The Fundamental Role of Pirouettes in Caenorhabditis elegans Chemotaxis

Dispersal behavior

As a prelude to studying chemotaxis, we examined the dispersion of worms in a spatially uniform concentration of attractant (2 mm NH₄Cl). Single, adult animals were started 22 mm from the center of a circular, agar-filled Petri plate and tracked for 20 min or until the worm reached the edge of the plate. The tracking system recorded the worm's position, speed, and turning rate at ∼1 sec intervals. Individual worms (n = 47) moved away from their starting location, leaving complex tracks (Fig.2A) similar to those reported previously (Ward, 1973;Croll, 1975a). Population behavior was visualized in a probability-density plot made by digitally superimposing individual tracks and computing, for each location in a 1 mm × 1 mm grid, the probability of an animal being observed there during a tracking assay. Probability was computed by dividing the amount of time that an animal was observed at a location by the total elapsed time during all 47 assays. For animals dispersing in a uniform concentration of attractant, probability density was highest at the starting position and decreased with distance from this point (Fig.2B). Similar dispersal behavior has been observed in other species, including other nematodes (Berg and Brown, 1972; Croll and Blair, 1973; Croll, 1975b; Dethier, 1976).

The tracks of individual worms and their population behavior suggest that dispersion in C. elegans may involve a random walk. To test this idea, we constructed a stochastic model of worm movement. A worm was simulated as a point (x_t ,y_t ) whose position in a simulated Petri plate was updated in 1 sec time steps, consistent with the sampling rate of the tracking system. At each time point t in the simulation, step length l_t and direction θ_t were calculated as: Equation 5 Equation 6

where v_t is the instantaneous speed, dθ_t/dt is the instantaneous turning rate (using the convention that dθ_t/dt > 0 is a right turn and dθ_t/dt < 0 is a left turn), δ is the turning bias, and Δt is the duration of the time step. A turning bias was chosen for each model worm by sampling randomly from the distribution of observed turning biases (Fig. 2E). Speed and turning rate were sampled randomly from their respective distributions (Fig.2C,D) obtained from tracking data (Fig.2B). In real worms, speed and the absolute value of turning rate showed a slight negative correlation (r = −0.290; SE = 0.0148; p < 0.001; data not shown), meaning that sharp turns slowed the animal down. To preserve this correlation in the model, we made use of the fact that, during tracking, values of speed and turning rate were recorded as pairs. The correlation was preserved by retaining the original pairing. As in the tracking experiments with real worms, the simulated worm was started 22 mm away from the center of the simulated Petri plate and allowed to move for the equivalent of 20 min or until it hit the edge of the plate.

The stochastic model reproduced worm behavior at the individual and population levels. Individual points made complex tracks leading away from the starting location (Fig. 2F) that looked like the tracks of real animals (Fig. 2A). Probability density for simulated worms (n = 47) was highest at the starting position (Fig. 2G) as in the case of real worms (Fig. 2B). Dispersion of model and real worms was compared by plotting probability density along the cross section defined by the line between points a andb in Figure 2, B and G. This comparison showed that model dispersion was similar to real dispersion, because the cross-sectional probability densities were nearly identical. Statistical tests on the population behavior of model (n = 500) and real (n = 47) worms showed that they were indistinguishable, because the percentage of animals reaching the center of the plate (real worms, 12.7%; model worms, 11.8%; Z = 0.049; p > 0.05) and the percentage of animals reaching the edge of the plate (real worms, 19.5%; model worms, 21.2%; Z = 0.27;p > 0.05) were not statistically different. As a further test of the population behavior of the model, we compared model and real worms in a standard population assay of dispersal behavior (Bargmann et al., 1993). Accordingly, model worms were started at the center of a simulated plate that contained two circular goals (radius = 5 mm), located at opposite sides of the plate, and allowed to wander for the equivalent of 1 hr. The percentage of model worms (n = 500) that reached either of the two goals (at least once) was counted and compared with the percentage for real animals (n = 270) in the same assay using previously published data from an independent laboratory (Bargmann et al., 1993). Percentages for the model and real animals were almost the same (real worms, 11.1%; model worms, 10.8%; Z = 0.127;p > 0.05). This result shows that the stochastic model accurately reproduced worm dispersal and that worm movement in a spatially uniform attractant is well described as a random walk. Below we use the stochastic model as a means of testing several theoretical mechanisms of C. elegans chemotaxis.

Chemotaxis

We performed chemotaxis assays by tracking individual worms in radial Gaussian-shaped gradients of two attractants, biotin (n = 36) and NH₄Cl (n = 37). As in previous studies (Ward, 1973; Bargmann and Horvitz, 1991), most animals reached the peak of the gradient (biotin, 55.6%; NH₄Cl, 59.5%), defined as a circular region with a radius of 5 mm located at the center of the plate (Fig. 3). Animals tested in a gradient were significantly more likely to reach the center than were control animals tested in a uniform concentration of attractant [2 mm NH₄Cl; 8.3%;n = 47; χ²₍₂₎ = 34.1;p < 0.0001]. This result shows that animals in the chemotaxis assays were performing well above chance level despite the fact that some animals did not reach the peak of the gradient.

