Migration of Gyrotactic Micro-Organisms in Water

: Understanding the swimming characteristics of micro-organisms is signiﬁcant for modelling the migration of motile cells and corresponding ecological risk assessments associated with harmful algae in oceans and estuaries. Presented in this paper is an experimental and numerical investigation of swimming characteristics of a typical gyrotactic micro-organism, Heterosigma akashiwo ( H. akashiwo ) in water, based on the technology of planar laser-induced ﬂuorescence and the ﬁnite volume method. Two-dimensional swimming velocity of algal cells are obtained by analyzing cells’ trajectories in the vertical plane, and three-dimensional swimming velocity is reconstructed based on the assumption that cells’ swimming is isotropic in the horizontal plane. Four important parameters are given to reﬂect the swimming characteristics of gyrotactic cells in still water, including the mean swimming speed ( V s = 146 µ m/s), the relative strength of reorientation by gravitational torque to rotational diffusion ( λ = 1.96), the time scale of reorientation ( B = 5.6 s), and rotational diffusivity ( D r = 0.046 rad 2 /s). A database of the ambient vorticity, mean swimming velocity and diffusivity tensor is established, by solving Fokker-Planck equation for the probability density function of cells’ swimming under the combined action of gravity, rotational diffusion, and the ambient vorticity. The mean swimming velocity and translational diffusion tensor of H. akashiw o are found to change with the horizontal and vertical vorticity. It is also shown that gyrotactic cells swim in a given direction for a weak horizontal vorticity, in contrast to cells’ tumbling and being trapped for a strong horizontal vorticity. all the performed a to

Intelligent Laser Applications of Germany, Aachen, Germany) with a resolution of 2048 × 2048 pixels, a lens (EF-S 60 mm f/2.8 USM, Canon, Tokyo, Japan), a 550 nm high-pass filter (Beijing JJAM Technology Limited corporation, Beijing, China), and an imaging analysis software (CamWare, Intelligent Laser Applications of Germany, Aachen, Germany). The injection system included a syringe pump (Pump 11 elite, Harvard Apparatus, Holliston, MA, USA) and injection syringes (Himilton, Reno, NV, USA) with various capacities, which was connected with the rubber tube to provide fluid for the test tube.
Water 2018, 10, x FOR PEER REVIEW 3 of 19 a lens (EF-S 60 mm f/2.8 USM, Canon, Tokyo, Japan), a 550 nm high-pass filter (Beijing JJAM Technology Limited corporation, Beijing, China), and an imaging analysis software (CamWare, Intelligent Laser Applications of Germany, Aachen, Germany). The injection system included a syringe pump (Pump 11 elite, Harvard Apparatus, Holliston, MA, USA) and injection syringes (Himilton, Reno, NV, USA) with various capacities, which was connected with the rubber tube to provide fluid for the test tube. The swimming speed of H. akashiwo measured in the present paper was larger than that measured by Hill and Häder [13], and the test tube in the present work was extended in the length and width compared with that of Vladimirov et al. [30] to suit the faster cells. The measurement area (14.23 mm × 14.23 mm) was located 140 mm away from the bottom of the test tube. The design of the test tube considered the swimming speed of the motile cells and the injection time to weaken the influence of the injection on the fluid in the test tube.
The trajectories of H. akashiwo could be recorded by camera due to its fluorescent characteristics under the illumination of a laser with an appropriate wavelength. One preliminary work in the present paper was to measure the fluorescence spectrum of H. akashiwo using a fluorescence spectrophotometer (F-7000 FL, HITACHI, Tokyo, Japan). The cells were stimulated by the laser with various wavelengths, and the optimum wavelength that made the fluorescent intensity of cells reach its maximum was selected from 430-470 nm, as shown in Figure 2. The horizontal and vertical axes respectively represent the wavelength and the logarithm of the normalized light intensity. The left five peaks were from the excitation light, and the peaks that almost coincide in the right were from the emission light by the algae. The enlarged peaks in the inset show that under the illumination of 450 nm laser, the emission of H. akashiwo has the maximum peak, and the peak of fluorescence spectrum locates at 685 nm. Thus, a laser with a wavelength 447 nm and 550 nm high-pass filter were selected in the present work, and all the experiments in present paper were performed in a dark environment to eliminate interference from other lights. The swimming speed of H. akashiwo measured in the present paper was larger than that measured by Hill and Häder [13], and the test tube in the present work was extended in the length and width compared with that of Vladimirov et al. [30] to suit the faster cells. The measurement area (14.23 mm × 14.23 mm) was located 140 mm away from the bottom of the test tube. The design of the test tube considered the swimming speed of the motile cells and the injection time to weaken the influence of the injection on the fluid in the test tube.
The trajectories of H. akashiwo could be recorded by camera due to its fluorescent characteristics under the illumination of a laser with an appropriate wavelength. One preliminary work in the present paper was to measure the fluorescence spectrum of H. akashiwo using a fluorescence spectrophotometer (F-7000 FL, HITACHI, Tokyo, Japan). The cells were stimulated by the laser with various wavelengths, and the optimum wavelength that made the fluorescent intensity of cells reach its maximum was selected from 430-470 nm, as shown in Figure 2. The horizontal and vertical axes respectively represent the wavelength and the logarithm of the normalized light intensity. The left five peaks were from the excitation light, and the peaks that almost coincide in the right were from the emission light by the algae. The enlarged peaks in the inset show that under the illumination of 450 nm laser, the emission of H. akashiwo has the maximum peak, and the peak of fluorescence spectrum locates at 685 nm. Thus, a laser with a wavelength 447 nm and 550 nm high-pass filter were selected in the present work, and all the experiments in present paper were performed in a dark environment to eliminate interference from other lights. The experimental materials included algal cells (H. akashiwo, Shanghai Guangyu Biological Technology Co., Ltd., Shanghai, China) and f/2 medium. The algal cells were incubated at 25 °C in f/2 medium, and the daily cycle was 12 h of white light (3000 lux) and 12 h in the dark. The cells used for experiments were taken from the upper layer of suspensions which were vibrated for 5 min and stood for 30 min. Cell suspensions were prepared by diluting algal culture with f/2 media to attain a cell concentration of ~5 × 10 4 cells/mL. All the experiments in this work were carried out during the first half of the algae's daytime to keep the same swimming characteristics. The tests were performed according to the following procedures. Firstly, the acrylic box and the test tube were filled with water and the filtered f/2 medium at the same temperature, respectively. After 1 h, the fluid in the tube was completely still. Then suspensions of cells (3.5 mL) were continuously injected into the tube by a discharge of 1.5 μL/s, which ensured mean rising velocity of the water surface in the test tube was no more than 7 μm/s. The algal cells arrived at the bottom of the test tube when the injection was finished, and we marked this moment as t = 0 min. The number of motile cells in the measured zone at t = 20 min is adequate to get reasonable statistics, and therefore, the first set of images (30 successive frames) were recorded at t = 20 min, with a 50 ms exposure time. Then images were taken every 5 min for an hour with the camera and laser keeping the same setting. The above operation was one test, and three tests were performed over the whole experiment.

