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First published online November 1, 2006
Journal of Experimental Biology 209, 4452-4463 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02537

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Interaction of two swimming Paramecia

Takuji Ishikawa^1,* and Masateru Hota²

¹ Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University, Aoba 6-6-01, Sendai 980-8579, Japan
² Department of Mechanical Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui 610-8507, Japan

View larger version (7K): [in a new window]   Fig. 1. Schematics of the experimental apparatus. Test fluid was placed between the bottom of the inner dish and the top of the outer dish.

View larger version (14K): [in a new window]   Fig. 2. Sequences showing the interaction of two swimming P. caudatum cells observed in the experiment. The background was subtracted from the figure.

View larger version (98K): [in a new window]   Fig. 3. Original image for a swimming P. caudatum cell in a water with a small amount of milk between flat plates.

View larger version (30K): [in a new window]   Fig. 4. Velocity vectors relative to the swimming velocity of P. caudatum, which were calculated by the PIV methods by assuming that the velocity field is axisymmetric and time-independent. The large arrow indicates the swimming direction of the cell.

View larger version (7K): [in a new window]   Fig. 5. Experimental results and an approximated curve defined by Eqn 1 for a surface velocity of P. caudatum. The coefficients in Eqn 1 are c1=1.707, c2=0.2400, c3=0.2472, c4=0.1506 and c5=0.1154.

View larger version (15K): [in a new window]   Fig. 6. Computational mesh for two interacting squirmers, in which 590 triangle elements are generated per squirmer. The mesh is finer in the near-contact region. Using the boundary element method, the computational mesh is generated only on the particle surfaces.

View larger version (10K): [in a new window]   Fig. 7. Sequences showing the biological reactions when two swimming P. caudatum (labeled 1, 2) experience a near-contact. Long arrows are added to schematically show cell motion. (A) Avoiding reaction (t=1/6 s). (B) Escape reaction (t=1/3 s). Scale bars, 500 µm.

View larger version (14K): [in a new window]   Fig. 8. Some sample sequences showing the hydrodynamical interactions when two swimming P. caudatum experience a near-contact. The time interval between each sequence is 1/3 s. Long arrows are added to schematically show cell motion. (A) Sample case 1; (B) sample case 2; (C) sample case 3; (D) sample case 4. Scale bars, 500 µm.

View larger version (51K): [in a new window]   Fig. 9. Sequences (A-F) showing the hydrodynamic interactions between two squirmers under the initial condition of in. At t=0, there is a distance of 0.3 in the perpendicular direction to the orientation vectors, where t is the dimensionless time and t=0 is the initial instant. The orientation vectors of the squirmers are shown as large arrows on the ellipsoids, and a thin solid line is added so that one can easily compare the angle between the two squirmers. d, explained in Fig. 12, is about 0.0 in this case. (A) t=0.0; (B) t=1.0; (C) t=1.5; (D) t=2.0; (E) t=3.0; (F) t=5.0.

View larger version (56K): [in a new window]   Fig. 10. Sequences (A-F) showing the hydrodynamic interactions between two squirmers under the initial condition of in2/3. The orientation vectors of the squirmers are shown as large arrows on the ellipsoids, and a thin solid line is added so that one can easily compare the angle between the two squirmers. d, explained in Fig. 12, is about 0.3 in this case. (A) t=2.0; (B) t=3.2; (C) t=4.0; (D) t=4.4; (E) t=5.0; (F) t=6.0.

View larger version (56K): [in a new window]   Fig. 11. Sequences (A-F) showing the hydrodynamic interactions between two squirmers under the initial condition of in/2. The orientation vectors of the squirmers are shown as large arrows on the ellipsoids, and a thin solid line is added so that one can easily compare the angle between the two squirmers. d, explained in Fig. 12, is about 2.0 in this case. (A) t=0.0; (B) t=2.4; (C) t=3.0; (D) t=4.5; (E) t=6.0; (F) t=9.0.

View larger version (7K): [in a new window]   Fig. 12. Definition of three kinds of angles, where e1 and e2 are the orientation vectors of cell 1 and 2, respectively. in is the angle between e1,in and e2,in when two cells surfaces are at a distance of L before the collision. out is the angle between e1,out and e2,out when two cell surfaces are at a distance of L after the collision. In order to describes the change of orientation of cell 2 relative to cell 1, a frame is fixed to cell 1 so that the frame rotates when cell 1 rotates, and d is defined as change in the angle of cell 2 relative to this rotating frame.

View larger version (7K): [in a new window]   Fig. 13. Change of the reaction rate with in. AR, avoiding reaction; ER, escape reaction. The data are classified according to the contact points. Head-tail, for instance, indicates that the collision occurs between the head of cell 1 and the tail of cell 2 (cf. Fig. 14).

View larger version (7K): [in a new window]   Fig. 14. Head-tail interaction, in which the collision occurs between the head of one cell and the tail of the other. A cell is divided into three equal length sections; head, body and tail, respectively, from the anterior end.

View larger version (6K): [in a new window]   Fig. 15. Temporal change of swimming velocity in the case of escape reaction. Error bars show the standard deviation of 63 escaping cells. Collision occurs at t=0, and Uin is the velocity when two cell surfaces are at a distance of L before the collision.

View larger version (10K): [in a new window]   Fig. 16. Correlation between in and d for the three contact positions. The broken line of slope one and the solid line of d=0 are added for comparison.

View larger version (11K): [in a new window]   Fig. 17. Comparison of the results of d between the experiments and the simulations. Gray symbols, experimental results; the numerical results are plotted by large circles and squares. The broken line of slope one and the solid lines of d=0 and d=in+0.4 are added for comparison.