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Fig. 3.

Chemotaxis behavior. A,B, Tracks of three worms moving in a radial Gaussian gradient of the attractants biotin (A) or NH₄Cl (B) originating at the center of the plate (peak). The gray circle indicates the region of the gradient peak used for statistical analysis. Starting points are indicated by an asterisk. Elapsed time is 20 min. Each animal was run separately. C,D, Probability-density plots for worms assayed in gradients of biotin (C) or NH₄Cl (D). The probability (C,D) scale is indicated in D. Scale bar:A–D, 1 cm.

Kinesis behavior

Two main forms of chemotaxis have been identified in animals that are limited to sensing chemical concentration at a single point in space (Dunn, 1990). The first is a kinesis mechanism in which the animal modulates its speed (orthokinesis) or turning rate (klinokinesis) as a function of attractant concentration. The second is a taxis mechanism in which the direction of locomotion is correlated with the direction of the gradient (∂C/∂x, ∂C/∂y) at the animal's location (Dunn, 1990).

To test whether C. elegans exhibits a kinesis mechanism in NH₄Cl gradients, we measured the concentration dependence of instantaneous speed and turning rate by tracking animals (n = 118) in agar-filled plates containing a uniform concentration of the attractant NH₄Cl at 2, 4, 6, 8, or 10 mm. These values cover the range of concentrations encountered by animals in our chemotaxis assays (data shown in Fig. 3D). Average instantaneous speed, plotted against concentration, showed a significant quadratic trend (Fig.4A; F = 8.89; df = 1; p < 0.01; n = 118). Average turning rate showed a significant linear trend (Fig.4B; F = 7.40; df = 1;p < 0.01; n = 118). Thus, average instantaneous speed and turning rate were weakly dependent on concentration. Are the observed concentration dependencies sufficient for chemotaxis? This question was addressed by imposing these dependencies on the stochastic model to see whether it now exhibited chemotaxis. The concentration dependence of speed was incorporated in the model by making the mean of the instantaneous speed distribution a function of the attractant concentration C. Thus, the concentration-dependent speed v_t^′at location (x, y) was given by the expression: Equation 7where v_t is an element randomly selected at time t from the observed instantaneous speed distribution in 2 mm attractant (Fig.2C), v₀ is the mean of the instantaneous speed distribution at 2 mm, and (C[x, y]) is the quadratic fit to the data shown in Figure 4A. Similarly, the concentration dependence of turning rate was incorporated by making the mean of the instantaneous turning-rate distribution a function of attractant concentration. Thus, the concentration-dependent turning rater_t^′ at location (x,y) was given by the expression: Equation 8where r_t is the value of the turning rate paired with v_t from the observed instantaneous turning-rate distribution in 2 mmattractant (Fig. 2D),r₀ is the mean of the instantaneous turning-rate distribution at 2 mm, (C[x, y]) is the linear fit to the data shown in Figure 4B, Δt is the time step, and δ is the turning bias selected as in the dispersion model. In Equations 7 and 8,C[x, y] was determined by the diffusion equation (see Materials and Methods).

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Fig. 4.

Kinesis behavior and model.A,B, Dependence of the average speed (A) and turning rate (B) of real worms on the absolute concentration of the attractant NH₄Cl. Numbersabove thecircles indicate the sample size for each concentration group. Error bars represent 95% confidence intervals. Thedashed line in A is the best-fitting quadratic function, and the dashed line inB is the best-fitting linear function. C,D, Probability-density plots for simulated worms in a radial gradient of NH₄Cl (C; 2–8 mm) and a uniform concentration of the same attractant (D; 2 mm). In model worms, speed and turning rate were determined by the concentration dependencies shown inA and B (n = 5000 for each plot). The probability scale is shown in C; aplus sign indicates the peak of the gradient inC. Scale bar: C, D, 1 cm.E, Probability-density cross sections frompointsa to b inC and D. The concentration dependencies of speed and turning rate were not sufficient to produce accumulation at the peak of the gradient. The inequality of areasunder the curves in E is a result of taking the cross sections of the probability-density plots.

Probability density for simulated worms (n = 5000) was plotted for two different conditions: in the presence of a gradient to test for chemotaxis (Fig. 4C) and in the absence of a gradient (spatially uniform attractant concentration, 2 mm NH₄Cl) to measure simple dispersion (Fig. 4D). The effect of the imposed concentration dependence was assessed by comparing cross-sectional probability density as described above. Cross-sectional probability for the two conditions was nearly identical, indicating that the observed concentration dependence of speed and turning rate was not sufficient to cause accumulation at the peak of the gradient. Thus, it is unlikely that C. elegans accumulates at the peak of the NH₄Cl gradient in our assays by a kinesis mechanism.

