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First published online November 17, 2006
Journal of Experimental Biology 209, 4652-4662 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02590

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Sensorimotor control during isothermal tracking in Caenorhabditis elegans

Linjiao Luo^1,*, Damon A. Clark^1,*, David Biron^1,2, L. Mahadevan^3,4, and Aravinthan D. T. Samuel^1,

¹ Department of Physics, Harvard University, Cambridge, MA 02138, USA
² Department of Biology, Brandeis University, Waltham, MA 02453, USA
³ Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
⁴ Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA

View larger version (35K): [in a new window]   Fig. 1. Isothermal tracks. (A) Representative isothermal tracks of wild-type worms (N2 strain) cultivated at 20°C navigating a radial thermal gradient on the surface of a 9-cm diameter agar plate. The temperature from center to edge is 18.5 to 21.5°C, corresponding to 0.7°C cm-1 steepness. The trajectory of each isothermal track on the spatial gradient is shown by open circles connected by black lines, with each circle indicating position of the worm centroid at 10 s intervals throughout each track. For purposes of presentation, the undirected movements before and after each isothermal track are not shown. (B) Histogram of the absolute temperatures of 400 isothermal tracks made by wild-type worms cultivated at 20°C navigating linear thermal gradients from 15-25°C. (C) Percentage of time spent in isothermal movement of wild-type worms navigating defined spatial thermal gradients, quantified for each data point by totaling the duration of each isothermal movement exhibited by 60 worms in 30 min of observation on linear thermal gradients, and then dividing by 1800 min. Error bars represent ± 1 s.e.m. (D) Histogram of the durations of 400 isothermal tracks made by wild-type worms navigating a radial thermal gradient with 0.7°C cm-1 steepness. The solid line shows a fit to an exponential function (=80 s; P>0.5). (E) Mean durations of isothermal tracks of wild-type worms navigating radial spatial thermal gradients with 0.2, 0.7, and 1.2°C cm-1 steepness (black circles). Each point represents data from 400 tracks. For comparison, the mean duration of 1000 runs exhibited by wild-type worms navigating an isotropic environment at 20°C is also shown (open circle at 0°C cm-1). Error bars represent ± 1 s.e.m. (F) Percentage of time spent in isothermal movement of wild-type (WT) and mutant worms navigating spatial thermal gradients with 0.5°C steepness, quantified for each data point by totaling the duration of each isothermal movement exhibited by 60 worms in 30 min of observation on linear thermal gradients, and then dividing by 1800 min. Error bars represent ± 1 s.e.m.

View larger version (24K): [in a new window]   Fig. 2. The effect of food and starvation on isothermal tracking. (A) Representative isothermal tracks of wild-type worms navigating radial thermal gradients with uniformly spread bacterial lawns. The temperature from center to edge of the plate was 18.5 to 21.5°C, corresponding to a steepness of 0.7°C cm-1. The time interval between data points is 10 s. (B) Histogram of the durations of isothermal tracks of wild-type worms navigating a radial thermal gradient with 0.7°C steepness in the presence of bacterial lawns. The solid line shows a fit to an exponential (=156±6 s; P>0.2). (C) Percentage of time spent in isothermal movement of wild-type navigating spatial thermal gradients with 0.5°C cm-1 steepness, after overnight cultivation at 20°C with bacterial food and being placed at 20°C without food for fixed periods of time. Each data point was quantified by totaling the duration of each isothermal movement exhibited by 60 worms in 30 min of observation on linear thermal gradients, and dividing by 1800 min. Error bars represent ± 1 s.e.m.

View larger version (13K): [in a new window]   Fig. 3. Movement preceding isothermal tracks. (A) Scatter plot indicating the duration and direction of the run preceding each isothermal track of wild-type worms navigating a linear thermal gradient with 0.7°C cm-1 steepness. Black arrow indicates direction of the thermal gradient. For presentation purposes, the start points of all isothermal tracks are centered at the origin, and the isothermal tracks corresponding to each data point are aligned to the direction of the gray arrow. Black circles indicate the start point of the run preceding each isothermal track relative to the start-point of that isothermal track. Radial distance from the origin represents the duration of the preceding run (see scale bar), and the orientation is indicated by the angle with respect to the directions of the linear thermal gradient. Data corresponding to 700 isothermal tracks are shown. Preceding runs <10 s in duration are excluded. (B) Histogram of the orientation of the run preceding each isothermal track, using data from Fig. 2A. The dip at 0° indicates that isothermal tracks are not typically preceded by runs with strong isothermal alignment in the opposite direction. Also, the dip at 180°C indicates occurs because preceding runs closely aligned with the isothermal track tend to be categorized as part of the same isothermal track. (C) Representative trajectories of C. elegans that left one isothermal track and rapidly began tracking a new isotherm at a different temperature. Open circles indicate worm centroid position at 5 s intervals.

