QUICK SEARCH:   [advanced] Author: Keyword(s):
Home     Help     Feedback     Subscriptions     Archive     Search     Table of Contents

First published online July 26, 2004
Journal of Experimental Biology 207, 3055-3066 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01141

This Article Summary Full Text Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Miyasaka, I. Articles by Azuma, A. Search for Related Content PubMed PubMed Citation Articles by Miyasaka, I. Articles by Azuma, A. Social Bookmarking                         What's this?

Functional roles of the transverse and longitudinal flagella in the swimming motility of Prorocentrum minimum (Dinophyceae)

Iku Miyasaka^*, Kenji Nanba, Ken Furuya, Yoshihachiro Nimura and Akira Azuma

Department of Aquatic Bioscience, Graduate School of Agricultural and Life Sciences, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-8657, Japan

View larger version (25K): [in a new window]   Fig. 1. Prorocentrum minimum motility described in Miyasaka et al. (1998). (A) P. minimum cell and directional terminology used in the present study. (B) Swimming trajectory. (C) Sutural view, (D) valval view, (E) anterior view, as indicated in A. Open arrows indicate wave directions. AE, anterior end; a1, amplitude of the longitudinal flagellum; ap, antero-posterior axis; at, amplitude of the transverse flagellum; LF, longitudinal flagellum; PE, posterior end; pn, the half-pitch corresponding to the adjacent parts to the ap axis of the transverse flagellum; pf, remote part; Pp, pitch of helical swimming trajectory; Rp, radius of helical swimming trajectory; S, suture of two valves; TF, transverse flagellum; the pitch angle of the cell against the axis of the swimming trajectory; p, the pitch angle of the cell against the axis of the swimming trajectory; 1, angle between the cellular antero-posterior axis and the center line of oscillation of longitudinal flagellum; 1, wavelength of the longitudinal flagellum; t, wavelength of the transverse flagellum.

View larger version (19K): [in a new window]   Fig. 2. Illustration of the coordinate systems and their transformation. (A) The helical swimming trajectory is shown on a cylinder. VX is the net displacement speed of the cell, where c is the angular speed of the cell, Rp is the radius of the cylinder and t is time. epara, etan and erad are the unit direction vectors where epara is parallel to the axis of the cylinder, erad is radial to a circular transections of the cylinder and etan is tangential to the circular transection whose plane is perpendicular to the cylinder's axis. (XI,YI,ZI) is the inertial frame fixed relative to the laboratory and (X',Y',Z') is a coordinate translating with the cell and rotating with the cell and is rotating about X' axis (identical to XI axis) at the angular speed of c. epara, etan and erad are the components parallel to the net displacement, tangential and radial to the swimming trajectory. (X',Y',Z')T=Rc+T1·(XI,YI,ZI)T, where superscript T indicates the transposed vector, Rc is the position of the cell in the inertial frame (see Equation 1) and T1 is a matrix indicating the rotational movement of the cell defined in Equation 3, respectively. (B) Transformations between the coordinates , where , and are the Eulerian angles indicating cell orientation and T2-T4 are defined in Equations 4-7, to define the cell frame (x,y,z). (C) Model of the P. minimum cell and flagella in the cell frame. a1, amplitude of longitudinal wave; at, amplitude of transverse wave; rt radius of the circle along which the transverse wave is propagating; xbt, x coordinate of the circle along which the transverse wave is propagating.

View larger version (26K): [in a new window]   Fig. 3. Two-pitches model of the transverse flagellar wave. (A) 0 (broken line) and (solid line) which alternate between 1 and 2. (B) cos (solid line) and sin (broken line). indicates the phase of this wave and it switches between 1 and 2. 1=0/2p is the equation for when 0<0<2, and 2=(0-2)/2(1-p) is the equation for when 2p''0<2, where p is the ratio of a half pitch corresponding to the remote part of the antero-posterior axis of the cell to a wavelength and 0 is the minimum non-negative value for 2(ftt-s/t)-2m, where t, s, ft, t and m are time, length along the circle where the transverse flagellum wave propagates, the frequency and wavelength of the helical wave and a positive integer that minimizes 0(s,t), respectively.

View larger version (48K): [in a new window]   Fig. 4. Schematic illustrations of the movement of the model cell (Table 1). Cell motility variables and abbreviations are given in Table 1. The cells are viewed along the YI axis except in G, which is viewed along the XI axis (see Fig. 2). The position and orientation of the cell is shown at intervals of 0.25 s, except F, where the time interval is 1 s. The spheres representing cells are divided into eight regions labelled (from left to right) 1-4 and 5-8 on the upper and lower hemispheres, respectively. Arrows indicate the swimming trajectory and direction of turning. (A) LF+h2TF, (B) LF+h9TF, (C) LF+Stf, (D) h2TF, (E) h9TF, (F) Stf, (G) LF.

View larger version (18K): [in a new window]   Fig. 5. Thrust made by flagellar shaft and flagellar hairs of the transverse flagellum in an hTF cell during a cycle of the transverse flagellum. Thick gray lines indicate the trajectory of a segment on the flagellum and the small arrows on the line indicate the direction and strength of the thrust. The change of force vectors are drawn at intervals of 1.4 ms. (A) hTF cell and flagellar trajectory's plane. Coordinate system , and are set as =x, and as the trajectory of the segment is included in - plane. (B) Flagellar movement and the generated force vectors in - plane. (C) Flagellar movement and the generated force vectors in - plane.

View larger version (18K): [in a new window]   Fig. 6. Temporal changes in the forces and moments generated the flagellum and cell. (A) Temporal changes of the thrust resulting in net displacement Fpara generated by various types of cells with a flagellum: as h2TF cell with a transverse flagellum of wavenumber of 4.5 (dot-dashed line), h2TF cell with a transverse flagellum of wavenumber of 3.5 (gray line), h2TF cell with a transverse flagellum of wavenumber of 4.0 (solid line) and LF cell with a longitudinal flagellum (broken line). (B) Temporal changes in force in the direction parallel to net displacement Fpara, in the hTF+LTF cell generated by the transverse flagellum (dot-dashed line), the longitudinal flagellum (broken line), sum of the two types of flagella (gray line) and drag force by the cell body (solid line). (C) Temporal changes in the moment around the antero-posterior axis of the cell Mpara, in the hTF+LTF cell generated by the transverse flagellum (dot-dashed line), the longitudinal flagellum (broken line), sum of the two types of flagella (gray line) and drag force by the cell body (solid line).

View larger version (14K): [in a new window]   Fig. 7. The cell swimming motion as a function of the amplitude-to-wavelength ratio at/t in model cells with a transverse flagellum. Solid, broken and dot-dashed lines indicate h2TF, h9TF and sTF cells, respectively. Horizontal and vertical lines show the average value for observed cells (Miyasaka et al., 1998; Table 1). (A) Speed ratio VX/(ftt). (B) Frequency ratio c. (C) Efficiency . at, amplitude of transverse flagellum wave; t, wavelength of transverse flagellum wave; VX, net displacement speed of cell; ft, frequency of transverse flagellum wave; c, angular speed of cell. See Table 1 for model cell abbreviations.

CiteULike

Complore

Connotea

Del.icio.us

Digg

Reddit

Technorati

Twitter