Analysis of the effects of turning bias on chemotaxis in C. elegans.

Abstract

C. elegans advances up a chemical gradient by modulating the probability of occasional large, course-correcting turns called pirouettes. However, it remains uncertain whether C. elegans also uses other behavioral strategies for chemotaxis. Previous observations of the unusual, spiral-shaped chemotaxis tracks made by the bent-head mutant unc-23 point to a different strategy in which the animal continuously makes more subtle course corrections. In the present study we have combined automated tracking of individual animals with computer modeling to test the hypothesis that the pirouette strategy is sufficient on its own to account for the spiral tracks. Tracking experiments showed that the bent-head phenotype causes a strong turning bias and disrupts pirouette execution but does not disrupt pirouette initiation. A computer simulation of disrupted pirouette behavior and turning bias reproduced the spiral tracks of unc-23 chemotaxis behavior, showing that the pirouette strategy is sufficient to account for the mutant phenotype. In addition, the simulation reproduced higher order features of the behavior such as the relationship between the handedness of the spiral and the side to which the head was bent. Our results suggest that the pirouette mechanism is sufficient to account for a diverse range of chemotaxis trajectories.

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In a radial gradient of chemical attractant, wild-type
nematodes Caenorhabditis elegans reach the gradient peak by
moving almost directly up the gradient (Ward, 1973; Pierce-
Shimomura et al., 1999). Statistical analysis of this behavior
shows that movement down the gradient raises the probability
of sharp turning events called pirouettes, and that movement
up the gradient lowers the probability of pirouettes (Pierce-
Shimomura et al., 1999). Pirouettes not only serve to stop the
animal from moving down the gradient, they also reorient the
animal up the gradient. Computer simulations of the pirouette
mechanism have shown that it is sufficient to reproduce
directed tracks like those of real worms.
Certain mutant strains exhibit chemotaxis trajectories that
differ markedly from wild-type trajectories, raising the
question of whether the pirouette mechanism is the only
behavioral strategy for chemotaxis in C. elegans. One such
strain is the so-called bent-head mutant (unc-23), in which the
worm’s head is permanently fixed at an angle to the main body
axis because of a defect in muscle attachment (Fig.·1A; Ward,
1973; Waterson et al., 1980). In a radial gradient, bent-head
mutants move up the gradient on a striking spiral trajectory
centered on the gradient peak (Ward, 1973). Visual inspection
of unc-23 mutants suggests that the bent head acts like a rudder
at the front of the worm, causing it to orbit in tight overlapping
curls as it progresses along the spiral (Ward, 1973). Thus the
chemotaxis trajectory of the unc-23 mutant is complex and it
is not immediately obvious whether such a track could be
produced by a strategy as simple as the pirouette mechanism.
Indeed, based on inspection of the tracks of unc-23 and other
mutants, an alternative to the pirouette mechanism has been
proposed whereby the animal continually aligns its head with
the direction of steepest ascent.
Here we use a combination of experiment and theory to
show that the pirouette mechanism is sufficient to reproduce
the spiral chemotaxis trajectory of unc-23 mutants. We
conclude that the pirouette mechanism explains a wider range
of orientation trajectories than previously recognized.
Materials and methods
Animals
The strain unc-23(e611) was obtained from The C. elegans
Genetics Center and raised as described by Brenner (1974).
This allele was the same one used in the original study of the
unc-23 mutant (Ward, 1973). Mutants were grown at room
temperature (19–24°C) in mixed-stage populations on agar-
filled plates seeded with OP50 bacteria for food. The
chemotaxis behavior of unc-23 mutants was compared with
The Journal of Experimental Biology 208, 4727-4733
Published by The Company of Biologists 2005
doi:10.1242/jeb.01933
C. elegans advances up a chemical gradient by
modulating the probability of occasional large, course-
correcting turns called pirouettes. However, it remains
uncertain whether C. elegans also uses other behavioral
strategies for chemotaxis. Previous observations of the
unusual, spiral-shaped chemotaxis tracks made by the
bent-head mutant unc-23 point to a different strategy in
which the animal continuously makes more subtle course
corrections. In the present study we have combined
automated tracking of individual animals with computer
modeling to test the hypothesis that the pirouette strategy
is sufficient on its own to account for the spiral tracks.