Taxis behavior

Finding no evidence of chemotaxis by a kinesis mechanism, we searched for correlations between chemosensory input and changes in orientation that might be consistent with a taxis mechanism. As a first step, we examined the tracks of individual worms during the chemotaxis assays of Figure 3, C and D. Like spontaneous locomotion, locomotion during chemotaxis comprised a series of runs punctuated by turns, as described previously (Ward, 1973). In our assays, turns generally resulted in a substantial change in a worm's orientation with respect to the gradient. Thus, if turns were triggered appropriately, they could function to orient a worm in a taxis mechanism. To test this idea, however, we needed an objective procedure to segment worm tracks into runs and turns. Development of the segmentation procedure involved three steps.

Analysis of turning rate

We developed an algorithm to identify sharp turns on the basis of the turning rate dθ/dt. First, we visually identified all sharp turns (n = 366) in a group of 15 animals randomly selected from the populations in Figure 3, C andD. A histogram of the absolute value of the turning rate ‖dθ/dt‖ associated with each visually identified sharp turn (Fig. 5A) showed a discontinuity at 50° sec⁻¹, with 98% of all sharp turns falling in the range 50–210° sec⁻¹. Thus, we classified periods in a worm's track in which ‖dθ/dt‖ > 50° sec⁻¹ as sharp turns; we classified periods in which ‖dθ/dt‖ ≤ 50° sec⁻¹ as swims. Using 50° sec⁻¹ as a threshold for detecting sharp turns, we analyzed the time series of ‖dθ/dt‖ for the tracking data of a different randomly selected set of 30 animals from the same population and found a 100% correspondence between threshold crossings and the occurrence of a visually identified sharp turn. This test showed that we could identify sharp turns with confidence simply by noting when ‖dθ/dt‖ exceeded the turning-rate threshold.

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Fig. 5.

The algorithm for segmentation of tracks.A, Histogram of turning rates associated with visually identified sharp turns (reversals and/or omega turns;n = 366). The histogram shows that 97.5% of all sharp turns were associated with a turning rate ≥50° sec⁻¹ (verticaldashedline). This value was used to identify sharp turns objectively. B, Histogram of swim durations. Swims were defined as track segments between sharp turns defined by the 50° sec⁻¹ cutoff identified inA. The swim-duration histogram is well fit by the sum of two exponentials (solid line) indicating the existence of distinct long- and short-swim states. The predicted long- and short-swim distributions are shown as dashed lines. Thevertical dashed line beneath the arrowindicates the critical swim duration (t_crit = 6.05 sec) that minimizes the probability of misclassifying a swim as long or short. Intervals whose duration was greater than or equal to t_critwere assumed to be long swims (later identified as runs); intervals whose duration was less than t_crit were assumed to be short swims. Episodes of one or more consecutive short swims (and the associated turns) are called pirouettes because they were brief and usually resulted in large changes in direction. Theshaded regionunder thedashedlines indicates the predicted number of misclassified swims. C, A track segmented according to t_crit. Long swims (runs) areblack; short swims and turns (pirouettes) aregray.

Analysis of tracks

Segmenting a track into runs and pirouettes was complicated by the fact that pirouettes often occurred as clusters or bursts of turns, each separated by a short interval of swimming. Thus, many of the short swimming intervals appeared to be components of pirouettes rather than of runs. To distinguish between runs and pirouettes, therefore, we computed the histogram of all swimming intervals (Fig. 5B) from the data of Figure 3, C and D. The histogram was well fit by the sum of two exponentials, suggesting a kinetic model in which there are two distinct swimming states: one in which the sharp-turn probability is low, yielding mostly long swims, and one in which the sharp-turn probability is high, yielding mostly short swims. In such a model both states are theoretically capable of generating long or short swims, so we used a probabilistic method to distinguish between long and short swims in the data. This method involved computing t_crit, the swim-duration threshold that minimized the number of mistakes made in classifying a swim as long or short (see Materials and Methods). Swims with durations greater than or equal to t_crit were assumed to be long swims; swims with durations less thant_crit were assumed to be short swims. Tracks were segmented into runs and pirouettes by defining runs as long swims and defining pirouettes as track regions composed of sharp turns and short swims (Fig. 5C).

