Functional roles of the transverse and longitudinal flagella in the swimming motility of Prorocentrum minimum (Dinophyceae)
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
* Author for correspondence (e-mail: miyasaka{at}aujaghi.fs.a.u-tokyo.ac.jp)
Accepted 14 June 2004
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Summary |
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Key words: Prorocentrum minimum, flagella, hydrodynamic resistive force theory, swimming, dinoflagellate
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Introduction |
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In a previous study (Miyasaka et al.,
1998), the motility of Prorocentrum minimum
(Fig. 1A) was investigated.
Briefly, P. minimum was found to swim along a helical trajectory with
the same side of the cell always facing the axis of the trajectory as, for
example, lunar motion with respect to the Earth
(Fig. 1B). Net swimming speed
was 95.3 µm and the Reynolds number of the motion was
1.1x10-3. The transverse flagellum encircles the anterior end
of the cell, and a helical wave is propagated along it
(Fig. 1A,E); this helical wave
shows different half-pitches between the nearer and farther parts relative to
the cellular antero-posterior axis (Fig.
1D). The longitudinal flagellum produces a planar sinusoidal wave
propagated posteriorly (Fig.
1C).
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In the present study, equations to describe the steady swimming motion of
P. minimum based on resistive force theory
(Gray and Hancock, 1955) are
presented and the roles of both flagella are elucidated from the resulting
calculations.
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Materials and methods |
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![]() | (1) |
where VX is net displacement speed, c
is the angular speed of cell revolution, RP the radius of
the path helix and t is time, and superscript T indicates the
transposed vector. The transformation to the cell frame from the inertial
frame is performed by successive transformations using the Eulerian angles
,
and
describing cell orientation
(Fig. 2A,B).
|
The transformation from the inertial frame
(XI,YI,ZI), to the
first frame (X',Y',Z'), is
performed using a matrix T1, describing the
rotation about the X1 axis at the angular speed
c as:
![]() | (2) |
where
![]() | (3) |
The transformation from the first frame,
(X',Y',Z'), to the second frame,
(X'',Y'',Z''), is performed using a
matrix T2, describing the orientation of the
cell about the Z' axis as:
![]() | (4) |
where
![]() | (5) |
The transformation from the second frame
(X'',Y'',Z''), to the third frame
(X''',Y''',Z'''), is performed using
a matrix T3, describing the orientation , of
the cell about the Y'' axis as:
![]() | (6) |
where
![]() | (7) |
The transformation from the third frame
(X''',Y''',Z'''), to the cell frame
(x,y,z), is performed using the matrix T4,
describing the orientation , of the cell about the X'''
axis as:
![]() | (8) |
where
![]() | (9) |
Therefore, the transformation from the inertial frame
(XI,YI,ZI), to the
cell frame (x,y,z), is performed as:
![]() | (10) |
The unit direction vectors relative to the swimming trajectory
epara, erad and
etan, (Fig.
2A) are defined as:
![]() | (11) |
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 and perpendicular to the cylinder's axis.
The swimming velocity vc, and rotational velocity
c, in the cell frame are transformed from those in the
inertial frame as:
![]() | (12) |
and
![]() | (13) |
so that
![]() | (14) |
Upper dots in ,
and
and indicate time derivatives. Time
derivatives of the Eulerian angles,
,
and
, are assumed to be negligible
compared with
c, because P. minimum cells are
observed to swim steadily along a helical trajectory with the same side always
facing the trajectory axis (Fig.
1B).
Formulae for the flagella
The flagellar waves of the transverse and longitudinal flagella
(Fig. 1A,B) are reconstructed
as modified helical and sinusoidal waves, respectively (Figs
2C,
3), using variables from
Miyasaka et al. (1998).
Flagellar motion is formulated in the cell frame. The coordinate's origin is
fixed at the cell's centre.
|
The cell's anterior end is represented by the intersection of the spherical cell and x axis; the valval suture plane is represented by plane x,y (Fig. 2C). While the basal parts of both flagella in the observed cell are attached to the anterior end of the cell, they are not included here in the flagellar model because their effects on the motion of the cell are thought to be small.
Transverse flagellum
The transverse flagellum encircles the anterior end of the cell and beats
in a helical wave. It has two different pitches depending on the distance from
the cellular antero-posterior axis (Fig.
