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

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.