The segmentation procedure revealed a striking correlation between chemosensory input and pirouettes. Figure6 shows the analysis of 4 min of representative tracking data from an animal performing chemotaxis. Sharp turns (Fig. 6B) were identified by noting the times at which dθ/dt (Fig. 6A) exceeded the turning-rate threshold. Runs and pirouettes, as defined by the segmentation procedure operating on the sharp-turn record, are shown in Figure 6C. Pirouettes tended to occur after episodes in which the animal moved down the gradient, visible as regions of negative slope in the plot of estimated attractant concentration versus time (Fig. 6D). The correlation between pirouettes and movement down the gradient can be seen more easily in thetrace showing the time derivative of estimated attractant concentration (Fig. 6E). In four out of five cases in the data shown, the pirouette occurred after an episode in which dC/dt < 0. We found no correlation between pirouettes and other measures such as C or d²C/dt². Thus, it appears that an episode in which dC/dt < 0 constitutes the sensory event that triggers a pirouette. An analogous reaction has been observed previously in the response of tethered worms to imposed decreases in attractant concentration (Dusenbery, 1980).

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Fig. 6.

The correlation between concentration changes and the initiation of pirouettes. These data (A–E) were recorded from a single animal during a typical chemotaxis assay. A, Instantaneous turning rate. Dashed horizontal lines indicate the 50° sec⁻¹ threshold for identifying sharp turns.B, Sharp turns identified as threshold crossings inA. Note that sharp turns often occur in bursts.C, Runs (white) and pirouettes (black) distinguished by the algorithm described in Figure 5. D, Estimated attractant concentration at the worm's location in the plate (see Materials and Methods).E, Rate of change of attractant concentration (dC/dt). The shaded regions indicate dC/dt < 0. Note that pirouettes were usually preceded by episodes in which dC/dt < 0 (dashed vertical lines).

To examine the reliability of the correlation between pirouettes and episodes in which dC/dt < 0, we computed the ensemble average of dC/dt values before each pirouette, which we call the “prepirouette average.” The prepirouette average of dC/dt shows the typical time course of dC/dt leading up to a pirouette. For animals in the biotin and NH₄Cl assays (Fig.7A,B), the averages were negative for ∼15 sec before each pirouette, consistent with a reliable correlation between pirouettes and episodes in which dC/dt < 0. At times >15 sec before each pirouette, the averages were positive, indicating movement up the gradient. The time course of the prepirouette averages of dC/dt suggests that worms moving up the gradient eventually drift off course, leading to an episode of dC/dt < 0 that triggers a pirouette. As a control, we tracked animals in a uniform concentration of attractant (2 mm NH₄Cl; data from Fig.2B) but computed the prepirouette average of dC/dt as if a typical biotin or NH₄Cl gradient were present. As expected, pirouettes in control animals were not correlated with dC/dt, because the prepirouette averages of dC/dt for control animals were flat (Fig.7A,B). Such pirouettes are most likely spontaneous in origin, consistent with previous observations (Croll, 1975a, 1976; Chiba and Rankin, 1990).

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Fig. 7.

Prepirouette averages. A, B, The prepirouette average of dC/dt for animals in biotin (A) and NH₄Cl (B) gradients (solid circles). The averages are well fit by single exponentials (solid lines) having time constants τ = 9.45 sec for biotin and τ = 10.0 sec for NH₄Cl. For comparison, the prepirouette average was also computed for animals tested in a uniform concentration of attractant (2 mmNH₄Cl; data from Fig. 2B) as if the same gradient were present (open squares).C,D, The prepirouette average of speed for the same animals in biotin (C) and NH₄Cl (D) gradients (solidcircles). The prepirouette average of speed for the animals in a uniform concentration of attractant (open squares) is shown for comparison inC and D. In A, C, andD, error bars representing SEM are smaller than some or all of the markers.

The prepirouette averages of dC/dt for worms in biotin and NH₄Cl gradients reached a minimum ∼4 sec before the pirouette and then rose slightly until the pirouette occurred (Fig. 7A,B). The rise in dC/dt could reflect a progressive improvement in orientation or simply a decrease in speed. Examination of the direction of locomotion with respect to the gradient peak for all worms in each of the 4 sec before a pirouette showed that orientation continued to worsen during this period (data not shown). This result suggests that the rise in dC/dt is not attributable to improved orientation. In contrast, examination of the prepirouette average of speed revealed a sharp deceleration in the 4 sec leading up to a pirouette (Fig. 7C,D). We conclude that the rise in dC/dt is attributable to a drop in speed before pirouettes. This result suggests that the animal may detect that it has drifted off course several seconds before the pirouette starts. Similar decelerations were observed in animals tested in the absence of a gradient (Fig. 7C,D). This observation suggests that changes in direction are the result of a two-step motor program in which the animal slows down and then does a pirouette.

We noted that the average speed of animals in the presence of biotin was higher than that in the absence of biotin (Fig. 7C). This result suggests the possibility that speed in biotin is concentration dependent. However, we found no correlation between instantaneous speed and biotin concentration (r = 0.006; n = 33,164; p > 0.05) in the tracking data for worms tested in a biotin gradient. This result argues against a kinesis effect in the biotin assay.

