Interface problems inspired by the
biofluids of reproduction
SAMSI
Sept 25, 2007
Lisa J. Fauci
Tulane University, Math Dept.
New Orleans, Louisiana, USA
Collaborators:
Ricardo Cortez
Tulane University
Mathematics
Robert Dillon
Washington State University
Mathematics
Charlotte Omoto
Washington State University
School of Biological Sciences
Michael Shelley
New York University
Mathematics
Joseph Teran
University of California, Los Angeles
Mathematics
Xingzhou Yang,
Tulane University
Center for Computational Science
Reproduction –
No better illustration of complex
fluid-structure interactions
See: Fauci and Dillon,
Ann. Rev. Fluid Mech.,
Vol. 38, 2006
A scanning electron
micrograph of hamster
sperm bound to a
zona pellucida.
ZP –glycoprotein layer
surrounding oocyte
courtesy of
P. Talbot, Cell Biol.
UC Riverside
•Transport of sperm to site of fertilization
•Transport of oocyte cumulus complex (OCC) to oviduct
•Transport and implantation of embryo in uterus
•Motile spermatozoa
C. Brokaw, CalTech
www.cco.caltech.edu/~brokawc/
•Motile spermatozoa
•Muscular contractions
•Ciliary beating
C. Brokaw, CalTech
www.cco.caltech.edu/~brokawc/
How likely is a successful sperm-egg
encounter?
•In mammals, ten to hundreds of millions sperm
are introduced.
•One tenth reach cervix
•One tenth penetrate cervical mucus to reach uterus
•One tenth make it through uterus to oviduct
•After progressing through narrow, tortuous mucuscontaining lumen –as few as
one sperm per oocyte complete this journey
M.A. Scott
Anim. Reprod. Sci. 2000
Are mammalian sperm
chemotactic?
Williams et al. 1993, Human reprod. -more sperm in ampullar region in ovulating oviduct
Are mammalian sperm
chemotactic?
Williams et al. 1993, Human reprod. -more sperm in ampullar region in ovulating oviduct
Is this sperm chemotaxis or hormonally
mediated mechanical activity of the oviductal
muscles or cilia?
Are mammalian sperm
chemotactic?
Williams et al. 1993, Human reprod. -more sperm in ampullar region in ovulating oviduct
Is this sperm chemotaxis or hormonally
mediated mechanical activity of the oviductal
muscles or cilia?
Kunz et al. 1997, albumin macrospheres introduced – more
end up in ovulating oviduct.
Are mammalian sperm
chemotactic?
Williams et al. 1993, Human reprod. -more sperm in ampullar region in ovulating oviduct
Is this sperm chemotaxis or hormonally
mediated mechanical activity of the oviductal
muscles or cilia?
Kunz et al. 1997, albumin macrospheres introduced – more
end up in ovulating oviduct.
Eisenbach 2004 conjectures that chemotaxis is a short-range
guidance mechanism.
Oviductal cilia can transport
particles
courtesy of
P. Talbot, Cell Biol.
UC Riverside
More is needed to allow OCC
to enter oviduct!
Adhesive interaction
between ciliary tips
and cumulus layer
Outermost layer of OCC –
Cumulus layer – cells bound
together by elastic matrix
courtesy of
P. Talbot, Cell Biol.
UC Riverside
•Peristaltic contractions of uterus/oviduct –
role in ovum/embryo transport?
• Role of fluid mechanics in
successful implantation of
embryo : in vitro
fertilization?
Yaniv, Elad, Jaffa, Eytan
2003, Ann.Biomed. Engr.
•Injection speed critical.
•Timing of injection with
peristaltic phase?
•Embryo not point particle!
Cilia/flagella
Viscous,
(Force
generators)
Incompressible
fluid
Emergent properties:
• Beat form
Micro
• Swimming
• Metachronism (i.e. synchronized ciliary
beating, phase-locking of sperm)
•Patterns of cell populations (bioconvection)
Macro
Scale
Video images of swimming patterns of bull sperm.
Each frame shows two images spaced 1/60 second apart.
(a)Regular beat pattern of activated sperm.
(b)Assymetric beat pattern of hyperactivated sperm.
(c)Hyperactivated sperm in thick, viscoelastic solution
Ho and Suarez 2001, Reprod. 122
Much progress has been made in the last 60
years:
1951 G.I. Taylor “Analysis of the swimming of microscopic
organisms”, Proc. R. Society, 209
Many others including Lighthill, Blake, Keller-Rubinow, Higdon,
Gueron, Liron,…
Approaches include:
•
•
•
Resistive Force Theory
Slender Body Theory
Boundary Integral Methods
p   u  0
 u  0
2007 E. Lauga “Propulsion in a viscoelastic fluid” Phys. Fluids 19
Fluid coupled with ‘elastic structure’
Flow is governed by the incompressible Navier Stokes equations:
Fk is a ‘delta function’ layer
of force exerted by the kth filament
on the fluid.
Immersed boundary framework
Xk(t)
fk(t)
Stokes flow
Transmit
fk(t) to grid
Direct
