# Migration of Gyrotactic Micro-Organisms in Water

^{*}

## Abstract

**:**

_{s}= 146 μm/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. akashiwo 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.

## 1. Introduction

_{s}, the rotational diffusion coefficient D

_{r}, as well as the time scale B for cells’ reorientation by the gravitational torque against the viscous torque caused by flow shear. Several ways have been adopted to obtain these parameters based on microscope-based tracking methods [13,27,28], laser-based tracking methods [29,30], and video-based tracking methods [12,21,31], in which a key point is to capture the trajectories of cells for statistical analysis. The microscope-based tracking method can provide a high spatial resolution for cells’ trajectories, while the laser-based and video-based tracking methods can offer a large field of view which is beneficial to track more trajectories each time. Hill and Häder [13] proposed a biased random walk model to calculate the swimming direction and rotational diffusion coefficient of micro-organisms. Leptos et al. [32] measured the swimming velocity and compared the diffusion of C. reinhardtii and Brownian motion of passive particles. Croze et al. [18] measured the gyrotactic parameters of D. salina in the still water. Recently, Sengupta et al. [26] investigated the time scale for reorientation of different strains of H. akashiwo.

_{r}, and (5) to calculate the mean swimming velocity and translational diffusion tensor in three-dimensional vorticity field.

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{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.

#### 2.2. Image Processing and Analysis Methods

_{x}, V

_{z}) is calculated by

_{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 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 (V

_{x}= V

_{y}), and the resultant speed (V) is defined as

_{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°~180°), respectively, as shown in Figure 4.

_{x}, and V

_{z}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,

_{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 × Δv, (i + 1) × Δv], 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 one-dimensional f(V

_{x}), f(V

_{z}), f(V) and two-dimensional probability density function f(V

_{x}, V

_{z}). f(V, θ) are obtained by use of the same computational method as above.

#### 2.3. Fokker-Planck Model

_{s}(equate to V in Section 2.2) is constant independent of the swimming direction

**p**. The mean swimming velocity

**V**can be computed as [2]

_{c}**p**) is probability density function of swimming direction of H. akashiwo. The cell’s translational diffusivity tensor

**D**is given by [2]

**p**), satisfies the Fokker-Planck (FK) equation:

**p**-space, $\dot{\mathit{p}}$ is the gyrotactic reorientation rate, and D

_{r}is the rotational diffusivity. For spherical cells (e.g., H. akashiwo), $\dot{\mathit{p}}$ is given, for the weak fluid acceleration, by

**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.

_{x}, w

_{y}, w

_{z}), to obtain the relationship of the mean swimming velocity and the translational diffusivity tensor.

## 3. Results

#### 3.1. Swimming Velocity and Direction Distribution

_{x}, V

_{z}locating at 143.57, 0.37, 104.54 µm/s. The f(V

_{x}) is approximately subjected to Gaussian distribution $f\left({V}_{x}\right)=1/\sqrt{2\pi}\sigma \xb7{e}^{-{V}_{x}^{2}/\left(2{\sigma}^{2}\right)}$, where σ = 62.82. The PDFs for V, V

_{x}and V

_{z}at different time are shown in Figure 5b–d, which change little over time (20–50 min).

_{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.

_{s}) are computed with Equations (17)–(19),

_{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.

#### 3.2. Relative Strength of Reorientation by Gravitational Torque to Rotational Diffusion

_{0}is λ/(4πsinhλ).

#### 3.3. Time Scale of Reorientation by Gravitational Torque

_{⊥}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.

_{0}[α(0)], represent turning speed of cells with different α(0), as shown in Figure 13, in which the −85° and −75° respectively represent the population of cells with α(0) at [−90°, −80°] and [−80°, −70°] when T = 0 s. For the population characterized by α(0) = −85°, the cells constantly rotate to adjust the swimming direction until the rotation angle nearly reaching 85°, and at that moment the turning amplitude does not increase with time any more, which means that the swimming directions have been upward and the cells have reached equilibrium. The turning amplitudes are almost constant for the cells α(0) = ±5°, because the initial positions of algal cells are close to vertical direction. The initial instant T = 0 is any moment for the algal cells, thus Figure 13 shows that at any time the cells constantly adjust their swimming direction to reach the equilibrium state.

_{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),

_{0}, a positive constant, is the drift coefficient. Thus we estimate B = d

_{0}

^{−1}= 5.6 s for H. akashiwo in present paper.

#### 3.4. Rotational Diffusion Coefficient

_{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.

_{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.

#### 3.5. Swimming Characteristics of H. akashiwo in Vorticity Field

**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.

_{x}= 0 and w

_{z}= 0, the horizontal swimming velocity and the off-diagonal components of

**D**are equal to zero, and V

_{z}/V

_{s}, D

_{xx}/(V

_{s}

^{2}τ), D

_{yy}/(V

_{s}

^{2}τ), and D

_{zz}/(V

_{s}

^{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.

## 4. Discussion

_{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.

