## SUMMARY

Equations describing the motion of the dinoflagellate *Prorocentrum minimum*, which has both a longitudinal and a transverse flagellum, were formulated and examined using numerical calculations based on hydrodynamic resistive force theory. The calculations revealed that each flagellum has its own function in cell locomotion. The transverse flagellum works as a propelling device that provides the main driving force or thrust to move the cell along the longitudinal axis of its helical swimming path. The longitudinal flagellum works as a rudder, giving a lateral force to the cell in a direction perpendicular to the longitudinal axis of the helix. Combining these functions results a helical swimming motion similar to the observed motion. Flagellar hairs present on the transverse flagellum are necessary to make the calculated cell motion agree with the observed cell motion.

## Introduction

Dinoflagellates are microorganisms that swim using two flagella: a transverse flagellum encircling the cellular antero-posterior axis and a longitudinal one running posteriorly. There are numerous reports about diurnal vertical migration of dinoflagellates and their survival strategy is deeply linked to their swimming motion (Ault,2000; Eppley et al.,1968; Franks, 1997; Horstmann, 1980; Kamykowski, 1981; MacIntyre et al., 1997; Olli et al., 1997; Olsson and Granéli,1991). It has conventionally been thought that the forward thrust for swimming is provided by the transverse flagellum and/or the longitudinal flagellum, that the transverse flagellum produces cell rotation and that the longitudinal flagellum controls cell orientation. These suggestions are confirmed by observations of water movement around the organism(Jahn et al., 1963; Peters, 1929). Further electron microscopic studies (Honsell and Talarico, 1985) gave rise to hypothesizes of mechanisms how a transverse flagellum generates thrust(Gaines and Taylor, 1985; LeBlond and Taylor, 1976; for reviews, see Levandowsky and Kaneta,1987; Sleigh,1991; Goldstein,1992). However, a quantitative examination of how the swimming motion and flagellar motion are linked is lacking, which makes it difficult to decide which flagellum is responsible for thrust generation during swimming. To answer the question, it is necessary to quantify the forces and moments generated by each flagellum and to relate them to the swimming speed,rotational speed, swimming trajectory and other swimming variables.

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 1.1×10^{-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).

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.

## Materials and methods

### The coordinate systems

*Prorocentrum minimum*(Pavillard) Schiller is here represented as a sphere of equivalent volume, moving steadily along a helical trajectory, with the variables describing the cell motion at first treated as unknowns. Two Cartesian coordinate systems or frames are established; one is the `inertial frame'(

*X*

_{I},

*Y*

_{I},

*Z*

_{I}), fixed relative to the laboratory, and the other is the `cell frame'(

*x,y,z*), fixed relative to the cell. When the cell moves along a helical trajectory, the position of the cell

*R*_{c}in the inertial frame is written in the inertial frame as:

where *V*_{X} is net displacement speed, Ω_{c}is the angular speed of cell revolution, *R*_{P} 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 anglesψ, Θ and Φ describing cell orientation(Fig. 2A,B).

*X*

_{I},

*Y*

_{I},

*Z*

_{I}), to the first frame (

*X*′,

*Y*′,

*Z*′), is performed using a matrix

*T*_{1}, describing the rotation about the

*X*

_{1}axis at the angular speedΩ

_{c}as:

*X*′,

*Y*′,

*Z*′), to the second frame,(

*X*″,

*Y*″,

*Z*″), is performed using a matrix

*T*_{2}, describing the orientation ψ of the cell about the

*Z*′ axis as:

*X*″,

*Y*″,

*Z*″), to the third frame(

*X*‴,

*Y*‴,

*Z*‴), is performed using a matrix

*T*_{3}, describing the orientation Θ, of the cell about the

*Y*″ axis as:

*X*‴,

*Y*‴,

*Z*‴), to the cell frame(

*x,y,z*), is performed using the matrix

*T*_{4},describing the orientation Φ, of the cell about the

*X*‴axis as:

*X*

_{I},

*Y*

_{I},

*Z*

_{I}), to the cell frame (

*x,y,z*), is performed as:

*e*_{para},

*e*_{rad}and

*e*_{tan}, (Fig. 2A) are defined as:

where *e*_{para} is parallel to the axis of the cylinder, *e*_{rad} is radial to a circular transections of the cylinder and *e*_{tan} is tangential to the circular transection and perpendicular to the cylinder's axis.