Image Processing and Analysis Methods
A sample of individual cells was recorded in each frame with a spatial resolution 6.95 μm/pixel, as shown in Figure 3a. The trajectories of cells shown in Figure 3b were obtained by overlaying 30 successive images. Each trajectory, containing several walks, is defined as a track, as shown in Figure  3c. In the present work, 13,684 tracks in 3 tests were collected, and 6292 tracks that contain at least 10 walks were selected to calculate the swimming velocity and direction. The experimental materials included algal cells (H. akashiwo, Shanghai Guangyu Biological Technology Co., Ltd., Shanghai, China) and f/2 medium. The algal cells were incubated at 25 • C in f/2 medium, and the daily cycle was 12 h of white light (3000 lux) and 12 h in the dark. The cells used for experiments were taken from the upper layer of suspensions which were vibrated for 5 min and stood for 30 min. Cell suspensions were prepared by diluting algal culture with f/2 media to attain a cell concentration of~5 × 10 4 cells/mL. All the experiments in this work were carried out during the first half of the algae's daytime to keep the same swimming characteristics. The tests were performed according to the following procedures. Firstly, the acrylic box and the test tube were filled with water and the filtered f/2 medium at the same temperature, respectively. After 1 h, the fluid in the tube was completely still. Then suspensions of cells (3.5 mL) were continuously injected into the tube by a discharge of 1.5 µL/s, which ensured mean rising velocity of the water surface in the test tube was no more than 7 µm/s. The algal cells arrived at the bottom of the test tube when the injection was finished, and we marked this moment as t = 0 min. The number of motile cells in the measured zone at t = 20 min is adequate to get reasonable statistics, and therefore, the first set of images (30 successive frames) were recorded at t = 20 min, with a 50 ms exposure time. Then images were taken every 5 min for an hour with the camera and laser keeping the same setting. The above operation was one test, and three tests were performed over the whole experiment.

