Lisa J. Fauci Tulane University, Math Dept. New Orleans, Louisiana, USA

Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA

Interface problems inspired by the biofluids of reproductionSAMSISept 25, 2007

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

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

C. Brokaw, CalTechwww.cco.caltech.edu/~brokawc/


•Muscular contractions

•Ciliary beating


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




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.





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


More is needed to allow OCCto enter oviduct!


Outermost layer of OCC –Cumulus layer – cells boundtogether by elastic matrix

Adhesive interaction between ciliary tips and cumulus layer

•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!

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

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

2007 E. Lauga “Propulsion in a viscoelastic fluid” Phys. Fluids 19

0

0

p

u

u

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.

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


•Muscular contractions

•Ciliary beating


Eucaryotic axoneme

3D schematicThe precise nature of the spatial and temporalcontrol mechanisms regulating various wavefomsof cilia and flagella is still unknown.

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

Transport of mucus layer

No mucus – green fluidmarkers

Elastic mucus layer

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).

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 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

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

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)

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

Modified Swimming Kinematicsa Stokes-OB winner

Stokes Stokes Oldroyd-B

Modified Swimming Kinematics Forward Motion Modified Kinematics

StokesStokes 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 ofan Oldroyd-B fluidwith Shelley, Teran CIMS

Documents

Lisa J. Fauci Tulane University, Math Dept. New Orleans, Louisiana, USA