**v**

_{c}, and rotational velocityω

_{c}, in the cell frame are transformed from those in the inertial frame as:

Upper dots in

_{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

*x*

_{bt},

*r*

_{t}cos(

*s*

_{t}/

*r*

_{t}),

*r*

_{t}sin(

*s*

_{t}/

*r*

_{t})]

^{T}(0≤

*s*

_{t}≤2π

*r*

_{t}), where

*x*

_{bt}and

*r*

_{t}are the

*x*coordinates and the radius of the circle, respectively. The coordinate of a point on the transverse flagellum

*r*

_{t}(

*s,t*) is formulated as:

*s*

_{t}is the length along the axis of the helix,

*a*

_{t}is the amplitude, λ

_{t}wavelength and

*n*

_{t}wavenumber of the helical wave, respectively(Fig. 2C). φ indicates the phase of this wave and two different pitches of the flagellum are expressed by the two alternative equations described below.φ

_{0}(

*s*

_{t},

*t*) is a non-negative real number and:

*f*

_{t}is the frequency of the helical wave. Therefore, when

*s*and

*t*vary, φ

_{0}varies within the range 0≤φ

_{0}<2π. φ switches as when 0≤φ

_{0}<2π

*p*:

*p*≤φ

_{0}<2π:

where *p* is the ratio of a part corresponding to the remote part, *p*_{f}, from the antero-posterior axis of the cell to the wavelength of the flagellum, λ_{t}, or *p*_{f}/λ_{t}(Fig. 1D) and ranges as 0<*p*<1. φ_{1} and φ_{2} indicate equations for φ in two ranges. As 2π(*f*_{t}*t*- *s*_{t}/λ_{t}) increases, φ_{0}changes in a saw-tooth-shaped wave with a period of 2π, and φ shows a saw-tooth-shaped wave with inclinations of 1/(2*p*) and 1/(2-2*p*) when φ=φ_{1} andφ=φ_{2}, respectively(Fig. 3A). When φ changes as described above, cosφ 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 *z*cos(*s*_{t}/*r*_{t})-*y*sin(*s*_{t}/*r*_{t})=0. The transverse flagellum is assumed to encircle completely the cellular antero-posterior axis (see Fig. 2C).

#### Longitudinal flagellum

*xy*plane whose centre line(Fig. 2C) is:

*x*

_{bl}and

*y*

_{bl}are the

*x*and

*y*coordinates of the point on this line where

*s*

_{1}=0. A point,

*r*

_{1}(

*s*

_{1},

*t*), on the waveform is formulated as:

*s*

_{1}is the length of the line along which the flagellum wave propagates,

*r*

_{1}(

*s*

_{1},

*t*) are coordinates of a point on the wave,

*a*

_{1}is amplitude, λ

_{1}wavelength,

*f*

_{1}frequency and

*n*

_{1}wavenumber of the flagellar wave, and θ

_{1}is the angle between the wave's centre line and cell's antero-posterior axis.

### 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).

*l*and having a relative velocity

**to the fluid, as:**

*V*

*V*_{N}and

*V*_{T}are the velocity components in the normal and tangential directions to the flagellar shaft, respectively.

*C*

_{N}and

*C*

_{T}are the drag coefficients in the normal and tangential directions to the flagellar shaft, respectively. They proposed that

*C*

_{N}and

*C*

_{T}for a smooth-surfaced flagellum were:

*d*is the diameter of the flagellum and μ is the fluid viscosity. Lighthill(1976) improved these equations as:

*C*

_{N}and

*C*

_{T}of such a hispid flagellum were given by the sum of the drag coefficients of the flagellar shaft and flagellar hairs:

respectively, where *l*_{h} is the length of flagellar hairs, *n*_{sec} is the number of rows of flagellar hairs in cross section, *n*_{len} is the number of rows of flagellar hairs per unit length of flagellum, and θ_{i} is the angle between the moving direction of the flagellar shaft and the *i*th flagellar hair. Superscripts f and h indicate the flagellum and flagellar hairs, respectively. The drag coefficients

