lisa j. fauci tulane university, math dept. new orleans, louisiana, usa biofluids of reproduction
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Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA
biofluids of reproduction
Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA
biofluids of reproduction -
(amazing simulations…)
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
•Transport of sperm to site of fertilization•Transport of oocyte cumulus complex (OCC) to oviduct•Transport and implantation of embryo in uterus
•Motile spermatozoa
•Muscular contractions
•Ciliary beating
Courtesy: Susan Suarez, Cornell U.
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
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 oviductalmuscles 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 oviductalmuscles or cilia?
Kunz et al. 1997, albumin macrospheres introduced – moreend 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 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.
Oviductal cilia can transport particles
courtesy ofP. Talbot, Cell Biol. UC Riverside
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
Used OCC’s can’t make the turn!
courtesy ofP. 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, Eytan2003, Ann.Biomed. Engr.
•Injection speed critical.•Timing of injection with peristaltic phase?•Embryo not point particle!
Swimmer/pumper
(Force generators)
(non) Newtonian
viscous fluid
Emergent properties:
• Beat form/wave form (kinematics)
• Swimming/pumping
Multiscale models that include biochemistry.
Description of complex fluids.
Full 3D models
How is force generated to produceswimming?
How is force generated to produce peristaltic waves in uterus?
30 mm
50 mm
Uterine walls surround a lumen of cervical mucus
1 mm
50 mm
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
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.
• Spring forces• Bending (hinge) forces
Forces are derived from an energy function of the form:
Driving function
2D Sperm motilityXk
Xk+
Kinematics from energetics: move a geometric deformation.
2
0( ) ( , )2b
b
ks s t ds
2
12
ll
kds
s
x
20( , ) sin( ( ))k s t ap p s t
2D sperm motility
LF and A. McDonald, 1994Bull. Math. Biol.
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
Leech swimmingCortez, Cowen, Dillon, FauciComp. Sci Engr. 2004
Numerical methods for Navier-Stokes -
( no black boxes)
•Motile spermatozoa
•Muscular contractions
•Ciliary beating
C. Brokaw, CalTechwww.cco.caltech.edu/~brokawc/
Eucaryotic axoneme
3D schematicThe precise nature of the spatial and temporalcontrol mechanisms regulating various wavefomsof cilia and flagella is still unknown.
Axonemal Dynein
Microtubule sliding
Open Question
How are the dyneins regulated to the
produce the variety of flagellar waveforms?
(ciliary beat, sinusoidal, hyperactivation, steering mechanism)
Eucaryotic axoneme
3D schematicThe precise nature of the spatial and temporalcontrol mechanisms regulating various wavefomsof cilia and flagella is still unknown.
Abstraction - Picasso
Chris Johnson, U. Utah
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.
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.
• 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
Synchronization of beating cilia
Yang, Dillon, FauciBull. Math. Biol.2008
Transport of mucus layer
No mucus – green fluidmarkers
Elastic mucus layer
Schematic of Model Sperm
Two superimposed families of dyneins
Simple motor – flagellar beat
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).
Threshold model
10 centipoise 1 centipoise
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.
Mesh of Maxwell elementsoverlaid on Newtonian fluid.
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
Immersed boundary framework
Transmit fk(t) to grid
Solve NS/OB on grid
Interpolate grid
velocity
Xk(t) fk(t)
Uk(t)
Xk(t+t) = Xk(t) + t Uk(t)
Navier-Stokes/Oldroyd-B
Advect and damppolymer stress σ(t)on grid
Stokes flow is reversible
Teran, Peskin2007
Oldroyd-B is irreversible
Teran, Fauci,Shelley 2008Phys. Fluids
Peristaltic pumping.
Waves of contraction moving to the right transport fluidto the right.
See Jaffrin-Shapiro, Ann. Rev. Fluid Mech.1971.
Two departures from Newtonian pumping…
• Pumping efficacy does not increase with occlusion ratio.
• Reversibility no longer holds.
Teran, Fauci,Shelley 2008Phys. Fluids
Occlusion ratio
Meanflow
Back and forth peristaltic pumpingStokes: no mixing
Oldroyd-B: fluid mixing
Teran, Fauci,Shelley 2008Phys. Fluids
Peristaltic pumping of solid particles in NS/Oldroyd-B
Newtonian particle spends more time going forward.Viscoelastic particles spend more time in each of their periods going backwards.
• Continuum model of non-Newtonian fluid
• Preferred beatform prescribed.
Low Re number swimming:
1951 G.I. Taylor “Analysis of the swimming of microscopic
organisms”, Proc. R. Society, 209
Many others including Lighthill, Blake, Keller, Rubinow, Higdon, Gueron, Phan-Thien, Cortez
Approaches include:• Resistive Force Theory
• Slender Body Theory
• Boundary Integral Methods
• Regularized Stokeslet Methods
• Immersed Boundary Methods
0
0
p
u
u
The classical problem: Swimming of a period sheet
Stokes -- G.I. Taylor, 1951 Stokes wins!
2007 E. Lauga “Propulsion in a viscoelastic fluid” Phys. Fluids 19
Comparison of computations with asymptotics
SOB swimmer speed/Stokes swimmer speed < 1
SOB swimmer speed/Stokes swimmer speed < 1
Large amplitude, finite swimmers with
accentuated tail beating can
actually swim faster in a Stokes/Oldroyd-B
fluid than a Newtonian Stokes fluid.
J. Teran, LF, M. Shelley,Phys. Rev. Letters, 2010
On the other hand…
Wi
Ratio
Modified Swimming Kinematics Forward Motion Modified Kinematics
StokesStokes OB
recoil phase peak forward velocity
To follow:
Regularized Stokeslets
Integrative models of undulatory swimmers.