Because the initiation of pirouettes was correlated with episodes in which dC/dt < 0, one might expect that the rate of pirouette initiation r would be higher for animals in a gradient than for animals in a uniform concentration of attractant, where dC/dt = 0. This was not the case, however, because an ANOVA revealed no significant differences in the mean r (F = 2.60; df = 2; p > 0.05) for animals in a biotin gradient (n = 36; r = 0.0234 ± 0.00205 sec⁻¹), an NH₄Cl gradient (n = 37; r = 0.0264 ± 0.00136 sec⁻¹), and a uniform concentration of attractant (n = 47; r= 0.0296 ± 0.00213 sec⁻¹). Plottingr against dC/dt (Fig.8) revealed a sigmoidal relationship in which r rises above the spontaneous rate (measured in a uniform concentration of attractant) in the region where dC/dt < 0 and r is suppressed below the spontaneous rate in the region where dC/dt > 0. The sigmoidal relationship shows that the worms detected increases as well as decreases in attractant concentration, as demonstrated previously (Dusenbery, 1980). The fact that r is suppressed when the worm is going up the gradient may explain the absence of the expected relationship betweenr and the presence or absence of a gradient.

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Fig. 8.

Pirouette initiation rate as a function of dC/dt. A,B, Black lines represent the instantaneous pirouette initiation rate as a function of the value of dC/dtoccurring 4 sec previously for animals in biotin (A) and NH₄Cl (B) gradients. Each mean spontaneous pirouette initiation rate is represented by a horizontal dashed line. The data have been smoothed with a box filter with a width of 4001 points. Gray lines represent the same data with minimal smoothing (box width = 101 points). There were 32,458 points in the biotin data set and 29,171 points in the NH₄Cl data set. Histograms (formed by black lines) show the number of data points as a function of dC/dt.

The function of pirouettes in chemotaxis

To determine how pirouettes contribute to chemotaxis, we studied the distribution of orientations before pirouettes in the tracking experiments of Figure 3, C and D. Orientation was defined in terms of the bearing B with respect to the peak of the gradient, where B = 0° means movement directly up the gradient and B = ±180° means movement directly down the gradient. The bearing was calculated from worm tracks as described in Materials and Methods. The distributions of bearings before pirouettes, B_before, had a peak near ±180° and a trough near 0° in the NH₄Cl and biotin groups (Fig.9A1,B1). To control for nonspecific bias in the bearing distributions, we also examined the distribution of bearings before pirouettes with respect to the center of the plate for worms in a uniform concentration of attractant using the data of Figure 2B. The distribution ofB_before was flat (Fig.9C1), indicating that the shape of theB_before distributions for animals tracked in a gradient does not reflect a response to nonspecific cues in the testing environment. Taken together, these results show that pirouettes were most likely to occur when an animal was headed down the gradient and least likely to occur when an animal was headed up the gradient. Thus, one function of pirouettes may be to terminate runs that have veered down the gradient. In agreement with this idea, we found a positive correlation between the chemotaxis performance of individuals (chemotaxis indexI^che; see Materials and Methods) and their average value of ‖B_before‖ (biotin,r = 0.402; p < 0.05; NH₄Cl, r = 0.651;p < 0.001).

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Fig. 9.

Analysis of bearing distributions. Histograms of bearings immediately before pirouettes (B_before), bearings immediately after pirouettes (B_after), and changes in bearing (ΔB) are shown in their respectivecolumns. A1–3, NH₄Cl group.B1–3, Biotin group.C1–3, Animals tested in a uniform concentration of attractant (2 mm NH₄Cl; data from Fig. 2B) but analyzed as if a gradient were present. Arrowheads indicate the angle of the vector average for each distribution. The accompanyingnumbers indicate the corresponding vector magnitude. The number of pirouettes are 972 for the biotin group, 816 for the NH₄Cl group, and 1099 for the control group.

Pirouettes could contribute to chemotaxis by randomizing the animal's orientation or, in a more complex mechanism, by correcting the animal's course (Daykin et al., 1965; Green, 1966; Berg and Brown, 1972). These two mechanisms can be distinguished by the shape of the distribution of bearings after pirouettes,B_after. A distribution with a peak near 0° would reflect a reorientation mechanism, whereas a flat distribution would reflect a random mechanism. TheB_after distributions (Fig.9A2,B2) had a significant peak near 0° [modified Rayleigh test (Zar, 1984); NH₄Cl,n = 816; u = 3.83; p < 0.001; biotin, n = 972; u = 4.87;p < 0.001), showing that chemotaxis in C. elegans involves course correction. TheB_after distribution for the control animals tested in a uniform concentration of attractant was flat (n = 1099; u = 0.351; p> 0.05; Fig. 9C2), arguing against a response to nonspecific cues in the testing environment. Thus, a second function of pirouettes is to reorient the animal. In agreement with this idea, we found a negative correlation between the chemotaxis performance of individuals and their average value of ‖B_after‖ (biotin,r = −0.384; p < 0.05; NH₄Cl, r = −0.575;p < 0.001).