In the present model, *C*_{N} and *C*_{T} 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

**v**

_{flag}is:

**represents**

*r*

*r*_{t}(

*s*

_{t},

*t*) or

*r*_{1}(

*s*

_{1},

*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

*r*

_{c}moves with a linear velocity of

**v**

_{c}and an angular velocity ofΩ

_{c}, the flow due to the cell translation

**v**

_{tran}and rotation

**v**

_{rot}, at point

**in the cell frame according to Stokes' law is:**

*r**r*is the distance from

**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,**

*r***v**

_{tran}and

**v**

_{rot}. Based on this assumption,the terms in Equation 21 are:

*s*represents

*s*

_{l}or

*s*

_{t}.

**indicates a unit tangential vector to the flagellar shaft as:**

*e***indicates total velocity of flagellar element relative to the fluid as:**

*V*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

^{-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

*r*

_{c}at velocity

**v**

_{c}and rotational velocity ω

_{c}as:

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 1.082×10^{3} kg m^{-3} and 1.021×10^{3} kg m^{-3}, respectively(Kamykowski et al., 1992). Gravitational and buoyant forces acting on the model cell are 8.23×10^{-12} N and 7.76×10^{-12} N, respectively,and their resultant force 4.7×10^{-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

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 *v*_{x}, *v*_{y}, *v*_{z}, Ω_{x}, Ω_{y} anΩ _{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 *V*_{X}, Ω_{c}, *R*_{p}, ψ,Θ and Φ.

*P*done by the entire flagellum against the hydrodynamic force is given by integrating an inner product of flagellar velocity vector

**and the hydrodynamic force**

*V**P*. The efficiency of the flagellar motion into swimming and rotation is given as:

_{path}and for its net travelling along a linear distanceη

_{linear}are given as:

respectively, where **v**_{para} is the component of **v**_{c} in the direction of *e*_{para}.

### 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 *V*_{X}/*f*_{t}λ_{t}, the ratio of rotational frequency to flagellar frequencyΩ _{c}/*f*_{t}, and efficiency η, as a function of the amplitude-to-wavelength ratioπ *a*_{t}/λ_{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).

## Results

### Movement of cells

The results of the calculations gave distinctively different movement patterns for each of the seven model cells(Table 1, Fig. 4). The traces of the swimming trajectories fall into three types. Cells with both transverse and longitudinal flagella move along a helical trajectory. Those with a transverse flagellum swim along a linear trajectory and rotate at more than twice the speed of the corresponding cell with a longitudinal flagellum(Table 1, Fig. 4D-F). The LF cell swims along a circular trajectory, rotating sideways and making no net displacement(Fig. 4G).

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 *C*_{T}/*C*_{N} 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 *e*_{para}, *e*_{rad} and *e*_{tan} 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 *F*_{para} to drive the cell, and all the longitudinal moment *M*_{para} 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 *F*_{para} was less than 10%, the flagellum generated the lateral force, *F*_{tan}, to make the swimming trajectory helical. In cells with only a transverse flagellum (h2TF, h9TF and sTF cells), the flagellum did not generate *F*_{tan}(Table 2). In the LF cell, the longitudinal flagellum generated *F*_{tan} and *M*_{para} but no *F*_{para}, and the cell swam along a circular path.

The net efficiencies η ranged from 2.3 to 7.3% among the seven model cells (Table 3). Comparison ofη with the travelling efficiency η_{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 η_{linear} is one-quarter of η_{path}, which is attributed a greater deviation from the travelling path. In the LF cell η_{linear} was zero because the cell swims along a circular trajectory without advancing.

### 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.

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 *r*_{c}/** 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 *V*_{X}/(*f*_{t}λ_{t}), frequency ratio Ω_{c}/*f*_{t} and net efficiency ηchange as functions of π*a*_{t}/λ_{t}, in h2TF, h9TF and sTF cells (Fig. 7). The net efficiency η peaks atπ *a*_{t}/λ_{t}≅0.7 in the h2TF cell and at π*a*_{t}/λ_{t}≅1.0 in h9TF and sTF cells (Fig. 7C).