1C-E; Miyasaka et al.,
1998). The waveform is formulated here as a helical wave whose
axis is a baseline circle of [xbt,
rtcos(st/rt),
rtsin(st/rt)]T
(0
st
2
rt), where
xbt and rt are the x
coordinates and the radius of the circle, respectively. The coordinate of a
point on the transverse flagellum rt(s,t) is
formulated as:
![]() | (15) |
![]() | (16) |
where ft is the frequency of the helical wave.
Therefore, when s and t vary, 0 varies
within the range 0
0<2
.
switches as when
0
0<2
p:
![]() | (17) |
as when 2p
0<2
:
![]() | (18) |
where p is the ratio of a part corresponding to the remote part,
pf, from the antero-posterior axis of the cell to the
wavelength of the flagellum, t, or
pf/
t
(Fig. 1D) and ranges as
0<p<1.
1 and
2 indicate
equations for
in two ranges. As 2
(ftt
- st/
t) increases,
0
changes in a saw-tooth-shaped wave with a period of 2
, and
shows a
saw-tooth-shaped wave with inclinations of 1/(2p) and
1/(2-2p) when
=
1 and
=
2, respectively
(Fig. 3A). When
changes
as described above, cos
alternates between two pitches in the ratio
p(1-p), as observed in the transverse flagellum in side view
(Figs 1D,
3B).
When time t advances, the wave is propagated along the transverse flagellum, and the flagellar segments move along a circular trajectory in the plane of zcos(st/rt)-ysin(st/rt)=0. The transverse flagellum is assumed to encircle completely the cellular antero-posterior axis (see Fig. 2C).
Longitudinal flagellum
The longitudinal flagellum moves as a wave in a plane perpendicular to the
valval suture plane (Fig. 1C).
The waveform is a sinusoidal wave in the xy plane whose centre line
(Fig. 2C) is:
![]() | (19) |
where xbl and ybl are the
x and y coordinates of the point on this line where
s1=0. A point,
r1(s1,t), on the waveform is
formulated as:
![]() | (20) |
Forces and moments acting on the flagella
The hydrodynamic forces and moments acting on the flagella are given by
hydrodynamic resistive force theory (Gray
and Hancock, 1955). The thrust and moment generated by a flagellar
segment are derived from its velocity relative to the fluid, resistive force
coefficients associated with the fluid viscosity and the length of the
flagellar segment (Gray and Hancock,
1955
). The relative velocity is calculated using the Stokes'
solution for the flow around a sphere
(Jones et al., 1994
) and the
resistive force is calculated for various configurations and arrangements of
flagellar appendages or hairs (Brennen,
1974
; Gray and Hancock,
1955
; Holwill and Sleigh,
1967
; Lighthill,
1976
).
Gray and Hancock (1955)
formulated the hydrodynamic force
generated by a
flagellar element of length dl and having a relative velocity
V to the fluid, as:
![]() | (21) |
where VN and VT are
the velocity components in the normal and tangential directions to the
flagellar shaft, respectively. CN and
CT are the drag coefficients in the normal and tangential
directions to the flagellar shaft, respectively. They proposed that
CN and CT for a smooth-surfaced
flagellum were:
![]() | (22) |
and
![]() | (23) |
respectively, where is the flagellar wavelength along the
flagellar shaft, d is the diameter of the flagellum and µ is the
fluid viscosity. Lighthill
(1976
) improved these
equations as:
![]() | (24) |
and
![]() | (25) |
Holwill and Sleigh (1967)
investigated the hydrodynamics of a Chrosophyte flagellum, which has small
thin rigid hairs attached perpendicularly to the flagellar shaft. They
proposed that the CN and CT of such a
hispid flagellum were given by the sum of the drag coefficients of the
flagellar shaft and flagellar hairs:
![]() | (26) |
and
![]() | (27) |
respectively, where lh is the length of flagellar
hairs, nsec is the number of rows of flagellar hairs in
cross section, nlen is the number of rows of flagellar
hairs per unit length of flagellum, and i is the angle
between the moving direction of the flagellar shaft and the ith
flagellar hair. Superscripts f and h indicate the flagellum and flagellar
hairs, respectively. The drag coefficients
,
,
and
are derived from Equations
24 and 25 using the dimensions of the flagellar shaft and hairs. The alignment
of the flagellar hairs has not been observed because they do not remain after
fixation for electron microscopy. Holwill and Sleigh
(1967
) hypothesised two types
of hispid flagellum having two and nine rows of flagellar hairs, or
and
in Equations 26 and 27.
