The fundamental role of pirouettes in Caenorhabditis elegans chemotaxis

Abstract

To investigate the behavioral mechanism of chemotaxis in Caenorhabditis elegans, we recorded the instantaneous position, speed, and turning rate of single worms as a function of time during chemotaxis in gradients of the attractants ammonium chloride or biotin. Analysis of turning rate showed that each worm track could be divided into periods of smooth swimming (runs) and periods of frequent turning (pirouettes). The initiation of pirouettes was correlated with the rate of change of concentration (dC/dt) but not with absolute concentration. Pirouettes were most likely to occur when a worm was heading down the gradient (dC/dt < 0) and least likely to occur when a worm was heading up the gradient (dC/dt > 0). Further analysis revealed that the average direction of movement after a pirouette was up the gradient. These observations suggest that chemotaxis is produced by a series of pirouettes that reorient the animal to the gradient. We tested this idea by imposing the correlation between pirouettes and dC/dt on a stochastic point model of worm motion. The model exhibited chemotaxis behavior in a radial gradient and also in a novel planar gradient. Thus, the pirouette model of C. elegans chemotaxis is sufficient and general.

Fig. 1.

Comparison of estimated and actual concentration in a radial Gaussian gradient. The gradient was formed by placing two 5 μl drops of 500 mm NH₄Cl at the center of an agar-filled plate, the first drop 20 hr and the second drop 3.5 hr before actual concentrations were measured. Concentrations were measured with a chloride-selective microelectrode at the indicated distances from the center of the plate (measured). Concentration estimates (theory) were made as described in Materials and Methods (Eqs. 1, 2).

Fig. 2.

Dispersion of real and simulated worms in the absence of a chemical gradient. A,Tracks of three real worms moving in a spatially uniform concentration of attractant (2 mm NH₄Cl). Individuals were allowed to wander for 20 min from their starting point (asterisk). Each animal was run separately.B, Probability-density plot for real worms (same conditions described in A). The grayscale (right) indicates the probability per unit area of finding a worm at a given location in the plate during a 20 min assay, computed from the tracking data of 47 individuals started from the indicated location in the plate.C–E, Distributions of instantaneous speed, turning rate, and turning biases for the animals tested inB. Arrowheads indicate the average speed [0.152 ± 0.0702 mm sec⁻¹ (± SD)] and turning rate [0.861 ± 38.9° sec⁻¹ (± SD)] for 41,954 samples and the turning bias [0.441 ± 2.12° sec⁻¹ (± SD)] for the 47 animals.F, Tracks of three simulated worms moving in a uniform concentration of attractant as described in A. G, Probability-density plot for 47 simulated worms (same conditions described in A) started from the indicated location in the plate. H, Probability-density cross sections from pointsa tob in B and G. The dispersion behavior of model and real animals was similar, as indicated by the degree of overlap of the cross sections. The inequality ofareasunder the curves inH is a result of taking the cross sections of the probability-density plots.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.