How do pirouettes correct the animal's course? One simple possibility is that pirouettes cause the animal to reverse course. Course reversal, coupled with the observed tendency to initiate pirouettes when heading down the gradient (Fig. 9A1,B1), would result in a B_after distribution with a peak at 0°. To examine this possibility, we analyzed the change in bearing, ΔB = B_after −B_before, associated with each pirouette for animals in the gradient and control groups. In the gradient groups (Fig. 9A3,B3), the ΔB distributions had a broad, flat peak centered near ±180°. The shape of the ΔB distributions indicates that large changes in bearing were more frequent than very small ones, a pattern that reflects a tendency to approximately reverse course. Note that the ΔB distributions for the control group (Fig.9C3) were similar to the ΔB distributions for the gradient groups, implying that the ΔB distribution is to some extent a property of the pirouette motor program itself and not a directed response to the chemical environment.

To determine whether an approximate course-reversal mechanism was sufficient to account for the degree of reorientation evident in theB_after distribution, we estimated numerically the B_after distribution that would result if the change in bearing associated with each pirouette were random. This was done by adding to each entry in the pooled B_before distribution for both gradient groups (Fig.10A) a randomly selected entry from the pooled ΔB distribution for the same groups (Fig. 10B) and repeating this process 10,000 times to obtain a distribution ofB_after values. In contrast to the pooled B_after distribution, the estimated B_after distribution was essentially flat (Fig. 10C, black line). This result indicates that the pattern of course reversal exhibited by real worms is not sufficient to correct the animal's course. It further suggests that the degree of course correction exhibited by real worms requires that the original association between each B_before and its corresponding ΔB be maintained.

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Fig. 10.

Analysis of pirouettes in terms of two possible course-correction mechanisms. A–C, Absence of a course-reversal mechanism. A, Histogram ofB_before values for the pirouettes of animals in the gradient groups (biotin and NH₄Cl combined).B, Histogram of ΔB values for the animals in A. C, Comparison of course-reversal (blackline; courserev) and observed (gray bars; obs) histograms of B_after values for the animals in A. The course-reversalB_after histogram was generated by sampling randomly from the ΔB histogram in B. Arrowheads indicate the angle of the vector average for each distribution; the accompanyingnumbersindicate the magnitude of the vector average. D,E, Presence of an error-compensation mechanism. Plots of ΔB against B_before for pooled data of animals tested in a gradient (D;n = 1788) and data for control animals (E; n = 1099) tested in a uniform concentration of attractant (2 mm NH₄Cl) are shown. Control data are from Figure 2B, analyzed as if a gradient were present. The ΔB axis is extended beyond ±180° so that it is easier to see the cluster of points centered near B_before = ±180° and ΔB = ±180°. Although the ΔBaxis is periodic, each data point is represented only once, resulting in the blanktriangularregionsabove and belowthe data.

To understand why this association was essential for course correction, we plotted ΔB againstB_before for the gradient and control groups. The plot for the gradient group (Fig. 10D) had a broad cluster of points centered nearB_before = ±180° and ΔB = ±180°, whereas the data for the control group were uniformly distributed (Fig. 10E). During chemotaxis, therefore, large changes in bearing were specifically associated with large values ofB_before. BecauseB_before measures the degree to which the animal was off course before the pirouette, this result shows that pirouettes correct the animal's course by compensating for large errors in orientation. No such error compensation was apparent in the range −90° < B_before < 90° in which orientation errors are small.

Computer simulations of chemotaxis

The foregoing analysis suggests a strategy in which chemotaxis is achieved by a series of pirouettes that reorient the animal to the gradient by a course-correction mechanism. To test whether this mechanism is quantitatively sufficient to produce chemotaxis in our assays, we constructed a computer model. If the course-correction mechanism were sufficient, it would bias the random dispersion of model worms toward the peak of the gradient. The chemotaxis model was made by superimposing the course-correction mechanism on the stochastic model of dispersion (Fig. 2F,G). In the model, pirouettes were triggered with a rate r that depended on the instantaneous value of dC/dt according to the sigmoidal relationships between r and dC/dt shown in Figure 8. Pirouettes were modeled as instantaneous changes in direction determined by sampling from the ΔB distribution to preserve the observed correlation between ΔB and B_before(see Materials and Methods). Chemotaxis in biotin and NH₄Cl gradients was modeled separately using ther versus dC/dt relationship obtained from real animals in the appropriate attractant.