Tracking experiments showed that the bent-head
phenotype causes a strong turning bias and disrupts
pirouette execution but does not disrupt pirouette
initiation. A computer simulation of disrupted pirouette
behavior and turning bias reproduced the spiral tracks of
unc-23 chemotaxis behavior, showing that the pirouette
strategy is sufficient to account for the mutant phenotype.
In addition, the simulation reproduced higher order
features of the behavior such as the relationship between
the handedness of the spiral and the side to which the head
was bent. Our results suggest that the pirouette
mechanism is sufficient to account for a diverse range of
chemotaxis trajectories.
Key words: chemotaxis, orientation, locomotion, nematode,
behavioral modeling, Caenorhabditis elegans.
Summary
Introduction
Analysis of the effects of turning bias on chemotaxis in C. elegans
Jonathan T. Pierce-Shimomura1,2,*, Michael Dores2and Shawn R. Lockery2
1Ernest Gallo Clinic and Research Center, Department of Neurology, Programs in Neuroscience and Biomedical
Science, University of California, San Francisco, 5858 Horton Street, Suite 200, Emeryville, CA 94608, USA and
2Institute of Neuroscience, University of Oregon, Eugene, OR 97403-1254, USA
*Author for correspondence (e-mail: jonp@egcrc.net)
Accepted 17 October 2005
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wild-type C. elegans (N2) from a previous study
(Pierce-Shimomura et al., 1999). Wild-type worms
(N=84) were raised identically to unc-23 mutants
(N=52) and tested in parallel. Analysis of this first
data set is presented in Figs·1Bi,Biii,Biv,C,D and 2.
A second set of unc-23 mutants (N=10) was tested
separately to compare with mutants from a previous
study (Ward, 1973). Analysis of this second data set
is presented in Figs·1Bii and 5A.
Chemotaxis assays and behavioral analysis
Animals were rinsed off food plates with assay
medium and placed on a foodless, agar-filled holding
plate for 0.5–2·h. The assay medium contained (in
mmol·l–1): NH4Cl 2, CaCl21, MgSO41, and KPO4
25, pH·6.5. The agar in the holding plate contained
only assay medium. Single adult hermaphrodites (1–3
days old) were then transferred to assay plates
(diameter=9·cm, agar depth=0.264·cm) containing
either a radial Gaussian-shaped gradient of the
attractant ammonium chloride (NH4Cl) (N=77) or a
uniform concentration of NH4Cl (N=57). The
animal’s centroid coordinates relative to the peak of
the gradient were recorded with an automated
tracking system that samples behavior at ~1·Hz as
described previously (Pierce-Shimomura et al.,
1999). The tracking system used an image analysis
program (ImagePro Plus, Media Cybernetics, Silver
Spring, MD, USA) to record the position of the
animal and a computer-controlled motorized stage
(Prior Scientific, Rockland, MA, USA) to reposition
the animal within the field of view as necessary.
Centroid coordinates and instantaneous velocity were
used to calculate the estimated sensory input (rate of
change of attractant concentration (dC/dt) at the
animal’s location), and the motor output (speed and
turning rate) with equations described previously
(Pierce-Shimomura et al., 1999). Bearing (B) was
defined as the angle between the worm’s
instantaneous velocity vector and the vector pointing
to the gradient peak. The times at which pirouettes
were initiated were identified by an empirically tested
algorithm based on the time series of turning rate
values (Pierce-Shimomura et al., 1999). The
probability of pirouette initiation (Ppiro) was
calculated by dividing the total number of pirouette
initiation events by the total number of time samples
in each assay. Turning bias ?biaswas defined as the
absolute value of an animal’s average instantaneous
turning rate over the course of an assay, excluding
periods during pirouettes.
Chemical gradients
Gradients were formed by adding drops of the
attractant NH4Cl to plates containing agar made from assay
medium. Two different types of Gaussian gradients were used.
The first type of gradient (Fig.·1Bii) was identical to the one
used in the original study of the unc-23 mutant (Ward, 1973).
