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First published online February 20, 2004
Journal of Experimental Biology 207, 1203-1216 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00881

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Measurement of cell velocity distributions in populations of motile algae

V. A. Vladimirov¹, M. S. C. Wu², T. J. Pedley^3,*, P. V. Denissenko¹ and S. G. Zakhidova¹

¹ Department of Mathematics, University of Hull, Cottingham Road, Hull HU6 7RX, UK
² Department of Biology, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
³ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

View larger version (16K): [in a new window]   Fig. 1. (A) Experimental setup. (B) Test tube.

View larger version (104K): [in a new window]   Fig. 2. (A) Typical image of swimming cells. The white vertical bars represent the walls of the test tube, which is 1 cm wide. (B) Composite image from the 21 images acquired in a burst: algal tracks are clearly seen; the short straight vertical tracks are made by sedimenting dust particles.

View larger version (42K): [in a new window]   Fig. 3. Typical scatter plot of vertical and horizontal velocities of the cells detected within a burst. Cells clearly swim upwards on average.

View larger version (24K): [in a new window]   Fig. 4. Time evolution of the cells' distribution by two-dimensional projections of velocities onto x–z plane. Contour plots of the probability density function (see Equation 5) at six time instants. The difference between colour levels is 0.00015 s2 µm–2. In the 20 min plot, where the number of motile cells that had reached the observation area is small, the peak corresponding to suspended dust particles is seen. On later images, where the number of active swimmers is large enough, this peak vanishes.

View larger version (64K): [in a new window]   Fig. 5. Time evolution of the three-dimensional cells' distribution by velocities. Surface plot of reconstructed three-dimensional probability density function f (Vr,Vz) (Equation 9), expressed as f (V,), at six time instants. The axes correspond to the cell absolute velocity and the cell velocity angle to the vertical =atan(Vr/Vz). A maximum observed at small V is related to the passive particles suspended in the medium. At about 40 min the number of active swimmers in the observation area becomes large compared to that of passive particles and this maximum vanishes.

View larger version (28K): [in a new window]   Fig. 6. Time evolution of averaged velocities (A) and their standard deviations (B) for the cells located in the camera field of view. Absolute projected velocity Vp, vertical velocity Vz and horizontal velocity Vx. Diamonds with error bars correspond to the instant values Vxr,b, Vzr,b, Vpr,b. Solid lines correspond to averaged values of Vp, Vz and Vx in A and to their standard deviations in B, obtained using an equation similar to Equation 5.

View larger version (22K): [in a new window]   Fig. 7. Averaged (A) absolute projected velocity Vp, (B) vertical velocity Vz and (C) horizontal velocity Vx, calculated separately for the tracks detected in the different quarters of the observation area (compare with Fig. 6). Broken bold lines correspond to the upper right quarter, broken thin line to the upper left quarter, solid bold line to the lower right quarter and solid thin line to the lower left quarter of the observation area. As expected, faster swimmers (ones with higher vertical and projected velocities) are located further from the injection point at the bottom of the test tube.

View larger version (27K): [in a new window]   Fig. 8. Average parameters based on different parts of tracks (compare with Fig. 6). Thin lines correspond to data based on the four-point track fragments from frames 4–6, 6–9, 9–12 of the bursts; thicker lines correspond to frames 12–15, 15–18, 18–21 (boldest). Observe that the three thin lines indicating mean velocities almost coincide, i.e. the photokinetic response starts to develop around 10 s after the laser is switched on. The symbols correspond to the average parameters measured in the runs with laser intensity ten times higher than in the standard experiments.

View larger version (27K): [in a new window]   Fig. 9. One-dimensional distributions based on analysis of all tracks detected in all runs. (A) Distributions of cells by Vx, Vz and Vp; (B) cells' distribution by swimming direction =atan (Vr/Vz), obtained from the three-dimensional axisymmetric distribution reconstructed in accordance with Equation 9; (C) cells' distribution by plotted in B divided by sin, i.e. versus cos. The slope of this graph corresponds to values of in the Fisher distribution (Equation 10) in the range 2–3.

View larger version (62K): [in a new window]   Fig. 10. Cell velocity distribution obtained on the basis of all tracks detected in all runs. (A) Surface plot of the two-dimensional probability density function F(Vx,Vz); (B) surface plot of the reconstructed three-dimensional probability density function F(V,) (compare with Fig. 5); (C) surface plot of ln[f (V,)]; (D) sections of ln[f (V,)] at different values of V, i.e. view of C from the tip of the V axis (compare with Fig. 11).

View larger version (42K): [in a new window]   Fig. 11. Time evolution of the reconstructed three-dimensional cells' distribution by velocities (Equation 9). Cross-sections of ln[f (V,)] are plotted for different values of V. The slopes of the lines correspond to in Equation 10. Bold lines correspond to values of V between 30 µm s–1 and 60 µm s–1.

View larger version (27K): [in a new window]   Fig. 12. (A) Averaged cell turning amplitude (Equation 11) versus time at different angles (0) from vertical; lines join points corresponding to (0)(0°,10°], (0)(10°,20°], (0)(20°,30°] etc; thin and bold lines are used to simplify visual comprehension. (B) Turning speed µ0() versus angle, compare with Equation 13: gradient of the linear approximation to graphs in A for <5 s. See text for further explanation.

View larger version (26K): [in a new window]   Fig. 13. (A) Variance of the turning amplitude (Equation 12) versus time for different angles from vertical (0); lines join points corresponding to (0)(0°,10°], (0)(10°,20°], (0)(20°,30°] etc. The origin of lines corresponding to different (0) are shifted vertically for convenience; thin and bold lines are used to simplify visual comprehension. (B) Rotational diffusivity Dr=02/2 versus angle (Equation 12): gradients of the linear approximation to graphs in A for 5 s. See text for further explanation.

View larger version (23K): [in a new window]   Fig. 14. Autocorrelation of the cells' swimming direction (projected onto the x–z plane) calculated for the tracks detected in seven experimental runs separately (Equation 16). The broken line corresponds to the run that turned out to be invalid. Solid lines connect points related to one run. The exponential decrease of correlation with time is typical for random walk processes.

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