Image Processing and Analysis Methods
A sample of individual cells was recorded in each frame with a spatial resolution 6.95 µm/pixel, as shown in Figure 3a. The trajectories of cells shown in Figure 3b were obtained by overlaying 30 successive images. Each trajectory, containing several walks, is defined as a track, as shown in Figure 3c. In the present work, 13,684 tracks in 3 tests were collected, and 6292 tracks that contain at least 10 walks were selected to calculate the swimming velocity and direction.
Consider a Cartesian coordinate system with x-axis parallel to the laser sheet, y-axis perpendicular to the plane of laser sheet, and z-axis vertical upward along the axis of the test tube. The swimming velocity of cell (V x , V z ) is calculated by where (x k , z k ) and (x k−1 , z k−1 ) are the pixel positions of algal cells on the k-th frame and k−1-th frame, K (6.95 µm/pixel) is the conversion coefficient between pixel and physical coordinates, and ∆t (0.5 s) is the time interval between two continuous images. We assume that the swimming velocity of cells in the x-y plane is isotropic (V x = V y ), and the resultant speed (V) is defined as where V r is the total swimming velocity in the x-y plane, and the r-axis is determined by (V x , V y , 0). The swimming direction of motile cells in r-z plane and x-z plane are defined as θ (0~180 • ) and α (−180 •~1 80 • ), respectively, as shown in Figure 4.
Water 2018, 10, x FOR PEER REVIEW 5 of 19 Consider a Cartesian coordinate system with x-axis parallel to the laser sheet, y-axis perpendicular to the plane of laser sheet, and z-axis vertical upward along the axis of the test tube. The swimming velocity of cell (Vx, Vz) is calculated by where (xk, zk) and (xk−1, zk−1) are the pixel positions of algal cells on the k-th frame and k−1-th frame, K (6.95 μm/pixel) is the conversion coefficient between pixel and physical coordinates, and Δt (0.5 s) is the time interval between two continuous images. We assume that the swimming velocity of cells in the x-y plane is isotropic (Vx = Vy), and the resultant speed (V) is defined as V, V x , and V z calculated by Equations (1)-(3) are divided into square bins of width ∆v = 10 µm/s to obtain probability density function (PDF) of swimming velocity, where k (the value is 10) is the number of walks in one track, N 0 is the number of tracks in Table 1, n i is the number of walks in the i-th bin for velocity ranging from [i × ∆v, (i + 1) × ∆v], and n ij is the number of walks that V x and V z located in the i-th and j-th bins. Discrete values of f i and f ij constitute the are obtained by use of the same computational method as above.
Water 2018, 10, x FOR PEER REVIEW 6 of 19 V, Vx, and Vz calculated by Equations (1)-(3) are divided into square bins of width Δv = 10 μm/s to obtain probability density function (PDF) of swimming velocity, where k (the value is 10) is the number of walks in one track, N0 is the number of tracks in Table 1, ni is the number of walks in the i-th bin for velocity ranging from [i × Δv, (i + 1) × Δv], and nij is the number of walks that Vx and Vz located in the i-th and j-th bins. Discrete values of fi and fij constitute the one-dimensional f(Vx), f(Vz), f(V) and two-dimensional probability density function f(Vx, Vz). f(V, θ) are obtained by use of the same computational method as above.

Fokker-Planck Model
We assume that the swimming speed of cells, Vs (equate to V in Section 2.2) is constant independent of the swimming direction p. The mean swimming velocity Vc can be computed as [2] = 〈 〉 = 〈 〉〈 〉 where f(p) is probability density function of swimming direction of H. akashiwo. The cell's translational diffusivity tensor D is given by [ where τ is the correlation time. In this work, the translational diffusion coefficient is computed based on the Equation (14). Here, the correlation time, for simplicity, is assumed to be constant following the work [2]. However, the constant correlation time is not a good approximation for all flow

Fokker-Planck Model
We assume that the swimming speed of cells, V s (equate to V in Section 2.2) is constant independent of the swimming direction p. The mean swimming velocity V c can be computed as [2] V c = V s p = V s p (12) · ≡ · f (p)d 2 p where f (p) is probability density function of swimming direction of H. akashiwo. The cell's translational diffusivity tensor D is given by [2] where τ is the correlation time. In this work, the translational diffusion coefficient is computed based on the Equation (14). Here, the correlation time, for simplicity, is assumed to be constant following the work [2]. However, the constant correlation time is not a good approximation for all flow conditions, especially for the flow with a high shear rate. A better approximation is given by 'generalised Taylor dispersion theory' [43][44][45][46][47]. Hwang and Pedley [3] gave detailed comments on the Equation (14) and the generalised Taylor dispersion theory. The probability density function of swimming direction, f (p), satisfies the Fokker-Planck (FK) equation: where is the gradient operator in p-space, . p is the gyrotactic reorientation rate, and D r is the rotational diffusivity. For spherical cells (e.g., H. akashiwo), . p is given, for the weak fluid acceleration, by where k is the unit vector in the vertically direction, B is the reorientation time of gyrotactic H. akashiwo, and ω is the vorticity of ambient flow. Equation (15) is solved based on the finite volume method [14]. The initial uniform distribution of probability density (1/4π) is specified, and the periodical boundary condition f (θ, 0, t) = f (θ, 2π, t), is specified. In the present paper, the vorticity is inputted as a parameter of Equation (15). Equation (15), with the same initial and boundary conditions, is solved for a set of vorticities (w x , w y , w z ), to obtain the relationship of the mean swimming velocity and the translational diffusivity tensor.

