# Flagellar Cooperativity and Collective Motion in Sperm

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fluid Model

#### 2.2. Flagellum Model

#### 2.3. Coupling the Flagella and Fluid Environment

#### 2.4. Model for Apical Hooks

#### 2.5. Computational Method

- Initialize flagella in desired configuration.
- Every time step:
- (a)
- Compute forces on all flagella, including tensile, bending, and planar restriction forces. If applicable, compute attachment or repulsion forces.
- (b)
- Find the velocity at each point along flagella using regularized Stokeslets.
- (c)
- Move flagellar points ${\mathbf{X}}_{m}^{i}$ according to velocity at that point, $\mathbf{u}\left({\mathbf{X}}_{m}^{i}\left(t\right)\right)$, using a forward Euler method with time step $\Delta t$.

#### 2.6. Parameters and Initial Configurations of Flagella

- Cylindrical: flagellar planes tangent to a cylinder;
- Radial: flagellar planes normal to a cylinder;
- Helical offset: cylindrical pattern of neighboring flagella set back along a cylindrical surface to form a helical shape; and
- Grid: all flagella are aligned with bodies in grid patterns.

- $\pi /2$: we will refer to this as a “normal plane” configuration. Neighboring flagella have beat planes that are normal or nearly normal to each other. Motion of neighboring flagella are neither in phase (aligned) nor in opposition.
- $\pi $: we will refer to this rotation as an “out of phase” configuration. Neighboring flagella will be completely out of phase or nearly out of phase, but with similar beat planes. Similar here means identical, parallel, or nearly parallel beat planes. Motions of neighboring flagella are generally expected to push and pull in opposition to each other.

**Table 1.**Parameter values. Sperm parameters were set to that of a typical mammalian sperm with active motility. Wherever possible, biologically-measured values were used. In absence of biologically-measured values, values from prior modeling work were used, if they exist.

Parameter | Parameter | Parameter | Remark/Reference(s) |
---|---|---|---|

$\u03f5$ | Regularization parameter | 1.3 $\mathsf{\mu}$m | [45] |

Diameter of sperm flagellum | 0.5 $\mathsf{\mu}$m | [52] | |

$\mu $ | Viscosity (water) | 10${}^{-3}$ kg m${}^{-1}$ s${}^{-1}$ | |

$\Delta s$ | Spatial (arc length) discretization | 1 $\mathsf{\mu}$m | Set to be $<\u03f5$ [25,32] |

L | Flagellum length | 100 $\mathsf{\mu}$m | [53] |

b | Amplitude | 10 $\mathsf{\mu}$m | [54,55] |

$\kappa $ | Wavenumber | $2\pi /L$ | [5] |

$\omega $ | Frequency | 20$\pi $ (10 Hz) | [54,55] |

${S}_{t}$ | Tensile stiffness | 2 pN $\mathsf{\mu}$m${}^{-3}$ | [25,32] |

${S}_{p}$ | Planar restriction stiffness | 1 pN $\mathsf{\mu}$m${}^{-3}$ | [32] |

${S}_{b}$ | Planar bending stiffness | 10 pN $\mathsf{\mu}$m${}^{-3}$ | [37] |

${S}_{r}$ | Repulsion stiffness | 5 aN $\mathsf{\mu}$m${}^{-3}$ | [32] |

${S}_{a}$ | Attachment stiffness | 20 fN $\mathsf{\mu}$m${}^{-3}$ | Set to be $<{S}_{t}$, this work |

${d}_{r}$ | Repulsion length | 3 $\mathsf{\mu}$m | Set to be $>2\u03f5$ [32] |

${d}_{a}$ | Attraction threshold | 10 $\mathsf{\mu}$m | This work |

a | Attachment length | 6 $\mathsf{\mu}$m | Close to sperm head dimensions [53] |

V | Average path velocity | $\approx {10}^{-5}$–${10}^{-4}$ m/s | [56] |

$\rho $ | Density (water) | ${10}^{3}$ kg m${}^{-3}$ | |

$Re$ | Reynolds number | $\approx {10}^{-2}$ |

## 3. Results

#### 3.1. In Phase Simulations

#### 3.2. Phase and Alignment Differences

#### 3.3. Apical Hooks and Sperm Trains

#### 3.4. Surface Interactions

## 4. Discussion

- Flagella with similar beat planes tend to attract initially. If their beat planes are identical, attraction is a stable long-term behavior. In all other geometries, attraction is transient.
- If beat planes are nearly identical, velocities and efficiencies decrease as flagella attract.
- Flagella in non-identical beat planes tend to swim apart in the long term.
- Flagella can rotate to have beat planes that are closer to nearby neighbors and this is often accompanied by a transient attraction phase, with decreases in velocities and efficiencies.
- Flagella do not tend to swim in higher efficiency configurations unless there is another mechanism for keeping flagella near one another, such as apical hook binding.
- Near surfaces, having more than one flagella present may promote aggregation unless sperm are able to bind to one another and swim as a collective with higher velocities and powers.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

