Lisa J. Fauci Tulane University, Math Dept. New Orleans, Louisiana, USA
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Transcript of Lisa J. Fauci Tulane University, Math Dept. New Orleans, Louisiana, USA
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Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA
Interface problems inspired by the biofluids of reproductionSAMSISept 25, 2007
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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
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Reproduction –No better illustration of complex
fluid-structure interactions
A scanning electronmicrograph of hamstersperm bound to a zona pellucida.
courtesy ofP. Talbot, Cell Biol. UC Riverside
ZP –glycoprotein layersurrounding oocyte
See: Fauci and Dillon,Ann. Rev. Fluid Mech.,Vol. 38, 2006
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•Transport of sperm to site of fertilization•Transport of oocyte cumulus complex (OCC) to oviduct•Transport and implantation of embryo in uterus
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•Motile spermatozoa
C. Brokaw, CalTechwww.cco.caltech.edu/~brokawc/
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•Motile spermatozoa
•Muscular contractions
•Ciliary beating
C. Brokaw, CalTechwww.cco.caltech.edu/~brokawc/
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How likely is a successful sperm-egg encounter?
•In mammals, ten to hundreds of millions spermare 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 mucus-containing lumen –as few as one sperm per oocyte complete this journey
M.A. ScottAnim. Reprod. Sci. 2000
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Are mammalian sperm chemotactic?
Williams et al. 1993, Human reprod. --
more sperm in ampullar region in ovulating oviduct
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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 oviductalmuscles or cilia?
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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 oviductalmuscles or cilia?
Kunz et al. 1997, albumin macrospheres introduced – moreend up in ovulating oviduct.
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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 oviductalmuscles or cilia?
Kunz et al. 1997, albumin macrospheres introduced – moreend up in ovulating oviduct.
Eisenbach 2004 conjectures that chemotaxis is a short-rangeguidance mechanism.
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Oviductal cilia can transport particles
courtesy ofP. Talbot, Cell Biol. UC Riverside
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More is needed to allow OCCto enter oviduct!
courtesy ofP. Talbot, Cell Biol. UC Riverside
Outermost layer of OCC –Cumulus layer – cells boundtogether by elastic matrix
Adhesive interaction between ciliary tips and cumulus layer
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•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, Eytan2003, Ann.Biomed. Engr.
•Injection speed critical.•Timing of injection with peristaltic phase?•Embryo not point particle!
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Cilia/flagella
(Force generators)
Viscous,
Incompressible fluid
Emergent properties:
• Beat form
• Swimming
• Metachronism (i.e. synchronized ciliary
beating, phase-locking of sperm)
•Patterns of cell populations (bioconvection)
Micro
MacroScale
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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
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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
2007 E. Lauga “Propulsion in a viscoelastic fluid” Phys. Fluids 19
0
0
p
u
u
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Fluid coupled with ‘elastic structure’
Flow is governed by the incompressible Navier Stokes equations:
Fk is a ‘delta function’ layerof force exerted by the kth filament on the fluid.
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Immersed boundary framework
Transmit fk(t) to grid
Solve Navier -Stokes on grid
Interpolate grid
velocity
Xk(t) fk(t)
Direct sumformula
Stokes flow
Uk(t)
Xk(t+t) = Xk(t) + t Uk(t)
Grid
-bas
ed
Grid
-fre
e
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Leech swimmingCortez, Cowen, Dillon, FauciComp. Sci Engr. 2004
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•Motile spermatozoa
•Muscular contractions
•Ciliary beating
C. Brokaw, CalTechwww.cco.caltech.edu/~brokawc/
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Eucaryotic axoneme
3D schematicThe precise nature of the spatial and temporalcontrol mechanisms regulating various wavefomsof cilia and flagella is still unknown.
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2-microtubule axoneme
• `Rigid’ links build the microtubules.
• Nexin links are modeled by passive inter-microtubule springs.
• Dynein motors – dynamic springs.
Dillon and Fauci, J. Theor. Biol., Vol. 207, 2000.
Dillon, Fauci, Omoto, Dyn.Cont.&Imp.Syst, Vol. 10, 2003.
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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.
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• 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.
Simple motor for power/recovery stroke
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Transport of mucus layer
No mucus – green fluidmarkers
Elastic mucus layer
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Two superimposed families of dyneins
Simple motor – flagellar beat
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Curvature control algorithm
The activation of each individualdynein depends upon the local curvature at the site of the dyneinat some lag time in the past.
Initially, the axoneme has a pair of bends.
C. Brokaw (1972), Hines and Blum (1978),C. Lindemann (2002), Murase (1992).
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Two state stochastic model
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Threshold model
10 centipoise 1 centipoise
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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.
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Locomotion in Visco-Elastic FluidsTeran, Fauci, & Shelley ‘07
Undulatory swimming at low Re in Newtonian fluids is fairly well understood:
Biofluids: typically non-Newtonian and viscoelastic.Fauci & DillonAnn. Rev. Fluids 2006
Ho and Suarez, 2007
Left panel: sinusoidalstroke pattern of bull spermin Newtonian fluid
Right panels: stroke pattern ofbull sperm in cervical mucus
C. elegans nematode swimming in water
Undulatory slender-body swimmer
Resistive force thry:Taylor, Lighthill, Purcell, …and many, many others, e.g.Hosoi et al on optimization of stroke for speed and eff.
Standard tools: Singularity and boundary integral methods for Stokes Eqs.
0
0
p
u
u
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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
and 0
( )
p
Wi
u S f u
S S I transport and damping of polymer stress
momentum and mass balance
T
f
; upper convected time deriv.
; Weissenberg number
; coupling strength of polymer stress
polymer viscosity(1) materia
solvent viscosity
( )
/
G /
/
p f
p
D
DtWi
Wi G O
SS u S S u
l const.
with
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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
2
0( ) ( , )2b
b
ks s t ds
2
12
ll
kds
s
x
20( , ) sin( ( ))k s t ap p s t
contours of trace(S)
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The classical problem: Swimming of a period sheet
O(a2) scaling of swimming speed with sinusoidal profile amplitude a
Stokes -- G.I. Taylor, 1951Stokes-OB -- E. Lauga, 2007 small amplitude analysis: Same scaling & Stokes wins.
stokes
stokes-OB
swimming speed
time
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Modified Swimming Kinematicsa Stokes-OB winner
Stokes Stokes Oldroyd-B
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Modified Swimming Kinematics Forward Motion Modified Kinematics
StokesStokes OB
recoil phase peak forward velocity
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Challenges
• Full 3D “9+2” modeling• Non-Newtonian fluid
regime• Multiciliary oviductal
arrays • Complete coupling of
ciliary beating, mechanical contrations, sperm motility
Peristaltic pumping ofan Oldroyd-B fluidwith Shelley, Teran CIMS