Swimming Velocity and Direction Distribution
The one-dimensional distribution of probability density function of swimming velocity in all tests is shown in Figure 5a with the average value of V, V x , V z locating at 143.57, 0.37, 104.54 µm/s. The probability density function of swimming direction, f(p), satisfies the Fokker-Planck (FK) equation: where ▽ is the gradient operator in p-space, ṗ is the gyrotactic reorientation rate, and Dr is the rotational diffusivity. For spherical cells (e.g., H. akashiwo), ṗ is given, for the weak fluid acceleration, by where k is the unit vector in the vertically direction, B is the reorientation time of gyrotactic H. akashiwo, and ω is the vorticity of ambient flow. Equation (15) is solved based on the finite volume method [14]. The initial uniform distribution of probability density (1/4 ) is specified, and the periodical boundary condition ( , 0, ) = ( , 2 , ), is specified. In the present paper, the vorticity is inputted as a parameter of Equation (15).
Equation (15), with the same initial and boundary conditions, is solved for a set of vorticities (wx, wy, wz), to obtain the relationship of the mean swimming velocity and the translational diffusivity tensor.

Swimming Velocity and Direction Distribution
The one-dimensional distribution of probability density function of swimming velocity in all tests is shown in Figure 5a   The distributions of f(Vx, Vz) in the x-z plane at 20-55 min are shown in Figure 6, in which the areas with high probability density are similar to the shape of mushrooms. The value of Vx is mainly within [−80, 80] μm/s with the symmetric axis around Vx = 0 μm/s, while Vz is basically above 80 μm/s.
The distributions of f (V x , V z ) in the x-z plane at 20-55 min are shown in Figure 6, in which the areas with high probability density are similar to the shape of mushrooms. The value of V x is mainly within [−80, 80] µm/s with the symmetric axis around V x = 0 µm/s, while V z is basically above 80 µm/s. It is obvious that the spatial distribution is less clustered at t = 55 min with the accumulated area surrounded by red, dashed line extending in the z-direction and narrowing in the x-direction, as shown in Figure 6a,h. It is obvious that the spatial distribution is less clustered at t = 55 min with the accumulated area surrounded by red, dashed line extending in the z-direction and narrowing in the x-direction, as shown in Figure 6a,h. The mean swimming velocities, standard deviations (σ), and deviation coefficient (Cs) are computed with Equations (17)-(19), where k (the value is 10) is the number of walks in one track, U is the variable to represent V, Vx, or Vz, and Ni is the number of tracks in the last line in Table 1. Figure 7a presents the mean swimming velocities at three tests, and the dashed lines in Figure 7a represent the mean swimming velocity for all the tests. The values of V and Vz decrease with time to some extent, while the value of Vx fluctuates around Vx = 0 μm/s. The variation of standard deviations of Vx and V are weak compared with that of Vz, as shown in Figure 7b. The deviation coefficients do not change greatly with time, as shown in Figure 7c, which indicates that the distribution patterns remain stable. The results illustrate that the swimming characteristics of cells in the x-y plane is time-independent, the differences of PDFs for Vz increase slightly with time due to the smaller Vz of the cells coming later, and the swimming ability changes a little in tests. The mean swimming velocities, standard deviations (σ), and deviation coefficient (C s ) are computed with Equations (17)-(19), where k (the value is 10) is the number of walks in one track, U is the variable to represent V, V x , or V z , and N i is the number of tracks in the last line in Table 1. Figure 7a presents the mean swimming velocities at three tests, and the dashed lines in Figure 7a represent the mean swimming velocity for all the tests. The values of V and V z decrease with time to some extent, while the value of V x fluctuates around V x = 0 µm/s. The variation of standard deviations of V x and V are weak compared with that of V z , as shown in Figure 7b. The deviation coefficients do not change greatly with time, as shown in Figure 7c, which indicates that the distribution patterns remain stable. The results illustrate that the swimming characteristics of cells in the x-y plane is time-independent, the differences of PDFs for V z increase slightly with time due to the smaller V z of the cells coming later, and the swimming ability changes a little in tests. The PDF of swimming direction in r-z plane is shown in Figure 8a. The f (θ) increases with the decreasing θ to reach the maximum value at θ = 0 • , i.e., the vertical direction (z-axis). Figure 8b presents more intuitively the variation of f (θ) with θ after injection, which changes little over time. The f (θ) rises obviously when 0 • < θ < 90 • , while keeps stable around 0.001 in the case that θ > 90 • . The results show that the H. akashiwo tends to swim upwards, and this conclusion is also verified by the trajectories of 100 cells within 5 s at t = 20 min and 55 min, as shown in Figure 9. The origin of each trajectory in Figure 9 has been moved to the same point to compare. Both the gravitaxis and randomness can be observed in Figure 9. It is noted that a number of downward trajectories have been observed at t = 55 min, and corresponding reasons are not clear. The PDF of swimming direction in r-z plane is shown in Figure 8a. The f(θ) increases with the decreasing θ to reach the maximum value at θ = 0°, i.e., the vertical direction (z-axis). Figure 8b presents more intuitively the variation of f(θ) with θ after injection, which changes little over time. The f(θ) rises obviously when 0° < θ < 90°, while keeps stable around 0.001 in the case that θ > 90°. The results show that the H. akashiwo tends to swim upwards, and this conclusion is also verified by the trajectories of 100 cells within 5 s at t = 20 min and 55 min, as shown in Figure 9. The origin of each trajectory in Figure 9 has been moved to the same point to compare. Both the gravitaxis and randomness can be observed in Figure 9. It is noted that a number of downward trajectories have been observed at t = 55 min, and corresponding reasons are not clear.   The PDF of swimming direction in r-z plane is shown in Figure 8a. The f(θ) increases with the decreasing θ to reach the maximum value at θ = 0°, i.e., the vertical direction (z-axis). Figure 8b presents more intuitively the variation of f(θ) with θ after injection, which changes little over time. The f(θ) rises obviously when 0° < θ < 90°, while keeps stable around 0.001 in the case that θ > 90°. The results show that the H. akashiwo tends to swim upwards, and this conclusion is also verified by the trajectories of 100 cells within 5 s at t = 20 min and 55 min, as shown in Figure 9. The origin of each trajectory in Figure 9 has been moved to the same point to compare. Both the gravitaxis and randomness can be observed in Figure 9. It is noted that a number of downward trajectories have been observed at t = 55 min, and corresponding reasons are not clear.