The
flagellum is
hypothesised to have two rows of flagellar hairs on the opposite side of the
flagellar shaft. The
flagellum is hypothesised to have nine rows, based on the idea that the
alignment of the hairs corresponds to that of the nine microtubule pairs in
the flagellar shaft.
In the present model, CN and CT are
obtained from Lighthill (1976)
and Holwill and Sleigh (1967
),
and the wavelength for each flagellum is calculated from Equations 15-20. The
dimensions of the flagella and flagellar hairs were measured from electron
micrographs of P. minimum in Honsell and Talarico
(1985
), which shows a
smooth-surfaced longitudinal flagellum and a transverse flagellum with
flagellar hairs. The longitudinal flagellum (LF) is regarded as
smooth-surfaced with a diameter of 0.4 µm. Three types of transverse
flagellum have been assumed, to allow for testing of the effect of the
existence of flagellar hairs and their alignment: smooth-surfaced without
flagellar hairs (sTF), bearing hairs in two rows (h2TF) and bearing hairs in
nine rows (h9TF), projected on the transverse flagellum. The diameter of the
transverse flagellum and the length and diameter of a flagellar hair are
assumed to be 0.2 µm, 0.8 µm and 0.06 µm, respectively. The density
of the flagellar hairs on the transverse flagellum is assumed to be eight
hairs per micrometer, based on the electron micrographs in Honsell and
Talarico (1985
). The flagellar
hairs on the transverse flagellum are assumed to be arranged at even angle
intervals, and one of the flagellar hairs is assumed to be oriented in the
direction of the movement relative to the cell frame. Therefore
,
where
. The
number of flagellar hairs around the flagellar transection
was assumed to be two for
h2TF and nine for h9TF.
The velocity of a flagellar element with reference to the cell frame
vflag is:
![]() | (28) |
where r represents
rt(st,t) or
r1(s1,t), with values
taken from Miyasaka et al.
(1998). The fluid velocity
around the cell body is described by the Stokes' flow because of its small
Reynolds number (Jones et al.,
1994
). When a sphere of radius rc moves with a
linear velocity of vc and an angular velocity of
c, the flow due to the cell translation
vtran and rotation vrot, at point
r in the cell frame according to Stokes' law is:
![]() | (29) |
and
![]() | (30) |
respectively, where r is the distance from r to
the origin of the cell frame, or the centre of the sphere
(Lamb, 1932). The passive
fluid velocities caused by the flagellar motion are assumed to be negligibly
small in comparison with those caused by the cell motion,
vtran and vrot. Based on this assumption,
the terms in Equation 21 are:
![]() | (31) |
![]() | (32) |
and
![]() | (33) |
where s represents sl or
st. e indicates a unit tangential vector
to the flagellar shaft as:
![]() | (34) |
and V indicates total velocity of flagellar element
relative to the fluid as:
![]() | (35) |
The force produced by the flagellar element is given by substitution of
Equations 24-35 into Equation 21 and the moment
generated by the
element is given as:
![]() | (36) |
Inertial, buoyant and gravitational forces and moments acting on the flagella, and inertial force and moment acting on the added mass of flagella, are assumed to be negligibly small in comparison with those produced via hydrodynamic resistance.
Forces and moment acting on the cell
The forces and moments acting on the cell body arise from the inertia of
the cell body, inertia of the added mass of the cell, hydrodynamic forces
caused by the cell, gravity and the buoyancy of the cell. However, the
Reynolds number of the swimming motion of the cell, which is
1.1x10-3, shows that the hydrodynamic force and moment
dominate the motion, and inertial forces and moments are negligibly small in
comparison of hydrodynamic ones. The hydrodynamic drag force and moment are
represented by the drag force
and moment
required by the
hydrodynamic resistance to move a sphere of radius rc at
velocity vc and rotational velocity
c as:
![]() | (37) |
and
![]() | (38) |
respectively, where µ is the viscosity of the fluid. The force arising
from gravity and buoyancy on the motion depends on the densities of the cell
body and medium, which are 1.082x103 kg m-3 and
1.021x103 kg m-3, respectively
(Kamykowski et al., 1992).
Gravitational and buoyant forces acting on the model cell are
8.23x10-12 N and 7.76x10-12 N, respectively,
and their resultant force 4.7x10-13 N is much smaller than
the hydrodynamic force acting on the cell moving in the fluid at the speed
around 100 µm s-1, which is in the region of 10-11 N.