## Appendix A. Role of Repulsion Force

**Figure A1.**Simulation results for a cylindrical configuration of 9 flagella with no repulsion forces present. Distance refers to minimal distance from flagellum to any neighbors. Color of curves indicates data for flagellum of matching color. Units in panels (

**a**–

**d**) are in $\mathsf{\mu}$m. See Supplemental Material Movie S14.

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**Figure 1.**Images taken from experimental work with sperm swimming in populations. In panel (

**a**), a trajectory of a single human sperm swimming in a population of sperm. The trajectory twists over time, creating a chiral ribbon. Arrow denotes progressive motion of sperm and inset shows a front view of the trajectory. In panel (

**b**), sperm from deer mouse form a sperm train using apical hooks on sperm heads to loop around other flagella. This is one type of collective or cooperative behavior observed in sperm.

**Figure 4.**Examples of initial configurations of simulations. For ease of understanding the 3D configurations, we also show the “shadows” of each flagellum on the bottom plane, which is simply a projection of the flagellum onto the $z=-50$ $\mathsf{\mu}$m plane. All units are in $\mathsf{\mu}$m and 9 flagella are shown.

**Figure 5.**Conceptual idea of configurations between neighboring flagella used to explore the role of phase and alignment. Centerline of each flagella is shown by a dotted line. Rotation of nearest neighbors is done by rotating about this centerline. Beat planes are represented by transparent rectangular shapes either at $z=0$ or $y=0$, depending on rotation angle. When these rotations are applied to groups of swimmers in a grid-like formation, the beat planes of neighboring swimmers will either parallel or normal to each other. For cylindrical and radial configurations of groups of swimmers, neighboring swimmers’ beat planes are rotated by an additional $2\pi /m$ for m flagella to maintain circular symmetry.

**Figure 6.**Other initial configurations used. Panels (

**a**–

**c**) shows phase differences between nearest neighbors. Panels (

**d**–

**f**) show configurations with a planar surface (wall) at $x=0$. All panels have 9 flagella, with the exception of (

**a**), in which 8 flagella are used to maintain the out of phase pattern amongst nearest neighbors. All units are in $\mathsf{\mu}$m.

**Figure 7.**Snapshots of locations of flagella over time in 2D, overlaid. Times shown are every 2.5 s (as labeled), equivalent to 25 beats. Head positions started near $x=100$ $\mathsf{\mu}$m (see Figure 4) and flagella move in the negative x-direction. All units are in $\mathsf{\mu}$m. Colors correspond to the flagellum of the same color from the initial configurations in Figure 4.

**Figure 8.**Paths of individual flagella over time, by plotting the head position once per beat over the simulation. Insets in panels (

**a**) and (

**b**) show how flagella move in the initial 1.4 s (14 beats) of the simulation. Blue squares denote the location of flagellum head where maximal efficiency is achieved and red triangles denote where minimal efficiency is achieved. Head positions start at $x=100$ $\mathsf{\mu}$m and flagella move in the negative x-direction, so flagellum heads start at the far right and end in the far left of each image. All units are in $\mathsf{\mu}$m. Colors of curves correspond to the flagellum of the same color from the initial configurations in Figure 4.

**Figure 9.**Ribbon path trajectories created by triangulation of head positions over each beat for the entire simulation. Due to the lateral oscillation of the flagellum head (initial point) within each beat, the head carves out a ribbon-like trajectory. This ribbon indicates the flagellar beat plane. Blue squares denote the location of flagellum head when maximal efficiency is achieved and red triangles denote the location of the flagellum head when minimal efficiency is achieved. Head positions start near $x=100$ $\mathsf{\mu}$m and flagella move in the negative x-direction, so flagella heads start at far right and end up in the far left of each image. All units are in $\mathsf{\mu}$m. Colors of ribbons correspond to the flagellum of the same color from the initial configurations in Figure 4.

**Figure 10.**Comparison of minimal distances to neighboring flagella (upper left plot in each panel), velocities (average path velocities), power, and efficiency in reference to an individual swimmer in free space (dotted line). Note that power and efficiency have been normalized in reference to an individual swimmer in free space. Colors of curves correspond to the flagellum of the same color from the initial configurations in Figure 4.

**Figure 11.**Efficiency normalized to a single swimmer in free space (dotted horizontal line), plotted with respect to minimal distance (in $\mathsf{\mu}$m) to other flagella. Each dot represents one flagella (corresponding to its color as depicted in Figure 4) for one beat. Dot size is used to effectively show overlap of data only and does not indicate any other metric. For reference, red circles denote initial beat and red x’s denotes final beat for each flagella, with label size varied to show overlap.

**Figure 12.**Snapshots of flagellar configurations for the cylindrical simulation from Figure 4a, for both maximal and minimal efficiencies observed. One flagellum is highlighted in red for visual emphasis of the geometry. See Supplemental Material Movie S1 for all configurations over time.

