# Hydrodynamics of Prey Capture and Transportation in Choanoflagellates

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Computational Fluid Dynamics

#### 2.1.1. Placement of Fixed Prey

#### 2.1.2. Governing Equations and Hydrodynamic Effects

**u**and p denote the flow velocity and pressure, respectively. The time independency allows for the flow to be resolved in a series of discrete time steps, as the response of the fluid to the motion of boundaries can be considered instantaneous [13]. One flagellum beat cycle is discretized into 25 time steps. Doubling the number of time steps introduces a variation of less than 1% in the results [5]. Although it suffices to solve the Stokes equations, the full Navier–Stokes equations are solved at each discrete point in time by the CFD software. In combination with using mesh-morphing techniques, this allows for the flagellum position to be updated in each time step in a computationally efficient way. Given the symmetry of prey placed in the same plane but on opposing sides of the collar, it is only necessary to model half of a flagellum beat cycle. However, to ensure that mesh morphing does not have a significant impact on the symmetry at hand, we still choose to simulate the entire beat cycle.

**F**, and torque,

**L**, acting on the fixed prey are obtained by integrating the stress tensor, $\mathit{\sigma}$, over the prey surface [13], S:

#### 2.1.3. Solution Procedure to Model Drifting Prey

## 3. Results and Discussion

#### 3.1. Static Analysis

#### 3.1.1. Prey Positions on the Collar Filter

#### 3.1.2. Prey Retention

#### 3.1.3. Loricate Effect

#### 3.1.4. Prey Size and Shape

#### 3.2. Dynamic Analysis

#### Freely Drifting Prey

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Morphology of Diaphanoeca grandis. (

**A**) Microscopic image of the choanoflagellate. Scale bar: $4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$. (

**B**) Model morphology picturing the cell (green), collar filter (red), flagellum (yellow), and lorica dome and chimney (blue) with ribs (gray) in the lower part of the lorica. The direction of the flow is indicated with the dashed arrows.

**Figure 2.**(

**A**) The 3D computational fluid dynamics (CFD) model morphology, shown with a $5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ wide flagellar vane (yellow) and a medium-sized prey (gray). (

**B**) A 2D schematic of the CFD model. The bold curves, which correspond to Equations (2) and (3), are revolved around the longitudinal center axis to create the model surfaces (pictured as opaque). The polar coordinate system, which is shown in degrees, serves as the reference frame for the prey’s placement along the filter collar. (

**C**) The three vertical prey positions on the collar. Each prey (gray) is equipped with a local coordinate system: normal (x) and tangent (y) to the collar surface. Pictured is a medium-sized prey (gray) in true relative scale.

**Figure 3.**Flowchart illustrating the iterative procedure used to model drifting prey. The chart covers the process for one arbitrary discrete time step. Here, k denotes an iteration counter.

**Figure 4.**Tangent force distribution on medium-sized prey during one beat cycle with period T. Here, negative forces are in the direction of the cell. The left-hand column of plots is simulated with the presence of the lorica, and the right-hand column of plots is simulated without. The first row of plots displays forces on the right (R) side of the the collar, and the second row displays forces on the back (B) of the collar.

**Figure 5.**Planar views of an instantaneous flow field velocity simulated without the presence of prey. (

**A**) Beat plane view with the lorica. (

**B**) Beat plane view without the lorica. (

**C**) Vane plane view with and without the lorica, respectively. (

**D**) Horizontal view through the collar at position $\theta ={50}^{\circ}$, pictured with and without the lorica, respectively.

**Figure 6.**Normal force distribution on medium-sized prey during one beat cycle with period T. Here, negative forces are orientated in the direction away from the collar. Both plots are simulated without the presence of the lorica. The left-hand plot shows the forces on prey placed on the right (R) side of the collar, and the right-hand plot shows forces on prey placed at the back (B) of the collar.

**Figure 7.**Resultant tangent velocity of fixed prey when using Equations (10) and (11) to convert simulated forces (Table 3) into velocities. Here, a negative velocity is directed towards the cell. The left plot is simulated with the lorica and the right plot without. Velocities are shown for prey placed in the beat plane on the right (R) side of the collar and in the vane plane on the back (B) side of the collar. For each vertical position on the collar ($\theta ={68}^{\circ},\phantom{\rule{0.166667em}{0ex}}{50}^{\circ},\phantom{\rule{0.166667em}{0ex}}{32}^{\circ}$), the velocity is shown for each type of prey considered: small (s), medium (m), large (l), and elongated (e). The two legends apply to both plots.

**Figure 8.**Prey velocity components in the normal and tangent directions to the collar from both the static and dynamic analyses. A prey was placed on the middle and left-hand side of the collar ($50L$), the middle and right-hand side ($50R$), the top and back side ($32B$), and the bottom and back side ($68B$). The left plot shows the results for a medium-sized prey, and the right plot shows the results for a large-sized prey. All results were simulated with the presence of the lorica and at the flagellum position pictured in Figure 1.