## 5. Conclusions

^{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

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Images of motile cells: (

**a**) algal cells (represented by dark spots) in one frame, (

**b**) trajectories of cells in 30 frames, (

**c**) one track/trajectory with dozens of walks, and three swimming patterns: (

**d1**) helical trajectories, (

**d2**) straight trajectories, and (

**d3**) trajectories combined helical and straight ones.

**Figure 5.**Distribution of probability density of swimming velocity: (

**a**) f (V), f (V

_{x}), and f (V

_{z}) in all tests, (

**b**) f (V

_{z}) at different time, (

**c**) f (V

_{x}) at different time, and (

**d**) f (V) at different time.

**Figure 6.**Distribution of f(V

_{x}, V

_{z}) at different time: (

**a**) 20 min, (

**b**) 25 min, (

**c**) 30 min, (

**d**) 35 min, (

**e**) 40 min, (

**f**) 45 min, (

**g**) 50 min, and (

**h**) 55 min.

**Figure 7.**Statistical parameter of the swimming velocity V, V

_{x}, and V

_{z}and at different time: (

**a**) mean velocity, (

**b**) standard deviations (σ), and (

**c**) deviation coefficient (C

_{s}).

**Figure 9.**Swimming trajectories (100 tracks) of H. akashiwo cells in x-z plane: (

**a**) t = 20 min, and (

**b**) t = 55 min.

**Figure 11.**Distribution of ln[f(V,θ)]: (

**a**) ln[f(V,θ)] versus V and cosθ, (

**b**) ln[f(V,θ)] versus cosθ for given V.

**Figure 12.**λ at different values of V with the red bars corresponding to the values of all cells in tests and the black bars corresponding to the values of cells that only walk upwards.

**Figure 18.**Swimming velocity, (

**a**) V

_{x}/V

_{s}, (

**b**) V

_{y}/V

_{s}, and (

**c**) V

_{z}/V

_{s}with w

_{x}, w

_{y}, and w

_{z}.

**Figure 19.**Translational diffusion tensor, (

**a**) D

_{xx}/V

_{s}

^{2}τ, (

**b**) D

_{yy}/V

_{s}

^{2}τ, (

**c**) D

_{zz}/V

_{s}

^{2}τ, (

**d**) D

_{xy}/V

_{s}

^{2}τ, (

**e**) D

_{xz}/V

_{s}

^{2}τ, and (

**f**) D

_{yz}/V

_{s}

^{2}τ with w

_{x}, w

_{y}, and w

_{z}.

Sets | 20 min | 25 min | 30 min | 35 min | 40 min | 45 min | 50 min | 55 min |
---|---|---|---|---|---|---|---|---|

1 | N_{11} | N_{12} | N_{13} | N_{14} | N_{15} | N_{16} | N_{17} | N_{18} |

2 | N_{21} | N_{22} | N_{23} | N_{24} | N_{25} | N_{26} | N_{27} | N_{28} |

3 | N_{31} | N_{32} | N_{33} | N_{34} | N_{35} | N_{36} | N_{37} | N_{38} |

Sum | N_{1} | N_{2} | N_{3} | N_{4} | N_{5} | N_{6} | N_{7} | N_{8} |

Algae | V (μm s^{−1}) | B (s) | D_{r} (rad^{2} s^{−1}) | Paper |
---|---|---|---|---|

C. nivalis | 63 | 3.4 | 0.067 | [1,2] |

C. nivalis | 55 | 2.7 | 0.084 | [13] |

C. nivalis | 38 | 6 | 0.036 | [30] |

D. salina | 62.7 ± 0.4 | 10.5 ± 1.3 | 0.23 ± 0.06 | [18] |

H. akashiwo | - | 2 | - | [7] |

H. akashiwo (CCMP3107) | - | 4.9 ± 1.5 | - | [26] |

H. akashiwo (CCMP452↑ ^{a}) | 74.5 ± 42.4 | 19.3 ± 13.5 | - | [26] |

H. akashiwo (CCMP452↓ ^{b}) | 73.8 ± 46.2 | −23.1 ± 10.2 | - | [26] |

H. akashiwo (CCMP452) | 49–66 | - | - | [11] |

H. akashiwo (CCAP934-1) | 88–119 | - | - | [11] |

H. akashiwo | 20–145 | - | - | [12] |

H. akashiwo (CCMP3107) | 25–58 | - | - | [23] |

H. akashiwo | 85–135 | - | - | [24] |

H. akashiwo (7 strains) | 33–115 | - | - | [25] |

H. akashiwo (GY-H24) | 117–156 | 5.6 | 0.046 | Present work |

^{a}The population of algae that swim upwards (negative gravitaxis).

^{b}The population of algae that swim downwards (positive gravitaxis).

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**MDPI and ACS Style**

Chen, X.; Wu, Y.; Zeng, L.
Migration of Gyrotactic Micro-Organisms in Water. *Water* **2018**, *10*, 1455.
https://doi.org/10.3390/w10101455

**AMA Style**

Chen X, Wu Y, Zeng L.
Migration of Gyrotactic Micro-Organisms in Water. *Water*. 2018; 10(10):1455.
https://doi.org/10.3390/w10101455

**Chicago/Turabian Style**

Chen, Xiao, Yihong Wu, and Li Zeng.
2018. "Migration of Gyrotactic Micro-Organisms in Water" *Water* 10, no. 10: 1455.
https://doi.org/10.3390/w10101455