The chemotaxis model qualitatively reproduced the behavior of individual animals in that it produced oriented movement up the gradient and dwelling at the peak (Fig.11A,D). The model also qualitatively reproduced the behavior of populations, because the probability distribution of the model was biased toward the peak of the gradient (Fig. 11B,E), like the probability distribution for real animals (Fig.3C,D) and unlike the probability distribution for dispersion in a uniform concentration of attractant (Fig. 2B,G). Statistical comparisons of cross-sectional probability for the chemotaxis model and the dispersion model (Fig. 11C,F) showed that the behavior of the chemotaxis model differed significantly from the behavior of the dispersion model (biotin, n = 18000; F = 268.0; df = 1; p < 0.001; NH₄Cl, n = 18000;F = 340.1; df = 1; p < 0.001). In addition, the average distance to the peak of the gradient for the duration of the assay for individuals in the chemotaxis model was significantly less than the average distance to the peak in the dispersion model [biotin, t_(2),198 = 17.2; p < 0.001; NH₄Cl,t_(2),198 = 11.4; p < 0.001]. These results show that the course-correction mechanism is sufficient to bias the random dispersion of worms toward the direction of the gradient peak.

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Fig. 11.

Quantitative reconstruction of chemotaxis behavior in C. elegans for radial gradients of attractant. A–C, Biotin group. D–F,NH₄Cl group. A,D,Tracks of three simulated worms moving in radial gradients. Chemotaxis was modeled by imposing the course-correction mechanism on the dispersion model of Figure 2, F andG. Simulated worms were allowed to move for the equivalent of 20 min. B, E, The average probability-density plots for 100 batches of simulated worms (number of worms per batch indicated by n) in radial gradients. Probability is displayed according to the grayscale on the right in B. C,F, Cross sections of probability density (from pointsa to b inB,E) for real animals (black lines) in Figure 3, C andD, and model animals (gray lines) in B and E in radial gradients; verticallines represent SD in probability density (model only). The cross-sectional probability for the model tested with no gradient is shown for comparison (dotted lines). The inequality of areasunder the curves in C andF is a result of taking the cross sections of the probability-density plots. Starting points are indicated byasterisks in A and D. Scale bar: A, B, D, E, 1 cm.

Comparison of the probability distributions (Fig.11B,E) and the cross-sectional probability (Fig. 11C,F) for the model and real worms showed that model worms underperformed real worms in quantitative terms. The discrepancy could have at least three sources. First, tests of the analysis procedure used to estimate the relationship between r versus dC/dtshowed that the procedure systematically underestimates r in the region where dC/dt < 0 if the data are sparse. Thus, the actual values of r in this region could be somewhat higher. In agreement with this idea, we found that when the model was modified such that for dC/dt < 0,r = 0.08, the model reproduced the behavior of the real animals in the NH₄Cl gradient (data not shown). Second, the model did not incorporate the low-pass filter (Fig.7A,B) that could enhance performance by preventing responses to sudden changes in dC/dt. Third, it is possible that one or more unidentified mechanisms act in parallel with the course-correction mechanism to enhance performance in real worms.

As a further test of the model, we asked whether it could predict the behavior of worms in a novel gradient. This was done by computing the tracks of animals chemotaxing in a planar gradient instead of the usual radial one. Model worms traveled up the simulated gradient, often at an angle to the line of steepest ascent (Fig.12A). Thus an all-points histogram (see Materials and Methods) of bearing data, pooled across all model worms (n = 50), revealed a broad peak at 0° (Fig. 12B, solid line) rather than a sharp one. After testing real animals (n = 28) in identical planar gradients, we found similar behavior at the level of the individual worms (Fig.12C) and the all-points histogram (Fig.12B, dotted line). Thus the model qualitatively and quantitatively predicts chemotaxis behavior in a novel planar gradient.

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Fig. 12.

Predicted and actual chemotaxis behavior in a novel planar gradient of attractant. A,Tracks of 7 simulated worms moving in a planar gradient (concentration ranged from 0 to 100 mm NH₄Cl over a distance of 10 cm). Individuals were allowed to wander for the equivalent of 20 min from their starting position (asterisk). B, An all-points histogram of bearing values for model (solid line) and real (dotted line) worms. Bearing values between ±90° (verticaldashed lines) indicate movement up the gradient. C, Tracks of 28 real worms moving in a planar gradient of NH₄Cl identical to the simulated planar gradient in A. Each animal was run separately. Linecolor(gray or black) is used merely to distinguish neighboring tracks. Scale bar: A, C, 1 cm.

