Analysis of the effects of turning bias on chemotaxis in C. elegans
1 Ernest 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
2 Institute of Neuroscience, University of Oregon, Eugene, OR 97403-1254,
USA
* Author for correspondence (e-mail: jonp{at}egcrc.net)
Accepted 17 October 2005
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Summary |
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Key words: chemotaxis, orientation, locomotion, nematode, behavioral modeling, Caenorhabditis elegans
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Introduction |
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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.
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Materials and methods |
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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). This gradient was
constructed by placing two 5 µl drops of 3 mol l1
NH4Cl at the center of the assay plate 12 and 3 h before the
experiment. Concentration was estimated as 250 mmol
l1, 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 l1 NH4Cl 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 µl drops of 1 mol
l1 NH4Cl 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
l1 s1. 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.
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Results |
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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 0.51.0 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). Ppiro is 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
Ppiro as 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 Ppiro in unc-23 mutants exhibited
both facilitation and suppression relative to the basal
Ppiro as 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, 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 Bbefore distribution 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 Bafter
distribution 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
t in the virtual gradient were updated according to the
equations:
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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
612 deg. s1. 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.
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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 0.51.0 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. Therelative 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).
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Does the simplified geometric analysis apply to the chemotaxis tracks of real worms, in which pirouettes occur with a variable latency? In the case of variable latency, the geometric analysis predicts that for positive chemotaxis with opposite spiral-bias handedness, pirouettes in quadrant I should dominate. To test this prediction, we examined the relative frequency of pirouettes in quadrants IIV for our unc-23 worms, all of which exhibited spiral tracks with opposite spiral-bias handedness. As predicted by the geometric analysis, the most frequent pirouettes were those in quadrant I (Fig. 5A). We conclude that the geometric analysis is consistent with the behavior of real unc-23 worms in our assay.
As a further test of the geometric analysis in the case of variable
pirouette latency, we asked whether the frequency of pirouettes observed for
real worms in each quadrant was sufficient to produce a spiral track with
opposite spiral-bias handedness in a simulated worm. We did this by noting the
time at which the simulated worm entered the down-gradient region of its
circular trajectory and, at that point, randomly selecting one of four
latencies according to the probabilities dictated by the observed frequencies
of quadrant IIV pirouettes shown in
Fig. 5A. Overall pirouette rate
was adjusted to match the average pirouette rate in real unc-23 worms
(0.04 s1; Fig.
2A). The simulated worm's direction after a pirouette was updated
by sampling randomly from the B distribution of
Fig. 2D. Because of the
stochastic nature of the model, we ran 50 simulated worms for the equivalent
of 90 min each. We found that all of the simulated worms exhibited spiral
tracks with opposite spiral-bias handedness (representative track in black and
all others tracks in gray in Fig.
5B). This result provides additional support for the geometrical
analysis.
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Discussion |
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Our tracking analysis confirms and extends many of the main observations of
the initial report on unc-23 chemotaxis behavior
(Ward, 1973). In agreement with
previous results (Ward, 1973
),
we found that whereas some unc-23 individuals had a wild-type turning
bias, other unc-23 individuals had a strong turning bias imposed by
their bent head. For individuals with a bent head, the head appeared to act
like a rudder that forces the mutant to move in a curling path. In an
extension of the earlier work, we found that the likelihood of spiral tracks
depends on the severity of the head bend. Only those individuals with a
moderate turning bias made spiral tracks; individuals with a weak turning bias
did not make spiral tracks, whereas individuals with a severe turning bias
made little progress away from the starting location.
Our analysis also revealed a key difference from the initial study of the
unc-23 mutant (Ward,
1973). We found that all animals (10 out of 10) made tracks that
had the opposite relation between the handedness of the spiral track and the
handedness of the turning bias. This seemed to conflict with the initial
study, which reported that the handedness of the spiral and the handedness of
the turning bias were the same (Ward,
1973
). However, we found that we could resolve this conflict by
considering how the latency of pirouettes interacts with turning bias to
produce spirals with the same or opposite handedness. A geometric
consideration of this interaction showed that it could yield tracks like those
in the original study, in which the spiral and turning bias had the same
handedness, or tracks like those in the present study, in which the spiral and
turning bias had opposite handedness.
The results reported here revise our assessment of the main chemotaxis
strategy in C. elegans. Based on qualitative inspection of the tracks
that wild-type and mutant animals left in an agar substrate, Ward
(1973) proposed that a worm
adjusts the extent of dorsal vs ventral body bends so as to minimize
the difference in the attractant concentration sensed on either side, somewhat
like a weathervane shifting in the wind. The weathervane strategy was proposed
as the primary mechanism to explain the relatively direct tracks made by
wild-type animals and the spiral tracks of the unc-23 mutant.
However, we find that the pirouette mechanism can explain both the elementary
and higher-order features of chemotaxis behavior in wild-type and
unc-23 animals without appeal to alternatives. We therefore propose
that the pirouette mechanism is the primary strategy for C. elegans
chemotaxis in laboratory assays, and that other mechanisms, including a
possible weathervane strategy, serve as secondary or alternative
strategies.
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Acknowledgments |
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References |
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