**Figure 13.**Snapshots of flagellar configurations for the radial simulation from Figure 4b, for both maximal and minimal efficiencies observed. One flagellum is highlighted in red for visual emphasis of the geometry. See Supplemental Material Movie S2 for all configurations over time.

**Figure 14.**Phase/alignment differences between nearby neighbors. Snapshots of locations of flagella over time in 2D, overlaid. Times shown are every 2.5 s or 25 beats, as labeled in panels. Head positions started near $x=100$ $\mathsf{\mu}$m (see Figure 6a–c) and flagella move in the negative x-direction. All units are in $\mathsf{\mu}$m. Color of curves indicates data for flagellum of matching color from initial configurations (see Figure 6a–c).

**Figure 15.**Phase/alignment differences between nearby neighbors, ribbon path trajectories created by triangulation of head positions over each beat for the entire simulation. Due to lateral oscillation of head within each beat, the head carves out a ribbon-like surface indicating the flagellar beat plane. Blue squares denote the location of flagellum head when maximal efficiency is achieved and red triangles denote the location of the flagellum head when minimal efficiency is achieved. Head positions started near $x=100$ $\mathsf{\mu}$m and flagella move in the negative x-direction, so flagella heads start at far right and end up in the far left of each image. All units are in $\mathsf{\mu}$m. Color of curves indicates data for flagellum of matching color from initial configurations (see Figure 6a–c).

**Figure 16.**Phase differences between nearby neighbors. Comparison of minimal distances to other flagella, velocities, power, and efficiency with reference to individual swimmer in free space (dotted line). Color of curves indicates data for flagellum of matching color from initial configurations (see Figure 6a–c). Note that panels (

**a**,

**b**) only have 8 curves in order to exploit symmetries of rotation.

**Figure 17.**Positions of flagella in sperm train model, starting from the cylindrical initial configuration depicted in Figure 4a. All units are in $\mathsf{\mu}$m. Panel (

**a**) shows 2D snapshots of locations of flagella over time in 2D, overlaid. Times shown are every 2.5 s or 25 beats. Initial and middle time points are shown in lighter colors, with the initial configuration to the far right. Final time point positions are shown in bolder colors towards the left. Head positions started near $x=100$ $\mathsf{\mu}$m and flagella move in the negative x-direction. Panel (

**b**) shows the positions of flagella at the end of the simulation, forming a phase-locked train. See Supplemental Material Movie S9.

**Figure 18.**Comparison of minimal distances to other flagella, velocity, power and efficiency of apical hook model. In panel (

**a**), we see all metrics plotted over time. Color of curves indicates data for flagellum of matching color from the initial configuration, shown in Figure 4a. In panel (

**b**), we show the configuration for maximal efficiency of the leading flagellum.

**Figure 19.**Distance between the head point of the flagellum and the wall over time for swimmers initialized in parallel planes to the wall. In (

**a**), individual flagella are initialized at various heights away from the wall, with the flagellar beat plane parallel to the wall (each curve is a simulation of one solitary flagellum and results are overlaid). All flagella tend to move away from the wall over time, regardless of how far they start from the wall. In (

**b**), we show the results of a single simulation of a group of 9 flagella, all initiated with beat planes parallel to the surface, in phase but separated from each other by a distance of 20 $\mathsf{\mu}$m. Here, positions of heads are plotted once per beat to remove the zig-zag pattern that can occur due to the periodic undulation of flagella (1 beat = 0.1 s). Notice that some sperm seem to become trapped near the wall, in contrast with the individual swimmers in panel (

**a**). Color of curves in panel (

**b**) indicates data for flagellum of matching color from Figure 6d.

**Figure 20.**Distance between the head point of the flagellum and the wall over time for 9 swimmers initialized in planes normal to the wall. In panel (

**a**), we show swimmers initialized in a grid but with planes normal to the wall. Here, as expected, we see the swimmers moving mostly towards the wall over time. We can contrast that with panel (

**b**), which are also initialized in planes normal to the wall but the flagella are in a cylindrical configuration, as in Figure 6f. Lastly, panel (

**c**) shows the results of swimmers initialized in a cylindrical configuration, as in Figure 6f, but with the ability to “hook” onto each other and form a train. Again, they initially swim towards the wall, but some swim away after about 1 s, or 10 beats. Color of curves indicates data for flagellum of matching color from Figure 6e–f.

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Simons, J.; Rosenberger, A.
Flagellar Cooperativity and Collective Motion in Sperm. *Fluids* **2021**, *6*, 353.
https://doi.org/10.3390/fluids6100353

**AMA Style**

Simons J, Rosenberger A.
Flagellar Cooperativity and Collective Motion in Sperm. *Fluids*. 2021; 6(10):353.
https://doi.org/10.3390/fluids6100353

**Chicago/Turabian Style**

Simons, Julie, and Alexandra Rosenberger.
2021. "Flagellar Cooperativity and Collective Motion in Sperm" *Fluids* 6, no. 10: 353.
https://doi.org/10.3390/fluids6100353