J. T. Pierce-Shimomura, M. Dores and S. L. Lockery
1 cm
*
1 cm
*
0.3
0.2
0.1
0
0.6 0.40.20
Speed (mm s–1)
WT
0.4
0.3
0.2
0.1
0
12840
Turning bias (deg. s–1)
WT
+
1 cm
*
No gradient Gradient
Wild type
unc-23
+
1 cm
*
1 mm
Time
(min)
0
20 or 90
Wild type
unc-23
1 mm
Curls
A
BiBii
Biii Biv
CD
Probability
Probability
unc-23
unc-23
Fig.·1. Effect of the unc-23 bent-head phenotype on movement. (A)
Morphology of typical wild-type and unc-23 worms. The bend in the head of
the unc-23 mutant is indicated by an arrow. (B) Comparison of dispersion and
chemotaxis behavior for wild-type (Biii,Biv) and unc-23 mutant (Bi,Bii)
animals. Shown are representative tracks made by individual unc-23 mutants
and wild-type animals in the absence of a gradient (Bi,Biii) and in the presence
of a radial gradient of chemical attractant (NH4Cl; Bii,Biv). The unc-23 track
comprises a series of curls visible in the expanded inset. Note that the
handedness of the curls, indicative of turning bias (clockwise), is opposite to
the handedness of the spiral track (counterclockwise). Pirouette initiation
events are indicated by orange crosses superimposed on the magnified track
within the inset. Animals were started near asterisks; the gradient peak is
indicated by the plus sign. Color represents time, as shown in the key. Elapsed
time was 20·min for tracks Bi, Biii and Biv, and 90·min for the track in Bii.
(C,D) Distributions of turning bias and instantaneous speed for wild-type (black
lines) and unc-23 mutant (solid gray bars) animals. Arrowheads indicate the
average for each distribution.
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Turning bias in C. elegans chemotaxis
This gradient was constructed by placing two 5·?l drops of
3·mol·l–1NH4Cl at the center of the assay plate 12 and 3·h
before the experiment. Concentration was estimated as
2–50·mmol·l–1, according to the diffusion equation for a point
bolus in a thin slab (eqn 1 and 2 of Pierce-Shimomura et al.,
1999; Crank, 1956).
The second type of gradient (Figs·1Biv,C,D, 2) was used to
compare the chemotaxis behavior of unc-23 mutants with a
previously studied group of wild-type animals (Pierce-
Shimomura et al., 1999) in which the gradient was formed by
placing 2 drops of 0.5·mol·l–1NH4Cl at the center of the plate
12 and 3·h before the experiment. Because unc-23 mutants
move slower than wild-type animals (Ward, 1973, and this
study), they normally experience a restricted range of dC/dt
values relative to wild-type animals assayed in identical
gradients. Therefore, to ensure that unc-23 mutants would
experience a similar range of dC/dt values, we assayed them
in a steeper gradient than the ones used for wild-type worms.
Accordingly, the unc-23 gradients were formed by placing two
5·?l drops of 1·mol·l–1NH4Cl at the center of the plate 12 and
3·h before the experiment. Plotting histograms of values for
dC/dt, we found that unc-23 and wild-type worms assayed in
these two gradients experienced similar ranges of dC/dt values
between –0.05 and 0.05·mmol·l–1·s–1. Thus, any difference
between pirouette behavior for the unc-23 mutant and wild-
type is less likely to be due to a difference in the range of dC/dt
sensed and more likely to be due to an effect of the unc-23
mutation.
Results
To study the locomotion of unc-23 mutants, we used an
automated tracking system to follow individual adult worms
crawling on agar-filled plates. The agar contained either a
uniform concentration of chemical attractant (2·mmol·l–1
NH4Cl) or a radial gradient of chemical attractant (NH4Cl,
range: 2–16·mmol·l–1).
In a uniform concentration of attractant, unc-23 mutants
made complex tracks consisting of overlapping curls that
contrasted with the comparatively smooth tracks made by wild-
type animals (Fig.·1Bi,Biii). Viewed from above, unc-23
mutant animals with heads bent to the right made clockwise
curls, whereas animals with heads bent to the left made
counterclockwise curls. Thus the tendency to make curls is
due to the bent-head phenotype, as reported (Ward, 1973).
