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Sperm pairing and measures of efficiency in planar swimming models

Sperm pairing and measures of efficiency in planarswimming models

Paul Cripe, Owen Richfield, and Julie Simons

Tulane University Center for Computational Science

January 6, 2016

Paul Cripe, Owen Richfield, and Julie Simons | Tulane University Center for Computational Science | January 6, 2016 1 / 14

Introduction

Monodelphis domestica

Figure: Wikimedia.org and Pizzari et al, PLoS Bio., 2008

Introduction

This fused “sperm pair” reach 23.8% higher velocities than a singleMonodelphis domestica sperm swimming alone (HD Moore et al,Biol. of Rep., 1995).

How will changing the geometry of fusion change these velocity gains?

How is the efficiency of the motion affected by this behavior?

Our goal: to answer these questions using a computational model.

Introduction

Mathematical Model: Flagellum

bending force

tensile forces

Figure: Preferred curvature model of a sperm flagellum with sinusoidal waveform(L. Fauci et al, J. of Comp. Phys., 1988)

.Paul Cripe, Owen Richfield, and Julie Simons | Tulane University Center for Computational Science | January 6, 2016 4 / 14

Mathematical Model: Fluid

Due to their microscopic size, sperm move in a viscous fluid with aReynolds number on the order of 10−4–10−2.

Use the incompressible Stokes equations to model the governing fluiddynamics:

µ∆u = ∇p − F(x)∇ · u = 0

where u is the fluid velocity, µ is the dynamic viscosity, p is pressure,and F is the external force density (force per unit volume).

Immersing our flagellum in this fluid and using the method ofRegularized Stokeslets (R. Cortez, SIAM J. of Sci. Comp, 2001),we may update flagellum position over time:

−0.5 0 0.5

−0.2

µ∆u = ∇p − F(x)∇ · u = 0

−0.5 0 0.5

−0.2

µ∆u = ∇p − F(x)∇ · u = 0

−0.5 0 0.5

−0.2

µ∆u = ∇p − F(x)∇ · u = 0

−0.5 0 0.5

−0.2

“Paired” Sperm Model

Figure: Fused head pair of swimmers (shown in antiphase).

Results

0 50 100-50

Results

0 20 40 60 80 10050

Angle θ (degrees)

Velocity

(µm/s)

0 20 40 60 80 1000.7

Angle θ (degrees)Efficien

cyβ/β0

Figure: Velocity and Efficiency vs. Angle of Fusion.

Angle of Fusion, θ

100 200 300 400 500 600 700 800 900 1000

Figure: The angle of fusion of two M. domestica sperm. Characteristic angle isapproximately 60 degrees (HD Moore et al, Biol. of Rep., 1995).

Conclusions

According to our model, paired sperm are 26.6% faster than singlesperm, similar to the findings of Moore et al.

Paired sperm swimming is also more efficient at an angle of fusion ofapproximately 60 degrees.

These findings may give some insight into why M. domestica spermfuse at this angle.

Conclusions

Acknowledgments

I would like to thank

Julie Simons and Paul Cripe for work on this project.

The Tulane University Center for Computational Science.

This work was supported in part by the National Science Foundationgrant DMS-104626

Energy Formulation

F(x) =

0f(X(s, t), t)φε(||x− X(s, t)||)ds. (2)

E(X, t) = Etens(X, t) + Ebend(X, t)

Etens =1

N∑j=2

Xj − Xj−1

∆s‖ − 1

Ebend =1

N−1∑j=2

((xj+1 − xj)(yj − yj−1)− (yj+1 − yj)(xj − xj−1)

∆s3− Cj(t)

Cj(t) = k2b sin(kj∆s − ωt + φ0). (3)

Energy Formulation

Etens =1

N∑j=2

Xj − Xj−1

∆s‖ − 1

Ebend =1

N−1∑j=2

((xj+1 − xj)(yj − yj−1)− (yj+1 − yj)(xj − xj−1)

∆s3− Cj(t)

Cj(t) = k2b sin(kj∆s − ωt + φ0). (4)

fj = − dE

u(x, t) =N∑j=1

fj(r2j + 2ε2) + (fj · (x− Xj)(x− Xj))

8πµ(r2j + ε2)

[2, 1, 3, 4]

R Cortez.The method of regularized stokeslets.SIAM Journal of Scientific Computing, 23(4):1204–1225, 2001.

Lisa J Fauci and Charles S Peskin.A computational model of aquatic animal locomotion.Journal of Computational Physics, 77(1):85–108, 1988.

HD Moore and DA Taggart.Sperm pairing in the opossum increases the efficiency of spermmovement in a viscous environment.Biology of reproduction, 52(4):947–953, 1995.

Tommaso Pizzari and Kevin R Foster.Sperm sociality: cooperation, altruism, and spite.PLoS biology, 6(5):e130, 2008.

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