## Discussion

Waveforms of *P. minimum* flagella were formulated and examined by means of a numerical model based on the hydrodynamic resistive theory. The motility of the observed cells was reproduced by the LF+h2TF cell(Table 1, Fig. 4A), and this model proved to be a potent device for quantitatively treating the motility of *P. minimum*.

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 *F*_{para}and *M*_{para} (Table 2), and the cells swim along a straight line(Table 1, Fig. 4). In the LF cell, the flagellum generates *F*_{tan} and *M*_{para}, but no *F*_{para}, and the cell makes no net displacement but rotates sideways(Fig. 4). While net efficiencyη is highest for the LF cell among the model cells, the advancing efficiency η_{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 *F*_{para} and all of *M*_{para}, while the longitudinal flagellum generates 75% of *F*_{tan}(Table 2). The longitudinal flagellum of this cell generates negative *M*_{para} and 4% of *F*_{para}, while that of the LF cell generates positive *M*_{para} and no *F*_{para}. 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 *F*_{para}. 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 η_{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 10μm 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 *V*_{X}, or lesser efficiencies (η, η_{path}and η_{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 η reaches an optimum whenπ *a*_{t}/λ_{t}≅0.7 in the h2TF cell andπ *a*_{t}/λ_{t}≅1.0 in h9TF and sTF cells, respectively (Fig. 7). The amplitude-to-wavelength ratioπ *a*_{t}/λ_{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 Ω_{c} than the actual cell, while the LF+h2TF cell has a Ω_{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.

*a*_{t}amplitude of the helix

- C
drag coefficient

- d
diameter of the flagellum

*e*_{para},*e*_{rad},*e*_{tan}unit direction vectors relative to the swimming trajectory

- f (superscript)
flagellum hair

- F
force

*f*_{1}frequency of the longitudinal flagellar wave

- \(F_{\mathrm{c}}^{\mathrm{H}}\)
drag force

- \(F_{\mathrm{f}}^{\mathrm{H}}\)
hydrodynamic force

*f*_{t}frequency of the transverse helical wave

- h (superscript)
flagellar hair

- l
length

- LF, 1 (subscript)
longitudinal flagellum

- M
moment

- \(M_{\mathrm{c}}^{\mathrm{H}}\)
drag moment

- \(M_{\mathrm{f}}^{\mathrm{H}}\)
moment generated by the flagellar element

*n*_{1}wavenumber of the longitudinal flagellar wave

*n*_{len}number of rows of the flagellar hair per unit length of flagellum

*n*_{sec}number of rows of flagellar hairs in cross section

*n*_{t}wavenumber of the transverse flagellar wave

- P
power

- p
ratio of a half pitch

- r
radius

- r
position vector of a point on a flagellum

*R*_{c}position of the cell in the inertial frame

*R*_{P}radius of the path helix

*s*_{1}length along the axis of the longitudinal flagellar wave

*s*_{t}length along the axis of the transverse flagellar wave

- T (superscript)
transposed vector

- T
matrix

- t (subscript)
transverse flagellum

- t
time

- TF
transverse flagellum

- V
relative velocity

**v**_{c}swimming velocity

**v**_{flag}velocity of a flagellar element

*V*_{X}net displacement speed

*X*_{I},*Y*_{I},*Z*_{I}Cartesian coordinates `inertial frame'

- x, y, z
Cartesian coordinates `cell frame'

- Θ
angle between the wave's centre line and cell's antero-posterior axis

- φ
phase of helical wave

- η
swimming efficiency

- Ξ, Ψ, Ζ
coordinates

- Ψ, Θ, Φ
Eulerian angles describing cell orientation

- ΘP
pitch angle of the cell against the axis of the swimming trajectory

- ωc
rotational velocity

- Ωc
angular speed of cell revolution

- ηlinear
advancing efficiency

- ηpath
travelling efficiency

- λt
wavelength of the helix

- μ
fluid viscosity

## Acknowledgements

The authors are grateful to anonymous referees for their helpful suggestions to an earlier version of this manuscript. We also thank Dr Y. Fukuyo for providing information on morphology and taxonomy of the dinoflagellates. The culture strain of *P. minimum* was a kind gift of Dr S. Yoshimatsu. This work was partly supported by the Sasakawa Scientific Research Grant from The Japan Science Society.

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