However, the severity of the bent-head phenotype varied
considerably, ranging from animals with a profound bend that
made tight (small radius) curls to animals with no visible bend
that did not make curls (data not shown). We quantified this
variability in terms of each animal’s turning bias, computed as
the animal’s average instantaneous turning rate over a 20·min
observation period. Turning bias histograms are shown in
Fig.·1C for unc-23 mutants and wild-type worms. The unc-23
distribution was shifted toward larger biases relative to the
wild-type distribution. We also quantified the effect of the bent
head on average instantaneous speed; unc-23 mutants were
approximately 2.5 times slower than wild-type animals
(Fig.·1D). Thus the two main effects of the unc-23 mutation
on locomotion are to increase the range of turning biases
between individuals and reduce speed.
In the case of a radial gradient, we found that some unc-23
individuals made spiral tracks (Fig.·1Bii) as previously
reported (Ward, 1973), whereas others did not. We found that
we could predict whether or not an unc-23 worm would make
a spiral track in a gradient by first observing the diameter of
the worm’s curls in the absence of a gradient. Only those
worms whose curls measured 0.5–1.0·mm in diameter
consistently made spiral tracks. Within this range of diameters,
the worm’s body occupies approximately one-third to two-
thirds of the circumference of a curl. Worms whose curl
diameter was below this range made little or no progress up
the gradient, whereas worms whose curl diameter was above
this range reached the gradient peak without making spirals,
i.e. their tracks were essentially wild-type in shape (data not
shown). These results indicate that the likelihood of a spiral
track depends on the severity of the bent-head phenotype.
Because the tracks of unc-23 mutants with bent heads were
so different from the tracks of wild-type animals, it was unclear
whether the pirouette mechanism operates in this strain. We
have shown previously that the probability of pirouette
initiation (Ppiro) in wild-type animals is related to the rate of
change of attractant concentration (dC/dt) at the animal’s
location as it moves in the gradient (Pierce-Shimomura et al.,
1999). Ppirois higher than baseline probability when the animal
moves down the gradient (facilitation for dC/dt<0) and lower
than baseline probability when the animal moves up the
gradient (suppression for dC/dt>0). Although the points of
pirouette initiation were difficult to discern within the unc-23
tracks because of their overlapping curls, these points were
easy to detect automatically with an algorithm based on the
time series of instantaneous turning rate values (example
Fig.·1Bii inset). To test whether facilitation and suppression
are regulated by dC/dt in unc-23 mutants as in wild-type
animals, we plotted Ppiroas a function of dC/dt for wild-type
and unc-23 mutants tested in a gradient (Fig.·2A). These data
were compared against separate groups of wild-type and
mutant animals tested in the absence of a gradient to measure
baseline levels of Ppiro. We found that Ppiroin unc-23 mutants
exhibited both facilitation and suppression relative to the
basal Ppiroas a function dC/dt similar to the facilitation and
suppression seen in wild type. We conclude that a pirouette
mechanism operates in unc-23 mutants.
We next asked whether pirouettes are initiated normally in
unc-23. This was done by plotting the distribution of
orientations relative to the gradient peak immediately before
pirouettes for the wild-type and unc-23 mutant animals.
Orientation was defined in terms of bearing (B) relative to the
peak of the gradient, where B=0° means movement directly up
the gradient and B=±180° means movement directly down the
gradient. Surprisingly, the distributions of B before pirouettes
(Bbefore) for the two strains were statistically indistinguishable
[Fig.·2B; Watson-Williams test (Zar, 1984): F1,1953=2.18,
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P=0.14]. The similarity of the wild-type and unc-23 Bbefore
distributions could have arisen because we happened to select
unc-23 worms with a mild phenotype. This explanation seems
unlikely, however, because the unc-23 Bbeforedistribution was
computed for the same sample of mutant worms shown in
Fig.·1C, which contains a significant number of individual
worms with a severe turning bias phenotype. Thus, unc-23
pirouettes are initiated normally in unc-23 mutants despite the
fact that the head is bent.
In wild-type animals, the average effect of a pirouette is to
reorient the animal up the gradient (Pierce-Shimomura et al.,
1999). To determine whether pirouettes could also reorient
unc-23 mutants normally, we compared the distribution of
bearings immediately after pirouettes (Bafter) for wild-type and
unc-23 mutant animals. The wild-type Bafterdistribution had a
peak at 0°, whereas the unc-23 distribution was essentially flat
[Fig.·2C; modified Rayleigh test (Zar, 1984): z=0.906,
d.f.=1128, P>0.20]. Thus, pirouettes did not reorient the unc-
23 mutants up the gradient as they did for wild-type animals.