Evidence of an alternative chemotaxis strategy

The pirouette model differs substantially from the C. elegans chemotaxis strategy proposed previously by Ward (1973) on the basis of behavioral analysis of mutants. Ward found thatunc-23 mutants, forced to swim in arcs or tight curls because their heads rest at an angle to the body, nevertheless travel fairly directly to the peak of the gradient. This result was interpreted to mean that wild-type animals perform chemotaxis by pointing the tip of the head up the gradient, like a weather vane pointing into the wind. We find no direct evidence of the weather vane strategy, which in its strongest form would result in long runs pointed straight up the gradient, because we failed to observe this in both radial and planar gradients. We cannot rule out the possibility, however, that a weak weather vane strategy operates in parallel with the course-correction model, making runs better oriented than they might otherwise have been. Indeed, perhaps the reason that the real worms performed better than model worms in the radial gradient is that real worms use both a course-correction strategy and a weather vane strategy. We also cannot rule out the possibility that the weather vane strategy exists as a default mechanism when the course-correction strategy is inactivated or when animals are in steeper gradients.

Comparison with bacterial chemotaxis

The pirouette strategy for chemotaxis in C. elegansresembles the strategy used by the bacterium E. coli (Berg and Brown, 1972), but the C. elegans strategy is more efficient. In both strategies, locomotion during chemotaxis consists of episodes of fairly straight swimming (called runs) punctuated by sudden, large turns (pirouettes in C. elegans and tumbles inE. coli). The two strategies are similar in that the probability of a turn depends on values of dC/dtin the recent past, but they differ in two important ways. First, for chemotaxis to attractive compounds, C. elegans exhibits both suppression and facilitation of turning, whereas E. coliexhibits only suppression. Second, tumbles produce a random change in the direction of bacterial locomotion (Berg and Brown, 1972), whereas pirouettes correct the animal's course by error compensation. PerhapsC. elegans chemotaxis represents a transitional stage between biased random locomotion and continuous alignment with the direction of the gradient (Hara, 1971).

Adaptations for effective chemotaxis

The prepirouette average of dC/dt inC. elegans, analogous to a spike-triggered average of input in neurons, reveals important features of sensory processing in theC. elegans chemotaxis system. In a linear system stimulated by white noise input, the spike-triggered average is proportional to the weighting function (or kernel) that, when convolved with the time course of the system's input, yields the time course of the firing probability of the system (Rieke et al., 1997). To the extent that the C. elegans chemotaxis mechanism functions linearly, the prepirouette average reflects a weighting function well suited for efficient chemotaxis, because the magnitude of the prepirouette average is maximal at times just before the pirouette and decays exponentially to a steady-state value at times well before the pirouette. Thus, pirouette probability depends strongly on values of dC/dt in the recent past and weakly on values in the distant past. This makes functional sense because dC/dt in the recent past is well correlated with the animal's current bearing, whereas dC/dt in the distant past is not. A weighting function that decays as a single exponential is characteristic of a low-pass filter, suggesting that in C. elegans, the probability of a pirouette depends on a low-pass filtered version of the sensory input. Such a filter may act to reduce the likelihood of responses to sudden, potentially spurious, changes in attractant concentration.

Implicit computations in chemotaxis

In the course of identifying the behavioral strategy for chemotaxis, we found evidence of two computations: differentiation and low-pass filtering. This result leads to the conceptual model depicted in Figure 13. Chemosensory input in the model, represented by the concentration of chemical attractantC(t), modulates pirouette probabilityP[q(t)] by a three-stage process in which the input is differentiated, low-pass filtered, and passed through a saturating nonlinearity. The saturating nonlinearity represents the likely constraint that pirouette rate has an upper limit. An alternative model is one in which the saturating nonlinearity precedes the low-pass filter.

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Fig. 13.

Conceptual model for C. eleganschemotaxis. Attractant concentrationC(t) modulates pirouette probabilityP[q(t)] by a three-stage process involving differentiation, low-pass filtering, and a nonlinearity that maps filter outputq(t) to pirouette probabilityP[q(t)]. Note that the order of the first two stages of processing could be reversed if the biological system is approximately linear.

Such models provide a well-defined theoretical framework for a neuronal analysis of chemotaxis in C. elegans, currently in progress (J. Pierce-Shimomura and S. Lockery, unpublished observations; M. Gallegos and C. Bargmann, personal communication). A preliminary neural network for chemotaxis is being delineated via laser ablation of identified chemosensory neurons (Bargmann and Horvitz, 1991; Bargmann et al., 1993) and closely associated interneurons and motor neurons (Gallegos and Bargmann, personal communication). It should be possible, therefore, to determine whether the neuronal mechanisms of chemotaxis in C. elegans are consistent with the three-stage model proposed in Figure 13 and, if so, to assign computational roles to identified neurons. These assignments could be made by recording from identified chemosensory neurons, interneurons, and motor neurons (Avery et al., 1995; Lockery and Goodman, 1998) in response to controlled changes in attractant concentration or by tracking the movements of individual worms in which a particular neuron has been removed from the network by laser ablation (Bargmann and Avery, 1995).