jmm1-slash-6 (7)

Sperm pairing and measures of efficiency in planar swimming models
Sperm pairing and measures of efficiency in planar
swimming 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 single
Monodelphis domestica sperm swimming alone (HD Moore et al,
Biol. of Rep., 1995).

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

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

Introduction
How is the eﬃciency of the motion aﬀected by this behavior?
Our goal: to answer these questions using a computational model.

Mathematical Model: Flagellum
b
s = 0
Xj(t)
∆s
s = L
bending force
tensile forces
Figure: Preferred curvature model of a sperm ﬂagellum with sinusoidal waveform
(L. Fauci et al, J. of Comp. Phys., 1988)

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

Use the incompressible Stokes equations to model the governing ﬂuid
dynamics:
µ∆u = p − F(x)
· u = 0
(1)
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).

dynamics:
µ∆u = p − F(x)
· u = 0
(1)
Immersing our flagellum in this fluid and using the method of
Regularized Stokeslets (R. Cortez, SIAM J. of Sci. Comp, 2001),
we may update flagellum position over time:

dynamics:
µ∆u = p − F(x)
· u = 0
(1)
Immersing our flagellum in this fluid and using the method of
Regularized Stokeslets (R. Cortez, SIAM J. of Sci. Comp, 2001),
we may update flagellum position over time:
−0.5 0 0.5
−0.2
0
0.2

“Paired” Sperm Model
θ h
Figure: Fused head pair of swimmers (shown in antiphase).

Results
0 50 100
-50
0
50
0 50 100
-50
0
50
0 50 100
-50
0
50
0 50 100
-50
0
50

Results
0 20 40 60 80 100
50
60
70
80
90
Angle θ (degrees)
Velocity(µm/s)
0 20 40 60 80 100
0.7
0.8
0.9
1
1.1
Angle θ (degrees)
Eﬃciencyβ/β0
Figure: Velocity and Eﬃciency vs. Angle of Fusion.

Angle of Fusion, θ
100 200 300 400 500 600 700 800 900 1000
0
100
200
300
400
500
600
Figure: The angle of fusion of two M. domestica sperm. Characteristic angle is
approximately 60 degrees (HD Moore et al, Biol. of Rep., 1995).

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

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

Conclusions
Paired sperm swimming is also more eﬃcient at an angle of fusion of
approximately 60 degrees.
These ﬁndings may give some insight into why M. domestica sperm
fuse at this angle.

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 Foundation
grant DMS-104626

Energy Formulation
F(x) =
L
0
f(X(s, t), t)φ (||x − X(s, t)||)ds. (2)
E(X, t) = Etens(X, t) + Ebend (X, t)
Etens =
1
2
St
N
j=2
Xj − Xj−1
∆s
− 1
2
∆s
Ebend =
1
2
Sb
N−1
j=2
(xj+1 − xj )(yj − yj−1) − (yj+1 − yj )(xj − xj−1)
∆s3
− Cj (t)
2
∆
Cj (t) = k2
b sin(kj∆s − ωt + φ0). (3)

Energy Formulation
Etens =
1
2
St
N
j=2
Xj − Xj−1
∆s
− 1
2
∆s
Ebend =
1
2
Sb
N−1
j=2
(xj+1 − xj )(yj − yj−1) − (yj+1 − yj )(xj − xj−1)
∆s3
− Cj (t)
2
∆
Cj (t) = k2
b sin(kj∆s − ωt + φ0). (4)
fj = −
dE
dXj
u(x, t) =
N
j=1
fj (r2
j + 2 2) + (fj · (x − Xj )(x − Xj ))
8πµ(r2
j + 2)
3
2
. (5)

[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 sperm
movement 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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