This observation was reinforced by the distribution of changes
in bearing (?B) produced by individual pirouettes in the two
strains (Fig.·2D). The wild-type ?B distribution had a
minimum near 0° and a broad peak near ±180°, indicating that
most pirouettes produced relatively large changes in bearing.
In contrast, the unc-23 ?B distribution had only a minor peak
near ±180° and a major peak at 0°, indicating that the most
common type of pirouette produced little or no change in
direction, rather than a 180° change in direction as in wild-type
worms. Thus a key defect in the unc-23 pirouette mechanism
is the tendency to continue moving in the same direction after
a pirouette. This defect is consistent with the anatomical
finding of head muscle attachment defects in unc-23 mutants
(Waterson et al., 1980), because such a defect could limit a
worm’s ability to make large course corrections.
We next considered how spiral tracks might be generated by
unc-23 mutants. Ward (1973) suggested that unc-23 spirals
result from the tendency to align the head up the gradient,
coupled with the ruddering effect of the head bend. However,
the fact that unc-23 exhibits pirouettes triggered specifically by
movement down the gradient suggests an alternative
explanation. Perhaps the spiral track is due to pirouettes
triggered when the worm moves along the down-gradient
segment of the curl. To test this idea, we created a
mathematical model of unc-23 chemotaxis behavior.
Worms were represented in the model as a point that moved
with real-worm statistics in a virtual gradient. The virtual
gradient was the same as the real gradient used in the
experiments of Fig.·1Bii. At each time point t in the simulation
(?t=1·s), the worm’s location (xt, yt) and direction ?tin the
virtual gradient were updated according to the equations:
?t+1=?t+ ?t?bias,
xt+1=xt+ vcos(?t) ,
yt+1=yt+ vsin(?t)·,
where ?biasis turning bias and v is instantaneous speed. To
explore the effect of turning bias on track trajectory, a single
value for ?biaswas randomly selected for each model worm
from a range of values (6–12·deg.·s–1) consistent with the subset
of the distribution obtained from tracking data of real unc-23
animals that did not overlap with wild-type values (Fig.·1C).
Speed for each model worm was constant and set to the average
speed from the distribution obtained from real animals
(0.07·mm·s–1; Fig.·1D). Each model worm made pirouettes
probabilistically according to the instantaneous value of Ppiro,
updated for each time point according to the empirically derived
relationship between dC/dt and Ppiroshown in Fig.·2A. When a
pirouette occurred, direction was updated for that time step by
sampling randomly from the ?B distribution of Fig.·2D. Model
worms were started 35·mm away from the center of the radial
gradient with a random initial direction and allowed to move
for the equivalent of 90·min.
We found that simulations of unc-23 pirouette behavior
always reproduced the overall spiral shape of chemotaxis
tracks of the unc-23 mutant for ?bias values in the range
J. T. Pierce-Shimomura, M. Dores and S. L. Lockery
0.135
0.154
–180 –90090 180 –180 –900 90 180
Bearing (degrees)
0.0446
0.100
0.15
0.10
0.05
0
0.10
0.05
0
–180 –90090 180
0.10
0.05
0
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A
Facilitation
DCB
Bbefore
Bafter
ΔB
0.06
0.04
0.02
0
Rate of pirouette
initiation (s–1)
–0.100.1
dC/dt (mmol l–1 s–1)
Suppression
Basal
Probability
Fig.·2. Analysis of pirouette initiation and execution. (A) Instantaneous pirouette rate as a function of the value of dC/dt for wild-type and
mutant animals in NH4Cl gradients. The data have been smoothed with a box filter (4001 points long). The gray line represents unc-23 data
(21·293 points); the black line represents wild-type data (29·171 points). Spontaneous or basal pirouette rates measured in the absence of a
gradient are represented by horizontal lines (black broken line, wild-type; gray dotted, unc-23). (B–D) Histograms of bearings immediately
before pirouettes (B), immediately after pirouettes (C), and changes in bearing (D) for the animals in A. The wild-type distributions (black lines)
contain 1129 pirouettes; the unc-23 distributions (solid gray bars) contain 816 pirouettes. Arrowheads indicate the angle of the vector average
for each distribution; numbers indicate the corresponding vector magnitude.