Relative Strength of Reorientation by Gravitational Torque to Rotational Diffusion
According to the theoretical analysis of gyrotactic micro-organisms given by Pedley and Kessler [2], f(V,θ) and ln[f(V,θ)] are subjected to

Relative Strength of Reorientation by Gravitational Torque to Rotational Diffusion
According to the theoretical analysis of gyrotactic micro-organisms given by Pedley and Kessler [2], f (V,θ) and ln[f (V,θ)] are subjected to f = µ 1 e λ cos θ (20) ln f = λ cos θ + ln µ 1 (21) where λ represents the relative relationship between the random diffusion and the preferred migration, and µ 0 is λ/(4πsinhλ).
The f (V,θ) of motile cells shows that the cells with V = 80-180 µm/s and θ = 0 • -60 • occupy the majority in all tests, as shown in Figure 10. The peaks of f (V,θ) drop to smaller values in the V-axis, and the small peak appearing in Figure 10d represents the slower cells that coming later. The distribution of ln[f (V,θ)] at different swimming velocity and swimming direction is shown in Figure 11a, and according to Equation (21) Figure 11b. It is shown that λ = 1.44-2.60, and that the average value of λ = 1.96. We also found that the linear relation is more obvious for the case of cosθ > 0, in which λ = 2.53-3.98 with the average value, 3.44, as shown in Figure 12. In addition, λ can be calculated from another equation presented by Pedley and Kessler [2]: where < > represents the ensemble average. The value of λ is 3.64 based on Equation (22).