Gravitational and buoyant forces acting on the cell do not generate moment to
rotate the cell body because the cell body is represented by a sphere with a
homogeneous density.
Equations of motion
The equations of motion used to simulate steady motion of the cell can be
written as:
![]() | (39) |
and
![]() | (40) |
where the inertial, gravitational and buoyant forces and moments are
neglected and there are no other external forces and moments. Equations 39 and
40 are solved to find vx, vy,
vz, x,
y an
z, and the hydrodynamic forces and moments generated by the
flagella and acting on the cell are evaluated. Equations 12-14 are solved for
variables describing the cell motility in the inertial frame
VX,
c, Rp,
,
and
.
Using the acquired solutions, the power P done by the entire
flagellum against the hydrodynamic force is given by integrating an inner
product of flagellar velocity vector V and the hydrodynamic
force as:
![]() | (41) |
The conversion efficiencies from the power done by flagellar movement
against the hydrodynamic force to cell's motion are given by a ratio of a sum
of power done by the flagellum (a) of the cell, P. The
efficiency of the flagellar motion into swimming and rotation is given as:
![]() | (42) |
where
and
are the hydrodynamic power for motion and rotation of the cell, respectively.
Efficiency for the cell's swimming along the swimming path
path and for its net travelling along a linear distance
linear are given as:
![]() | (43) |
and
![]() | (44) |
respectively, where vpara is the component of vc in the direction of epara.
Model simulations
Seven model cells are considered in simulation: a cell with a longitudinal
flagellum (LF), with a hispid transverse flagellum (h2TF and h9TF), with a
smooth transverse flagellum (sTF), with a longitudinal flagellum plus a hispid
transverse flagellum (LF+h2TF and LF+h9TF) or with a longitudinal flagellum
plus a smooth transverse flagellum (LF+sTF). Cells with a transverse flagellum
are examined for changes in the ratio of swimming speed to wave propagation
speed VX/ftt, the
ratio of rotational frequency to flagellar frequency
c/ft, and efficiency
, as a
function of the amplitude-to-wavelength ratio
at/
t, to allow direct comparisons
with data obtained for other flagellated organisms in previous studies (Chwang
and Wu, 1971
,
1974
;
Coakley and Holwill, 1972
;
Higdon, 1979
;
Holwill, 1966
;
Holwill and Burge, 1963
;
Holwill and Sleigh, 1967
;
Lighthill, 1976
). All
calculations were performed using a Macintosh G3 equipped with Mathematica
version 4.1 (Wolfram Research, IL, USA).
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Results |
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Flagellar hairs on the transverse flagellum determined the direction of cell rotation and the speed of cell displacement and rotation. Cells that have hairs on the transverse flagellum rotated in a right-handed direction, i.e. in the same direction as the wave propagation of the transverse flagellum (Table 1, Fig. 4A,B,D,E), while cells LF+sTF and sTF rotated in a left-hand direction (Fig. 4C,F). Swimming speed decreased from h2TF, through h9TF and sTF for cells without a longitudinal flagellum. Addition of a longitudinal flagellum does not change the order. Cells with a larger value of CT/CN for the transverse flagellum swam faster (Table 1).
The force and moment vectors generated by each flagellum were also calculated and decomposed into the components in the epara, erad and etan directions (Fig. 2A and Equation 11), according to the thrust and moment function (Table 2). The transverse flagellum provided over 90% of the thrust force Fpara to drive the cell, and all the longitudinal moment Mpara to rotate the cell and the longitudinal flagellum in the LF+h2TF and LF+h9TF cells. While the contribution of the longitudinal flagellum to the thrust Fpara was less than 10%, the flagellum generated the lateral force, Ftan, to make the swimming trajectory helical. In cells with only a transverse flagellum (h2TF, h9TF and sTF cells), the flagellum did not generate Ftan (Table 2). In the LF cell, the longitudinal flagellum generated Ftan and Mpara but no Fpara, and the cell swam along a circular path.
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The net efficiencies ranged from 2.3 to 7.3% among the seven model
cells (Table 3). Comparison of
with the travelling efficiency
path indicates a nearly
one-third reduction in efficiency due to rotation in the h2TF and LF+h2TF
cells. In the LF+sTF cell, the advancing efficiency
linear is
one-quarter of
path, which is attributed a greater deviation
from the travelling path. In the LF cell
linear was zero
because the cell swims along a circular trajectory without advancing.
