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

1. biofluids of reproduction Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA

2. biofluids of reproduction - (amazing simulations…) Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA

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 • Muscular contractions • Ciliary beating Courtesy: Susan Suarez, Cornell U.

6. 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

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

8. 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?

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? Kunz et al. 1997, albumin macrospheres introduced – more end up in ovulating oviduct.

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. Eisenbach 2004 conjectures that chemotaxis is a short-range guidance mechanism.

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

12. 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

13. Used OCC’s can’t make the turn! 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 Swimmer/pumper (Force generators) (non) Newtonian viscous fluid • Emergent properties: • Beat form/wave form (kinematics) • Swimming/pumping

16. Current challenges: Multiscale models that include biochemistry. Description of complex fluids. Full 3D models

17. How is force generated to produceswimming?

18. How is force generated to produce peristaltic waves in uterus? Uterine walls surround a lumen of cervical mucus 30 mm 1 mm 50 mm 50 mm

19. 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

20. 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.

21. 2D Sperm motility Xk • Spring forces • Bending (hinge) forces Xk+1 Forces are derived from an energy function of the form: Driving function

22. Kinematics from energetics: move a geometric deformation.

23. 2D sperm motility LF and A. McDonald, 1994 Bull. Math. Biol.

24. 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)

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

27. Numerical methods for Navier-Stokes - ( no black boxes)

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

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

30. Axonemal Dynein

31. Microtubule sliding

32. Open Question How are the dyneins regulated to the produce the variety of flagellar waveforms? (ciliary beat, sinusoidal, hyperactivation, steering mechanism)

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

34. Abstraction - Picasso Chris Johnson, U. Utah

35. 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.

36. 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.

37. 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.

39. Synchronization of beating cilia Yang, Dillon, Fauci Bull. Math. Biol. 2008

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

41. Schematic of Model Sperm

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

43. 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).

44. Threshold model 10 centipoise 1 centipoise

45. Threshold model 10 centipoise 1 centipoise

46. 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.

47. Mesh of Maxwell elements overlaid on Newtonian fluid.

48. 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

49. Immersed boundary framework Navier-Stokes/Oldroyd-B Xk(t) fk(t) Transmit fk(t) togrid Advect and damp polymer stress σ(t) on grid Solve NS/OB on grid Interpolate gridvelocity Uk(t) Xk(t+Dt) = Xk(t) + Dt Uk(t)

50. Stokes flow is reversible Teran, Peskin 2007