View larger version (10K): [in a new window]   Fig. 4. Isothermal tracking does not affect the speed, wavelength, or frequency of propulsive undulations. (A) Using video microscopy, we quantified the propulsive undulations of C. elegans. Here, we show the representative path traced by the head of a worm crawling along an isotherm near 20.1°C on a spatial thermal gradient with 0.7°C cm-1 steepness. Based on the video images and movies of C. elegans crawling freely on agar surfaces and tracking defined spatial thermal gradients, we quantified the wavelength of sinusoidal bending (B), the temporal frequency of the propulsive undulation (C), and the center- of-mass speed of the movement of the animal's body. In (D), black circles represent direct measurements of the center-of-mass speed, and the white circles are the product of the wavelength and frequency measurements from (B) and (C). Each data point corresponds to measurements with 20 animals. Error bars represent ± 1 s.d., which reflect the worm-to-worm variabilities in the speed and geometry of propulsive undulations.

View larger version (50K): [in a new window]   Fig. 5. Spatiotemporal thermal gradients. Representative trajectories of worms navigating spatiotemporal thermal gradients with fixed steepness in the radial direction and with superposed, spatially uniform, sine-wave temporal variation. Each trajectory shows a single period of persistent forward movement exhibited by a worm navigating these gradients, with positions along each run indicated at 10 s intervals. (A) When worms navigate a spatiotemporal thermal gradient with 0.4°C cm-1 spatial steepness and sine-wave temporal variation with 0.05°C amplitude and 60 s period, prolonged floral trajectories emerge as the worm follows each isotherm during cycles of contraction and expansion. (B) Floral trajectories of worms navigating a spatiotemporal thermal gradient with 0.4°C cm-1 spatial steepness and sine-wave temporal variation with 0.1°C amplitude and 120 s period. (C) Floral trajectories are largely replaced by looping trajectories when worms navigate a spatiotemporal thermal gradient with 0.4°C cm-1 spatial steepness and sine-wave temporal variation with 0.2°C amplitude and 120 s period. (D) Floral trajectories of worms navigating a spatiotemporal thermal gradient with the same temporal variation as in Fig. 4C, but with a 0.7°C spatial steepness. (E) The relative appearance of floral and looping trajectories on spatiotemporal gradients with fixed 0.4°C cm-1 spatial steepness depends on the steepness of the superposed temporal gradient. The steepness of the temporal gradient is proportional to its amplitude (T0) and frequency (). The floral trajectories dominate when T0 is small (above the dotted line), and are replaced by looping trajectories when T0 is large (below the dotted line). The shading of each circle corresponds to the fraction of floral trajectories in a dataset corresponding to 300 tracks. The dotted line indicates the approximate boundary between loop-like and flower-like trajectories, occurring as approaches 1; the dotted line shows 0.4.

View larger version (10K): [in a new window]   Fig. 6. Analysis of the mathematical model. (A) Worm trajectories calculated using the model represented by Eqn 2 and Eqn 3 with and without a spatial thermal gradient. The black arrow indicates the direction of worm movement, the thick grey arrow indicates the direction of the thermal gradient, and the dotted line indicates an isotherm. The black line is the calculated trajectory without the gradient. The grayed trajectory shows the calculated trajectory in response to the perturbation caused by the gradient. The two labeled points in the magnified region indicate the extrema of both (t) and . At point x, ||2 is greater than at point y, so the worm curves more vigorously at x, leading to a slight right turn that corrects isothermal alignment. (B) Functional forms of f(), all of which can generate calculated trajectories consistent with the movements of real C. elegans responding to spatiotemporal thermal gradients. The requisite feature for the qualitative behavior is that f() is monotonic in ||. The thick line corresponds to the function f()=g||2, the functional form we use in the analytical discussion in Results. (C) Plot of as a function of and . The values of are listed to the right of their curves. Black dots represent stable fixed points, the white dot represents an unstable point, and the gray circle switches from unstable to stable when (t)>1. (D) Plots of the fixed point as a function of . When undergoes small oscillations about 0, so does . When approaches 1, the fixed point is able to move to different branches of the graph, corresponding to the looped trajectories exhibited by C. elegans responding to steep superposed temporal gradients.

View larger version (52K): [in a new window]   Fig. 7. Comparison of the model with experimental results. (A-D) Calculated trajectories of simulated worms using Eqn 2 and Eqn 3 and the same parameters of spatiotemporal thermal gradient as the experiments shown in Fig. 5 (E,F) The movements of real and simulated C. elegans responding to spatiotemporal thermal gradients are plotted as in Fig. 6D. The solid red line indicates the prediction of the model, , while the connected points correspond to the crawling trajectories exhibited real C. elegans (E) and calculated trajectories for simulated C. elegans (F) on a gradient of 0.4°C cm-1. In each graph, black points correspond to superposed sine-wave temporal thermal gradients with 0.1°C amplitude and 120 s period (Fig. 5B and Fig. 7B), and the blue points correspond to superposed sine-wave temporal thermal gradients with 0.2°C amplitude and 120 s (data taken from Fig. 5C and Fig. 7C). The larger amplitude oscillation allows the trajectories to move to different branches, creating the looped trajectories observed in experiment and simulation. The isothermal tracking performance of real C. elegans and simulated C. elegans may be quantified by the r.m.s. deviation between the direction of forward movement and the instantaneous direction of the isotherm. For the flowerlike (black) trajectories of E and F the r.m.s. deviation for real and simulated C. elegans is 6° and 10°, respectively. For the looping (blue) trajectories of E and F, the r.m.s. deviation is 34° and 22°, respectively.

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