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Characterization of the transverse flagellum
The mechanism of thrust generation by the transverse flagellum, which is
the main forward thrust generator (Table
2), was investigated and we describe the result of the simulation
for the h2TF cell (Fig. 5A) as
a simplest case.
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The motion and thrust generation of a flagellar segment of a given unit length are described as follows. When a transverse flagellum propagates a quasi-helical wave around the cell body, the flagellar segment moves along a planar circular trajectory (Fig. 5A). The thrust vector generated by the flagellar segment depends on the phase of the wave. The integration of the thrust over a period gives forward thrust, because the thrust strength is asymmetric between forward and backward directions (Fig. 5B,C). There are two reasons for this assmmetry. One is the Stokes' flow field caused by the cell translation and rotation. This attenuates the hydrodynamic force generated adjacently to the cell body. The hydrodynamic thrust force decreases in strength by the term containing rc/r in Equations 29 and 30. The forward thrust is generated at a remote part of the cell surface and becomes larger than the backward thrust, which is generated at a nearby part of the cell.
The second is the asymmetry of the waveform, introduced to the model by Equations 5 and 18. Because of this asymmetry, the thrust generated during the backward motion of the flagellar segment is larger than that of the forward motion (Fig. 5C). Therefore, the integrated hydrodynamic force in the direction x component results in a forward thrust in the hTF cell.
The component tangential to the baseline circle causes the moment around
the cell's antero-posterior axis to rotate the cell
(Fig. 5C); this and the radial
component balance each other between the counterpart of the transverse
flagellum. Simulations were made of the relationship between the wavenumber
and the resultant thrust. When there are four waves in the transverse
flagellum, the thrust and moment are constant because most of the force
components in the radial direction counterbalance each other
(Fig. 6A). While this does not
change if the wavenumber is an odd number, it does change when the wavenumber
is not an integer. The forward thrust generated by a transverse flagellum with
wavenumbers of 3.5 and 4.5 oscillates, depending the phase of the wave
(Fig. 6A). The forward thrust
by a cell with a longitudinal flagellum also fluctuates. It fluctuates,
however, when the wavenumber on the longitudinal flagellum is an integer
(Fig. 6A). The fluctuation of
the forward thrust by a cell with a longitudinal flagellum is apparently a
result of the center line of the longitudinal flagellum not penetrating the
center of the spherical cell (Fig.
6A). The ratio of fluctuation to the mean thrust of the
longitudinal flagellum is larger than that of the transverse flagellum, i.e.
the transverse flagellum provides a stable force and moment. This feature of
the transverse flagellum is attributed to its radial symmetry around the
cellular antero-posterior axis. This feature makes the transverse flagellum
unable to generate a force to change the swimming direction of the cell. It is
reasonable that the longitudinal flagellum works to change the cell
orientation while the transverse flagellum is at rest
(Miyasaka et al., 1998).
|
Simulations were made of the relationship between the wavelength and the
resultant speed and rotational frequency. The speed ratio
VX/(ftt), frequency
ratio
c/ft and net efficiency
change as functions of
at/
t, in
h2TF, h9TF and sTF cells (Fig.
7). The net efficiency
peaks at
at/
t
0.7 in the h2TF cell and
at
at/
t
1.0 in h9TF and sTF
cells (Fig. 7C).
|
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Discussion |
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What are the functions of the two flagella in swimming? The results of the
calculations lead to the following conclusions. In cells with only a
transverse flagellum, the flagellum generates Fpara
and Mpara (Table
2), and the cells swim along a straight line
(Table 1,
Fig. 4). In the LF cell, the
flagellum generates Ftan and
Mpara, but no Fpara, and
the cell makes no net displacement but rotates sideways
(Fig. 4). While net efficiency
is highest for the LF cell among the model cells, the advancing
efficiency
linear is zero for this cell
(Table 3). The motion of the
LF+h2TF cell appears to be the sum of the two types described above: the
transverse flagellum contributes 96% of Fpara and
all of Mpara, while the longitudinal flagellum
generates 75% of Ftan
(Table 2). The longitudinal
flagellum of this cell generates negative Mpara and
4% of Fpara, while that of the LF cell generates
positive Mpara and no
Fpara. This indicates that the central line of the
longitudinal flagellum is kept stable by its angle with the antero-posterior
axis, and this stability enables the longitudinal flagellum to generate
Fpara. The roles of the two flagella in LF+h9TF and
LF+sTF cells can be explained similarly, while the motion of the LF+sTF cell
(Fig. 6C) and its low
travelling efficiency
linear
(Table 3) also resemble those
of the LF cell because the sTF generates less force and moment than h2TF or
h9TF does, allowing the properties of the longitudinal flagellum to dominate
(Table 2). To summarise, the
transverse flagellum provides thrust to move the cell along the longitudinal
axis of the helical swimming path and rotates the cell about its
antero-posterior axis. The longitudinal flagellum makes the swimming
trajectory helical, and retards cell rotation.
