimulation, but decreases as the cell begins negative orientation, (c) o>, remains negative, indicating a
left-hand helix, until the cell is halfway through the
hairpin turn, when it becomes primarily positive, indicating a right-hand helix, (d) Speed drops nearly to
zero when the beam is turned on, indicating a stop
response; however, it rapidly returns to nearly normal.
See text for complete explanation.
Smyth, 1980; Crenshaw, 19936). The result
is that a flagellar response that causes positive phototaxis for a left-hand helical trajectory causes negative phototaxis after
transition to a right-hand helical trajectory.
617
ROTATING AND TRANSLATING IN MICROORGANISMS
Beam of light
Right-hand
Helix
si
Closest to
Light
Furthest
from Light
Oosest to
Light
Furthest
from Light
FIG. 10. Switching the handedness of the helix reverses the phase of the stimulus, and thus reverses the direction
of orientation, (a) C. reinhardtii initially swims with its eyespot pointing away from the axis of the helix;
however, when the cell switches to a right-hand helix, the eyespot points toward the axis of the helix. (The axis
of these helical trajectories is pointing out of the page.) (b) The perceived stimulus intensity, determined by the
orientation of the eyespot relative to the beam of light, reverses phase, going from being highest when the cell
is nearest the source of the light to being highest when the cell is furthest from the source of light. This reversal
of the phase of the stimulus will cause the cell to switch from positive to negative orientation. (Note that there
is no agreement in the literature as to how the eyespot points in unstimulated C. reinhardtii—Foster and Smyth,
1980; Ruffer and Nultsch, 1985; Witman, 1993.)
Again, it is clear that w changes as a
function of stimulus intensity. However, we
are unable to state definitively that these
changes in the components of w effect orientation. The resolution of our analyses do
not yet allow us to state that the changes in
the components of w are responsible for
alignment because we are unable to determine how stimulus intensity is modulated
by changes in the orientation of the cell
(which determines the orientation of the directional photoreceptor). Nevertheless, as
with the sperm of A. punctulata, all results
to date are consistent with the hypothesis
that C. reinhardtii orients to beams of light
by helical klinotaxis.
SUMMARY
Analysis of the 3D motion of microorganisms has provided new insights into
their orientation behaviors. Importantly, helical klinotaxis offers an explanation for
several orientation behaviors that have
eluded more complete explanation, such as
chemotaxis in the sperm of sea urchins and
phototaxis in Chlamydomonas. Circumstantial evidence strongly suggests that chemotaxis in ciliates arises from helical klinotaxis (Crenshaw, 1989a, b). Additionally,
Brokaw (1958, 1974) provides evidence
that bracken spermatozoids orient to chemical concentration gradients via helical klinotaxis, and Jennings (1904), in his first descriptions of orientation by helically swimming microorganisms, describes similar behaviors in the flagellate, Euglena, and in
rotifers. Thus, helical klinotaxis may be
employed by many, diverse micro-organisms.
3D analysis of the motion of microorganisms requires the development of new
techniques for both data collection and data
analysis. Such technical hurdles have, to
date, largely dissuaded scientists from pursuing 3D analyses. Nevertheless, microorganisms move in a 3D world in which the
constraints of solid surfaces and gravity
rarely restrict motion in any single direction
or plane of motion. To understand their motion, 3D analysis is required.
ACKNOWLEDGMENTS
I wish to thank my collaborators for assistance throughout this work. Most important among these are Leah Edelstein-Keshet
(University of British Columbia) for assistance in the early stages of the development
of the mathematical descriptions of helical
klinotaxis and a number of undergraduates
at Duke University whose assistance has
618
HUGH C. CRENSHAW
been invaluable and whose enthusiasm has
been inspiring—Anna Brant, Sonja Burbano, Marc Harvey, K. C. Kim, Tonia Korves,
Tina Kraljevic, Rahul Rathod, Bill Schloss,
Anna Wulfsberg, and Rob Zemble. Finally,
two anonymous reviewers offered several
helpful suggestions for the manuscript. This
work has been supported, in part, by funds
from the U.S. National Science Foundation
(DCB-8819271).
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