Time Scale of Reorientation by Gravitational Torque
The swimming direction of H. akashiwo is reoriented due to the torque caused by the displacement of the center of mass relative to the geometric center [1]. This intrinsic feature, resulted from bottom heaviness, can be described by the time scale of reorientation, B, which is determined by [48], where µ is the dynamic fluid viscosity, ρ is the density of cells, g is the gravitational acceleration, α ⊥ is a dimensionless constant, and h is the displacement of the center of mass relative to the geometric center. Kessler [34] proposed the range of h is 0-0.1a and estimated h = 0.1 µm, B = 3.4 s for C. nivalis, where a is the average radius of cells. It is not easy to estimate B according to Equation (24) is not so convenient to be used for other kinds of algal cells. Hill and Häder [13] built a biased random walk model to estimate the parameters of algal cells, and proposed B = 2.7 s for C. nivalis based on the swimming trajectories obtained by microscope-based tracking methods. For the experimental results in the present paper, the swimming directions of cells are subjected to where α(0) is the initial swimming angle at T = 0 s, which corresponds to t = 20 min in all tests, and α(0) = 0 represents that the cells swim vertically upward. In one shot, 30 successive frames are of algal cells, and proposed B = 2.7 s for C. nivalis based on the swimming trajectories obtained by microscope-based tracking methods. For the experimental results in the present paper, the swimming directions of cells are subjected to where α(0) is the initial swimming angle at T = 0 s, which corresponds to t = 20 min in all tests, and α(0) = 0 represents that the cells swim vertically upward. In one shot, 30   The slopes in Figure 13, µ0[α(0)], are proportional to [α(0)], as shown in Figure 14. The curve in Figure 14 shows that the more algal cells deviate from their equilibrium position, the greater the turning speed. The function µ0(α) represents the turning speed of motile cells from the initial position to the position of α = 0, which is subjected to Equation The slopes in Figure 13, µ 0 [α(0)], are proportional to [α(0)], as shown in Figure 14. The curve in Figure 14 shows that the more algal cells deviate from their equilibrium position, the greater the turning speed. The function µ 0 (α) represents the turning speed of motile cells from the initial position to the position of α = 0, which is subjected to Equation (26), where d 0 , a positive constant, is the drift coefficient. Thus we estimate B = d 0 −1 = 5.6 s for H. akashiwo in present paper.
Water 2018, 10, x FOR PEER REVIEW  13 of 19 where d0, a positive constant, is the drift coefficient. Thus we estimate B = d0 −1 = 5.6 s for H. akashiwo in present paper.

Rotational Diffusion Coefficient
The variance of the turning amplitude, Var[α(t)−α(0)], is proportional to time, 2 Figure 14. Turning speed of cells at different initial swimming angles.

Rotational Diffusion Coefficient
The variance of the turning amplitude, Var[α(t)−α(0)], is proportional to time, where σ 0 2 [α(0)] is the variance of turning speed, and the relationship is shown in Figure 15. The slopes for various initial angles, ranging from 0.023-0.156, are depicted in Figure 16. Thus, the rotational diffusion coefficient, D r , which equals to σ 0 2 /2, varies in the scope of 0.012-0.078.

Rotational Diffusion Coefficient
The variance of the turning amplitude, Var[α(t)−α(0)], is proportional to time, where σ0 2 [α(0)] is the variance of turning speed, and the relationship is shown in Figure 15. The slopes for various initial angles, ranging from 0.023-0.156, are depicted in Figure 16. Thus, the rotational diffusion coefficient, Dr, which equals to σ0 2 /2, varies in the scope of 0.012-0.078.
According to Equation (28), Dr = 1/(2Bλ) = 0.034-0.062 rad 2 /s with a mean value 0.046 rad 2 /s for all cells, and Dr = 0.022-0.035 rad 2 /s with a mean value 0.026 rad 2 /s for the cells only walking vertically upwards, and Dr = 0.025 rad 2 /s if λ is calculated by Equation (22). The values of Dr calculated by Equation (28) are consistent with that calculated according to Figure 16.

Swimming Characteristics of H. akashiwo in Vorticity Field
The swimming velocity V and translational diffusivity tensor D of H. akashiwo were found to change with the horizontal vorticity. For the case of wx = 0 and wz = 0, the longitudinal swimming velocity Vx increased firstly to a maximum and decreased to zero with the increase of wy, the vertical swimming velocity Vz decreased with the increasing wy, and the lateral velocity Vy kept constant, as shown in Figure 17a. When |Bwy| > 0.9, Vx is greater than Vz. For the weak horizontal vorticity, algal cells can swim with an angle against the vertical direction. However, for the strong vorticity, the cells cannot swim upwards in a fixed angle, as shown in Figure 17a, because the maximum gravitational torque due to the difference of cells' buoyance center and mass center cannot balance the viscous The rotational diffusivity can also be calculated by the equation given by Pedley and Kessler [1] According to Equation (28), D r = 1/(2Bλ) = 0.034-0.062 rad 2 /s with a mean value 0.046 rad 2 /s for all cells, and D r = 0.022-0.035 rad 2 /s with a mean value 0.026 rad 2 /s for the cells only walking vertically upwards, and D r = 0.025 rad 2 /s if λ is calculated by Equation (22). The values of D r calculated by Equation (28) are consistent with that calculated according to Figure 16.