For microorganisms, there are two advantages of active swimming over
passive movement by gravity and buoyancy: faster movement and the ability to
search for a more suitable place for survival. The former increases the rate
of diffusion between the cell surface and the matrix fluid, by means of which
it exchanges dissolved substances. For example, when a spherical microorganism
with a diameter of 10 µm moves relative to the matrix fluid at speeds of 10
µm s-1 and 100 µm s-1, the flux of dissolved
substances across the cell surface increases by 2% and 40%, respectively,
relative to a stationary cell (Lazier and
Mann, 1989). A moving organism can also search for appropriate
concentration gradients. For this purpose, a helical swimming path is more
useful than a straight one in spite of the longer distance for the same
displacement. This is because a helical swimming path enables detection of
three-dimensional components of a gradient whereas a straight path allows
detection of only one dimension (Crenshaw,
1996
). For a P. minimum cell, the transverse flagellum
enables the cell to achieve a high swimming speed. Addition of a longitudinal
flagellum to the h2TF cell did not cause it to swim faster or more
efficiently, as shown in smaller net displacement speed
VX, or lesser efficiencies (
,
path
and
linear) in the LF+h2TF cell than in the h2TF cell (Tables
1,
3). The longitudinal flagellum,
however, gives a cell the ability to search in the fluid, because it makes the
swimming trajectory helical, allowing the cell to swim in a three-dimensional
gradient and widening the fluid volume through which the cell passes. Turning
the cell in a favourable direction also requires a longitudinal flagellum
(Hand and Schmidt, 1975
;
Miyasaka et al., 1998
).
How does the waveform of the transverse flagellum work in the observed cell
motility? The net efficiency reaches an optimum when
at/
t
0.7 in the h2TF cell and
at/
t
1.0 in h9TF and sTF
cells, respectively (Fig. 7).
The amplitude-to-wavelength ratio
at/
t for the optimum efficiency is
larger than those found in past studies on flagella of spermatozoa or bacteria
(Anderson, 1974
;
Holwill and Burge, 1963
;
Holwill and Peters, 1974
;
Holwill and Sleigh, 1967
).
This feature of the transverse flagellum is attributed to its position, which
is so close to the cell surface that the contribution of the no-slip condition
of the fluid caused by the Stokes' flow field is significant. When the model
does not include the no-slip condition on the cell surface, as in the case of
the flagellum being sufficiently remote from the cell surface, the resultant
linear velocity is a half of the observed swimming speed. This suggests that
the no-slip condition on the cell surface contributes to effective propulsion
by the transverse flagellum.
Our results clearly demonstrate in terms of hydrodynamics that the
existence of flagellar hairs on a transverse flagellum reverses the cell's
rotational direction, as previously noted by Gaines and Taylor
(1985). The smooth-surfaced
transverse flagellum generates less thrust and moment than the observed cells
(Table 1). The LF+h9TF cell has
a smaller
c than the actual cell, while the LF+h2TF cell has
a
c close to the real cells
(Table 1). Although the
arrangement of flagellar hairs in P. minimum has not yet been
published, the simulations suggest that the transverse flagellum possesses
flagellar hairs arranged to form two rows in a cross section of the flagellum
projecting perpendicularly to the direction of the flagellar movement.
In conclusion, we propose the functions of the two flagella of P. minimum are as follows: the transverse flagellum acts as a propulsion device, to move the cell along the longitudinal axis of the helical swimming path and rotate it about its antero-posterior axis; the longitudinal flagellum acts as a rudder, to produce a helical swimming trajectory, and controls the orientation of the cell. Flagellar hairs on the transverse flagellum are probably present because they are necessary to produce simulated cell motion, in agreement with that observed in P. minimum. This is the first numerical evaluation of the functions of the transverse and longitudinal flagella of a dinoflagellate.
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List of symbols and abbreviations |
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Acknowledgments |
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References |
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