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Turning bias in C. elegans chemotaxis
6–12·deg.·s–1. Like the real unc-23 mutants, model worms
advanced up the gradient in a spiral track composed of a series
of overlapping curls (Fig.·3A,B). Points in the track where
pirouettes occurred were evident by truncated curls, usually at
a point where the track curled down the gradient (crosses
superimposed on the magnified track in Fig.·3A). Thus, the
pirouette mechanism is sufficient to account for spiral-shaped
chemotaxis tracks in a model that contained no free
parameters.
The unc-23 simulation also reproduced a key detail
concerning the handedness of the spiral tracks of real unc-23
chemotaxis. Inspection of the tracks of those real unc-23
worms that spiralled (i.e. those that had a curl diameter
between 0.5–1.0·mm) revealed a consistent relationship
between the handedness of the turning bias and the handedness
of the spiral; in 10 out of 10 cases, the handedness of the bias
and spiral were opposite. We found the same relationship in
the model. For instance, the simulation shown in Fig.·3A had
a clockwise turning bias and made a counterclockwise spiral.
Moreover, all runs of the simulation (N=50) made tracks with
this relationship (Fig.·3B). Thus, although the pirouette
mechanism is conceptually simple, it appeared to predict the
chemotaxis behavior of real unc-23 mutants in terms of both
the overall spiral track and the relationship between the
handedness of the spiral relative to the handedness of the bias.
Our observation that the animal’s turning bias and spiral
track have opposite handedness appeared to conflict with the
original observations of the unc-23 mutants in which the
handedness of the turning bias and spiral were reported to be
the same (Ward, 1973). This apparent conflict raised the
possibility that animals from the two studies may have used
different chemotaxis strategies. We have found, however, that
a simple geometrical analysis of unc-23 pirouette behavior
provides a way to reconcile these different observations.
Consider the case of an unc-23 worm with a significant
turning bias and a simplified pirouette behavior such that
pirouettes result in direction changes of either 0 or 180°. These
angles correspond to the two maxima in the unc-23 ?B
distribution of Fig.·2D. Simplifying pirouette behavior in this
way is valid because the angles intermediate between 0 and
180° occur in complementary pairs (e.g. +90° and –90°),
whose effects cancel out over the long term. Such a worm will
move along a circular path punctuated by occasional
pirouettes. Pirouettes of 0° have no effect, so the worm stays
on its circular course. In contrast, pirouettes of 180° cause the
worm to embark on a new circular trajectory, tangent to the
original one at the point where the pirouette occurred. Because
pirouette initiation depends probabilistically on the history of
dC/dt (Dusenbery, 1980; Pierce-Shimomura et al., 1999; Miller
et al., 2005), pirouettes will occur with variable latency with
respect to the instant at which the animal enters the down-
gradient half of its circular trajectory. Thus pirouettes can
occur in each of the four quadrants shown in Fig.·4. The
relative frequency of pirouettes in the four quadrants will
depend on such factors as the speed of the worm and the
steepness of the gradient, but for now let us consider the simple
case in which pirouettes occur with a fixed latency such that
they always occur in the same quadrant (I, II, III or IV).
For the case where pirouettes occur with the shortest latency
(quadrant I pirouettes) the track spirals toward the peak
(positive chemotaxis) with a handedness that is opposite to
the handedness of the turning bias (Fig.·4B). This track
corresponds to the unc-23 tracks reported here (e.g. Fig.·1Bii).
Pirouettes occurring with longest latency (quadrant IV
pirouettes) also yield positive chemotaxis, but now the
Fig.·3. Mathematical model of unc-23 behavior. (A) Representative
track of a simulated unc-23 mutant in a radial gradient. Note that the
simulated worm had a clockwise turning bias (7·deg.·s–1) and made a
counterclockwise spiral, as do real unc-23 mutants (Fig.·1Bii). The
gradient peak is indicated by the plus sign. Color represents time (see
color scale). Pirouette initiation events are indicated by orange crosses
superimposed on the magnified track within the inset. (B) Population
behavior of simulated unc-23 worms in a radial gradient. The track
of one simulated worm is highlighted in black; all other tracks are
gray (N=50). Each simulated worm had a right-handed turning bias
and made a left-handed spiral. The gradient peak is indicated by
intersection of dotted lines. Simulations in A and B were started near
the asterisks.
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