Swimming Characteristics of H. akashiwo in Vorticity Field
The swimming velocity V and translational diffusivity tensor D of H. akashiwo were found to change with the horizontal vorticity. For the case of w x = 0 and w z = 0, the longitudinal swimming velocity V x increased firstly to a maximum and decreased to zero with the increase of w y , the vertical swimming velocity V z decreased with the increasing w y , and the lateral velocity V y kept constant, as shown in Figure 17a. When |Bw y | > 0.9, V x is greater than V z . For the weak horizontal vorticity, algal cells can swim with an angle against the vertical direction. However, for the strong vorticity, the cells cannot swim upwards in a fixed angle, as shown in Figure 17a, because the maximum gravitational torque due to the difference of cells' buoyance center and mass center cannot balance the viscous torque caused by the flow shear [7]. Figure 17b presents the variation of translational diffusivity with lateral vorticity. It is shown that the diagonal components of translational diffusivity are much greater than the off-diagonal components. For the weak horizontal vorticity, the translational diffusion are found to be anisotropic. With the increase of horizontal vorticity, the anisotropic translational diffusion tends rapidly to the isotropic translational diffusion with the diagonal components equal to 1/3. upwards, and Dr = 0.025 rad 2 /s if λ is calculated by Equation (22). The values of Dr calculated by Equation (28) are consistent with that calculated according to Figure 16.

Swimming Characteristics of H. akashiwo in Vorticity Field
The swimming velocity V and translational diffusivity tensor D of H. akashiwo were found to change with the horizontal vorticity. For the case of wx = 0 and wz = 0, the longitudinal swimming velocity Vx increased firstly to a maximum and decreased to zero with the increase of wy, the vertical swimming velocity Vz decreased with the increasing wy, and the lateral velocity Vy kept constant, as shown in Figure 17a. When |Bwy| > 0.9, Vx is greater than Vz. For the weak horizontal vorticity, algal cells can swim with an angle against the vertical direction. However, for the strong vorticity, the cells cannot swim upwards in a fixed angle, as shown in Figure 17a, because the maximum gravitational torque due to the difference of cells' buoyance center and mass center cannot balance the viscous torque caused by the flow shear [7]. Figure 17b presents the variation of translational diffusivity with lateral vorticity. It is shown that the diagonal components of translational diffusivity are much greater than the off-diagonal components. For the weak horizontal vorticity, the translational diffusion are found to be anisotropic. With the increase of horizontal vorticity, the anisotropic translational diffusion tends rapidly to the isotropic translational diffusion with the diagonal components equal to 1/3.   For the case of wx = 0 and wz = 0, the horizontal swimming velocity and the off-diagonal components of D are equal to zero, and Vz/Vs, Dxx/(Vs 2 τ), Dyy/(Vs 2 τ), and Dzz/(Vs 2 τ), are equal to 0.71, 0.21, 0.21, and 0.08, respectively, which means that the single vertical vorticity cannot change the swimming direction and translational diffusion.
In the weak horizontal vorticity, wz slightly weakens the horizontal swimming velocity and components of D, while the strong horizontal vorticity enhances the gyrotaxis, as shown in Figures  18 and 19.

Discussion
Some swimming patterns of gyrotactic micro-organisms found in previous studies [11,12,49,50] have also been observed in this work, such as helical trajectories, straight trajectories, and the combination of helical and straight trajectories, as shown in Figure 3d. In the present work, the Equation (3) for computing three-dimensional velocity of cells was based on the assumption that the swimming velocity of cells in x-and y-directions is isotropic, which has been testified by previous studies for gyrotactic micro-organisms [17].
The swimming velocity of H. akashiwo, to some extents, depends on external environments and its strains, as shown in Table 2. Harvey and Menden-Deuer [12] found that the swimming speed of H. akashiwo increased by 38% in the presence of a predator, while Strom et al. [23] found that the presence of a predator resulted in an increase of 22% and a decrease of 58% of swimming speed below