sum
formula
Solve Navier Stokes on grid
Interpolate grid
velocity
Uk(t)
Xk(t+t) = Xk(t) + t Uk(t)
Leech swimming
Cortez, Cowen, Dillon, Fauci
Comp. Sci Engr. 2004
•Motile spermatozoa
•Muscular contractions
•Ciliary beating
C. Brokaw, CalTech
www.cco.caltech.edu/~brokawc/
Eucaryotic axoneme
3D schematic
The precise nature of the spatial and temporal
control mechanisms regulating various wavefoms
of cilia and flagella is still unknown.
2-microtubule axoneme
• `Rigid’ links build the
microtubules.
• Nexin links are modeled
by passive intermicrotubule springs.
• Dynein motors –
dynamic springs.
Dillon and Fauci, J. Theor. Biol., Vol. 207, 2000.
Dillon, Fauci, Omoto, Dyn.Cont.&Imp.Syst, Vol. 10, 2003.
Dynein Arms
• Dyneins are modeled by
dynamic inter-microtubule
springs.
• Dynein connectivity is
reassessed at each time step.
Depending on the amount of
microtubule sliding, the dyneins
might “ratchet” from one site of
attachment along neighboring
microtubule to another.
Simple motor for power/recovery
stroke
• Power stroke: all dyneins are activated.
When shear has reached a given
threshold, terminate power stroke.
• Recovery stroke: Activate opposing family
of dyneins from base up to the point of
maximum curvature.
Transport of mucus layer
No mucus – green fluid
markers
Elastic mucus layer
Simple motor – flagellar beat
Two superimposed families of dyneins
Curvature control algorithm
The activation of each individual
dynein depends upon the local
curvature at the site of the dynein
at some lag time t in the past.
Initially, the axoneme has
a pair of bends.
C. Brokaw (1972), Hines and Blum (1978),
C. Lindemann (2002), Murase (1992).
Two state stochastic model
Threshold model
10 centipoise
1 centipoise
We have developed the framework and methodology
for a coupled fluid-axoneme model that:
• Provides information concerning the local
curvature and spacing between the
microtubules at each dynein site.
• Facilitates stochastic models of dynein
activation due to the discrete
representation of the dyneins.
• May be used as a test-bed for different
activation theories.
Locomotion in Visco-Elastic Fluids
Teran, Fauci, & Shelley ‘07
Undulatory swimming at low Re in Newtonian fluids is fairly well understood:
p   u  0
 u  0
Resistive force thry:
Taylor, Lighthill, Purcell, …
and many, many others, e.g.
Hosoi et al on optimization
of stroke for speed and eff.
C. elegans nematode swimming
in water
Undulatory slender-body
swimmer
Biofluids: typically nonNewtonian and viscoelastic.
Fauci & Dillon
Ann. Rev. Fluids 2006
Standard tools: Singularity
and boundary integral
methods for Stokes Eqs.
Right panels:
stroke pattern of
bull sperm in
cervical mucus
Left panel: sinusoidal
stroke pattern of bull sperm
in Newtonian fluid
Ho and Suarez, 2007
Stokes-Oldroyd-B
• Standard viscoelastic flow models; balance of solvent and polymer
stresses.
• Derives from a microscopic theory of dilute suspension of polymer coils
acting as Hookean springs
• Model of a “Boger” elastic fluid (normal stresses, no shear thinning),
but can excessively strain harden in extensional flow
p  u      S  f and   u  0
Wi S  (S  I )
momentum and mass balance
transport and damping
of polymer stress
with  S  DS  (u  S  S u T ) ; upper convected time deriv.
Dt
 Wi  t p / t f ;
Weissenberg number
   Gt f /  ;
coupling strength of polymer stress
polymer viscosity
   Wi  Gt p /  
 O(1) material const.
solvent viscosity
Kinematics from energetics: move a geometric
deformation in a sheet via curvature-based
energy, couple sheet to Stokes-OB Eqs. via
Immersed Boundary Method Peskin & McQueen ‘89
k
b  b
2
  (s)   0 (s, t )  ds
2

2
l 
kl  x


1
ds



2   s

k 0 ( s, t )  ap 2 sin( p( s  t ))
contours of trace(S)
The classical problem: Swimming of a period
sheet
2
O(a ) scaling of swimming speed with
sinusoidal profile amplitude a
Stokes -- G.I. Taylor, 1951
Stokes-OB -- E. Lauga, 2007
small amplitude analysis:
Same scaling & Stokes wins.
swimming speed
stokes
stokes-OB
time
Modified Swimming Kinematics
a Stokes-OB winner
Stokes
Stokes Oldroyd-B
Modified Swimming Kinematics
Forward Motion Modified Kinematics
Stokes
Stokes OB
recoil phase
peak forward velocity
Challenges
• Full 3D “9+2” modeling
• Non-Newtonian fluid
regime
• Multiciliary oviductal
arrays
• Complete coupling of
ciliary beating,
mechanical contrations,
sperm motility
Peristaltic pumping of
an Oldroyd-B fluid
with Shelley, Teran CIMS