**Figure 9.**Planar views of the flow field velocity near a prey, showing the differences between the static and dynamic simulation approaches. Pictured is the large prey positioned on the left side of the filter around the middle (${\mathrm{Prey}}_{l50L}$) using the local prey coordinate system. All figures show the instantaneous flow field corresponding to the flagellum position pictured in Figure 1. (

**A**) The local flow field seen from a frontal perspective when the prey is fixed. (

**B**) The local flow field seen from a frontal perspective when the prey freely drifts. (

**C**) The local flow field seen from a top perspective when the prey is fixed. (

**D**) The local flow field seen from a top perspective when the prey freely drifts.

**Table 1.**Characteristic flagellum parameters specific to D. grandis. A is the wave amplitude, L is the projected length of the flagellum onto the center axis y, f is the wave frequency, $\lambda $ is the wavelength, and W is the width of the flagellar vane.

$\mathit{A}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | $\mathit{L}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | $\mathit{f}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{Hz}\right]$ | $\mathit{\lambda}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ | $\mathit{W}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ |
---|---|---|---|---|

2.8 | 8.3 | 7.3 | 8.6 | 5.0 |

**Table 2.**Characteristic parameters of the four different prey types considered: small, medium, large, and elongated, denoted by s, m, l, and e, respectively. Dimensions are given in μm. Only ${\mathrm{Prey}}_{s}$ is smaller than the filter openings.

${\mathbf{Prey}}_{\mathit{s}}$ | ${\mathbf{Prey}}_{\mathit{m}}$ | ${\mathbf{Prey}}_{\mathit{l}}$ | ${\mathbf{Prey}}_{\mathit{e}}$ | |
---|---|---|---|---|

Major radius | 0.18 | 0.25 | 0.50 | 0.50 |

Minor radius | - | - | - | 0.25 |

**Table 3.**Time-averaged tangent forces on prey during one flagellum beat cycle for both the loricate and the non-loricate choanoflagellate. One position in the beat plane (R) and one position in the vane plane (B) are shown. Column-wise, the table depicts the effects of prey size and shape variations. Row-wise, the effects of the longitudinal position on the collar ($\theta $) can be seen.

Average Tangent Force per Beat Cycle $\phantom{\rule{0.166667em}{0ex}}{[10}^{-3}\phantom{\rule{0.166667em}{0ex}}\mathbf{pN}]$ | ||||||
---|---|---|---|---|---|---|

Case | $\phantom{\rule{0.166667em}{0ex}}\mathbf{\theta}$ | ${\mathbf{Prey}}_{\mathit{s}}$ | ${\mathbf{Prey}}_{\mathbf{m}}$ | ${\mathbf{Prey}}_{\mathit{L}}$ | ${\mathbf{Prey}}_{\mathbf{e}}$ | |

Beat plane (R) | with lorica | 68 | 4.2 | 9.0 | 38.8 | 11.3 |

50 | 4.1 | 7.5 | 49.4 | 9.4 | ||

32 | −2.1 | −4.0 | 12.1 | −5.6 | ||

without lorica | 68 | 0 | 0.2 | 1.1 | 0.2 | |

50 | −2.3 | −4.0 | −2.9 | −5.1 | ||

32 | −0.9 | −1.7 | 2.4 | −2.4 | ||

Vane plane (B) | with lorica | 68 | 5.2 | 10.7 | 43.0 | 13.3 |

50 | 4.6 | 8.3 | 57.2 | 10.5 | ||

32 | −0.3 | −1.2 | 28.2 | −1.8 | ||

without lorica | 68 | 0.5 | 0.8 | 1.7 | 1.1 | |

50 | −4.0 | −6.9 | −5.8 | −8.8 | ||

32 | −2.7 | −4.9 | −1.6 | −6.4 |

**Table 4.**Minimum contact force between microvilli tentacles and prey required to counteract the simulated hydrodynamic normal forces in the non-loricate D. grandis.

${\mathbf{Prey}}_{\mathit{s}}$ | ${\mathbf{Prey}}_{\mathit{m}}$ | ${\mathbf{Prey}}_{\mathit{l}}$ | ${\mathbf{Prey}}_{\mathit{e}}$ | |
---|---|---|---|---|

Min. contact force [pN] | 0.03 | 0.05 | 0.10 | 0.08 |

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**MDPI and ACS Style**

Sørensen, S.; Asadzadeh, S.S.; Walther, J.H.
Hydrodynamics of Prey Capture and Transportation in Choanoflagellates. *Fluids* **2021**, *6*, 94.
https://doi.org/10.3390/fluids6030094

**AMA Style**

Sørensen S, Asadzadeh SS, Walther JH.
Hydrodynamics of Prey Capture and Transportation in Choanoflagellates. *Fluids*. 2021; 6(3):94.
https://doi.org/10.3390/fluids6030094

**Chicago/Turabian Style**

Sørensen, Siv, Seyed Saeed Asadzadeh, and Jens Honoré Walther.
2021. "Hydrodynamics of Prey Capture and Transportation in Choanoflagellates" *Fluids* 6, no. 3: 94.
https://doi.org/10.3390/fluids6030094