Discussion
Some swimming patterns of gyrotactic micro-organisms found in previous studies [11,12,49,50] have also been observed in this work, such as helical trajectories, straight trajectories, and the combination of helical and straight trajectories, as shown in Figure 3d. In the present work, the Equation (3) for computing three-dimensional velocity of cells was based on the assumption that the swimming velocity of cells in xand y-directions is isotropic, which has been testified by previous studies for gyrotactic micro-organisms [17].
The swimming velocity of H. akashiwo, to some extents, depends on external environments and its strains, as shown in Table 2. Harvey and Menden-Deuer [12] found that the swimming speed of H. akashiwo increased by 38% in the presence of a predator, while Strom et al. [23] found that the presence of a predator resulted in an increase of 22% and a decrease of 58% of swimming speed below and above halocline, respectively. Bearon et al. [11] noted that the gross swimming speeds of H. akashiwo in light phase are 49-66 µm/s and 88-119 µm/s for strain CCMP452 (Provasoli Guillard Center for Culture of Marine Phytoplankton, Maine, USA) and CCAP934-1(Culture Collection of Algae and Protozoa, North Sea, Norway), respectively. Havey et al. [25] also found similar variations of swimming velocity and turning rate due to strains. Recently, Sengupta et al. [26] revealed that different strains of H. akashiwo cells have different responses to turbulence cues. The algal cells in present work are originated from Zhoushan, China, where H. akashiwo frequently blooms, and the swimming velocity (measured at the 2nd-4th hour in the light phase) is relatively high compared with previous results. Parameters of various cells from previous studies and present work are listed in Table 2.
The effect of laser on the swimming direction of cells is studied by analyzing the swimming velocity in different frames; 1935 cells, walking through all the frames, were selected from all tests. The average values of V z are 100.81, 105.15, and 104.24 µm/s for 1-10, 10-20, 20-30 frames, respectively, and the average values of V are 128.97, 132.55, 134.11 µm/s for 1-10, 10-20, 20-30 frames, respectively. V z and V have a small fluctuation about 6 µm/s within 30 frames, much less than the mean swimming speed, which means that the effect of laser on the swimming direction of motile cells are negligible.
The swimming velocity of H. akashiwo probably depends on various factors, such as species, temperature, illumination, culture media, etc. With regard to the difference from previous works, some probable reasons we think come from the tiny difference of algal species, temperature of culture media, and the diel light cycle. The cells in the work [26] are cultivated at 21 • C or 18 • C under the diel light cycle (14 h light: 10 h dark), while the cells are the present experiment are cultivated at 25 • C under the diel light cycle (12 h light: 12 h dark). Furthermore, the cells (GY-H24, H. akshiwo, from Zhoushan, China, provided by Shanghai Guangyu Biological Technology Co., Ltd., Shanghai, China) used in the present work differ from those in the works [11,[23][24][25][26]. The range of swimming velocity of H. akshiwo ranges from 20 to 156 µm/s [11,12,[23][24][25][26], and the present velocity is closed to these results in References [12,24,26]. Besides, the present reorientation time of H. akshiwo is close to the results found in Reference [26]. The vertical migration of gyrotactic cells depends on the gyrotactic parameters of algal cells, hydrodynamic characteristics, and other external conditions, such as light, nutrients, temperature, etc. Mashayekhpour et al. [41] proposed that the high-gyrotaxis swimmers may reach the free water surface to form a high-concentration area, but the thermal stratification in lakes and oceans may disrupt cells' motility and hinder corresponding accumulation [37]. Strong horizontal vorticity can make cells tumble [7], resulting in some interesting ecological phenomena, for example thin phytoplankton layer. In this study, it was shown that the vertical vorticity can weaken the effect of the strong horizontal vorticity on the vertical swimming velocity, as shown in Figure 19, which means that the gyrotactic swimmers can swim more easily towards the free surface in the flow with the strong vertical vorticity than in the flow with the weak vertical vorticity. The circulation with the strong vertical vorticity exists in the realistic flows due to topography and external factors, such as wind. However, further investigations needs to be performed to understand the effect of the combined actions of horizontal and vertical vorticity on the distribution of microorganisms in a specific flow.

Conclusions
The swimming characteristics of H. akashiwo in still water and a three-dimensional vorticity field were investigated based on the laser-based tracking method and the finite volume method. The results showed that the peak of fluorescence spectrum of H. akashiwo locates at 685 nm under the illumination of 450 nm laser. The swimming ability of H. akashiwo cells is independent of time for a given period about 50 min, and most motile cells swim vertically upwards due to the negative gravitaxis, though several cells walk down due to randomness. For all motile cells, the dimensionless parameter λ, which represents the relative strength of reorientation by gravitational torque to rotational diffusion, is in the range of 1.44-2.60, while for the cells only walking upward, corresponding range is 2.53-3.98. The reorientation time and rotational diffusivity are 5.6 s and 0.046 rad 2 /s, respectively. The swimming velocity and translational diffusion tensor of H. akashiwo change with the horizontal and vertical vorticity. Algal cells are able to keep the swimming direction in the weak horizontal vorticity, while they cannot swim towards a given direction in the strong horizontal vorticity. In the presence of horizontal vorticity, the effect of vertical vorticity enhances with the increase of w x and w y , though a single vertical vorticity would not change the swimming direction and diffusion characteristics.
Author Contributions: X.C. and Y.W. were responsible for experiments of cell swimming, X.C. and L.Z. performed and analyzed numerical simulations, X.C. and L.Z. wrote the paper.