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

1. Interface problems inspired by the biofluids of reproduction SAMSI Sept 25, 2007 Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA

2. 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

3. Reproduction –No better illustration of complexfluid-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. courtesy of P. Talbot, Cell Biol. UC Riverside ZP –glycoprotein layer surrounding oocyte

4. Transport of sperm to site of fertilization • Transport of oocyte cumulus complex (OCC) to oviduct • Transport and implantation of embryo in uterus

5. Motile spermatozoa C. Brokaw, CalTech www.cco.caltech.edu/~brokawc/

6. Motile spermatozoa • Muscular contractions • Ciliary beating C. Brokaw, CalTech www.cco.caltech.edu/~brokawc/

7. 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 mucus- • containing lumen –as few as • one sperm per oocyte complete this journey M.A. Scott Anim. Reprod. Sci. 2000

8. Are mammalian sperm chemotactic? Williams et al. 1993, Human reprod. -- more sperm in ampullar region in ovulating oviduct

9. 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?

10. 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.

11. 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.

12. Oviductal cilia can transport particles courtesy of P. Talbot, Cell Biol. UC Riverside

13. More is needed to allow OCCto 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

14. 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!

15. Coupled System 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

16. Video images of swimming patterns of bull sperm. • Each frame shows two images spaced 1/60 second apart. • 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

17. 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

18. Fluid coupled with ‘elasticstructure’ 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.

19. Immersed boundary framework Xk(t) fk(t) Transmit fk(t) togrid Stokesflow Grid-based Direct sum formula Solve Navier -Stokes on grid Grid-free Interpolate gridvelocity Uk(t) Xk(t+Dt) = Xk(t) + Dt Uk(t)

21. Leech swimming Cortez, Cowen, Dillon, Fauci Comp. Sci Engr. 2004

22. Motile spermatozoa • Muscular contractions • Ciliary beating C. Brokaw, CalTech www.cco.caltech.edu/~brokawc/

23. Eucaryotic axoneme 3D schematic The precise nature of the spatial and temporal control mechanisms regulating various wavefoms of cilia and flagella is still unknown.

24. 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.

25. 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.

26. 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.

29. Transport of mucus layer No mucus – green fluid markers Elastic mucus layer

30. Simple motor – flagellar beat Two superimposed families of dyneins

31. Curvature control algorithm The activation of each individual dynein depends upon the local curvature at the site of the dynein at some lag timet in the past. Initially, the axoneme has a pair of bends. C. Brokaw (1972), Hines and Blum (1978), C. Lindemann (2002), Murase (1992).

32. Two state stochastic model

33. Threshold model 10 centipoise 1 centipoise

34. 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.

35. Locomotion in Visco-Elastic Fluids Teran, Fauci, & Shelley ‘07 Undulatory swimming at low Re in Newtonian fluids is fairly well understood: 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. Undulatory slender-body swimmer C. elegans nematode swimming in water Biofluids: typically non-Newtonian and viscoelastic. Fauci & Dillon Ann. Rev. Fluids 2006 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

36. 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 momentum and mass balance transport and damping of polymer stress with

37. 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 contours of trace(S)

38. The classical problem: Swimming of a period sheet • O(a2) 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

39. Modified Swimming Kinematicsa Stokes-OB winner Stokes Stokes Oldroyd-B

40. Modified Swimming Kinematics Forward Motion Modified Kinematics Stokes Stokes OB peak forward velocity recoil phase

41. 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