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

1. Calcium-driven dynamics of undulatory swimmers: a tale of two Reynolds numbers. Lisa J. FauciTulane University, Math Dept.New Orleans, Louisiana, USA

2. (Undulatory) Collaborators: Avis Cohen University of Maryland, Biology Eric Tytell Tufts University, Biology Chia-yu Hsu National Taiwan University Thelma Williams St. George’s Univ. of London, Basic Medical Sciences Phillip Holmes Princeton University, Mathematics Lex Smits Princeton University, Mechanical Engr. Megan Leftwich George Washington University, Mechanical Engr. Sarah Olson Tulane/WPI, Mathematics Susan Suarez Cornell University, Veterinary Sciences

3. Today’s cast of characters • Mammalian spermatozoa (Re 10 -2) • Lamprey (Re 103)

4. Neuroscience: How does the nervous system create a complete behavior(i.e. swimming)?

5. How is force generated to produceswimming?

6. Free swimming of lamprey

7. Fluid Dynamics Sensory system Musculature Neuronal network

8. Fluid Dynamics Musculature Neuronal network

9. Neural activation and fictive swimming in lamprey From Fish and Wildlife. The central pattern generator (CPG) of lamprey is a series of ipsi- and contralaterally coupled neural oscillators distributed along the spinal notocord. In “fictive swimming” in vitro, contralateral motoneurons burst in antiphase and there is a phase lag along the cord from head to tail corresponding to about one full wavelength, at the typical 1-2 Hz burst frequency. This has been modeled as a chain of coupled oscillators. The model can be justified by phase response and averaging theory: [Cohen, Holmes, Rand, J. Math Biol. 13, 345-369, 1982]

10. Neural activation and fictive swimming in lamprey From Fish and Wildlife. R7 R17 L7 L17

11. Methods Avis Cohen and Eric Tytell, UMD

12. Ichthyomyzon unicuspis (silver lamprey) + host (trout). Photo: Avis Cohen. .

13. In lamprey and many other fish, the wave of electrical activation travels faster than the observed mechanical wave. Williams , J. Exp. Biol., 1989 Grillner, Exp. Brain Res., 1974 Wardle, Videler, Altringham, J. Exp. Biol., 1995

14. Output of CPG Ultrastructure of vertebrate skeletal muscle Action potential bursts - Calcium release from SR – Calcium binds to the muscle filaments , causing conformational changes in thick filaments which form cross bridges – force is generated . Calcium then resequestered by SR, and muscle relaxes.

15. Output of CPG Ultrastructure of vertebrate skeletal muscle Action potential bursts - Calcium release from SR – Calcium binds to the muscle filaments , causing conformational changes in thick filaments which form cross bridges – force is generated . Calcium then resequestered by SR, and muscle relaxes.

16. Output of CPG Ultrastructure of vertebrate skeletal muscle Action potential bursts - Calcium release from SR – Calcium binds to the muscle filaments , causing conformational changes in thick filaments which form cross bridges – force is generated . Calcium then resequestered by SR, and muscle relaxes.

17. Output of CPG Ultrastructure of vertebrate skeletal muscle Action potential bursts - Calcium release from SR – Calcium binds to the muscle filaments , causing conformational changes in thick filaments which form cross bridges – force is generated . Calcium then resequestered by SR, and muscle relaxes.

18. Mass action equations d[c]/dt = k1[cs] - k2[c][s] - k3[c][f] d[cf]/dt = k3[c][f] - k4[cf][f] d[cs]/dt = -k1[cs] + k2[c][s] d[f]/dt = -k3[c][f] + k4[cf][f] d[s]/dt = k1[cs] - k2[c][s] While the stimulus is on, k2=0; While the stimulus is off, k1=0. [c] : unbound calcium [cs]: calcium bound SR sites [s]: unbound SR sites [f]: unbound filament sites [cf]: calcium bound filament sites

19. From motoneurons to muscles via calcium dynamics The Hill muscle model produces forces according to This model is a modified version of that fitted to single myotome data; it incorporates nonlinear length and velocity dependence. [Williams, Bowtell & Curtin, J. Exp. Biol. 201, 869-875, 1998]

20. Lamprey anatomy: spinal cord and muscles We replace the multiscale biological complexity of actin, myosin, crossbridges, etc. by simple discretized dampers, springs, and active force generators based on A.V. Hill’s muscle model (1938). [Peters & MacKay, J. Anatomy 95 (4), 575-585, 1961.]

21. Illustration of the simplified lamprey model structure • Some muscle segments on either side are shown in red. McMillen, Williams and Holmes, PLOS Computational Biology,1-16,2008 McMillen and Holmes, Mathematical Biology,53:843-886, 2006

22. Can we build a ‘compu-lamprey’?

23. Can we build a ‘compu-lamprey’? • Takes a wave of action potential as input and…

24. Can we build a ‘compu-lamprey’? • Takes a wave of action potential as input and… • Generates realistic muscle forces from this input and…

25. Can we build a ‘compu-lamprey’? • Takes a wave of action potential as input and… • Generates realistic muscle forces from this input and… • Reflects passive elastic properties of real lamprey and...

26. Can we build a ‘compu-lamprey’? • Takes a wave of action potential as input and… • Generates realistic muscle forces from this input and… • Reflects passive elastic properties of real lamprey and... • Swims?

27. Body plan. Lamprey built out of three filaments, with points connected by springs. Each spring has a particular restlength. How does this translate to macroscopic bend modulus? Lim and Peskin, 2004

28. Energy= .5 A κ2 L

29. What are the forces exerted by model lamprey? • Passive elastic forces (linear springs). • Active muscle contraction forces. • May include muscle damping forces.

30. What are the forces exerted by model lamprey? • Passive elastic forces (linear springs). • Active muscle contraction forces. • May include muscle damping forces. A system of ODE’s governing calcium dynamics and muscle forces are solved on each muscle segment!!!

32. Re (body) = 7900 Re (tail) = 127 St = .65 Wave/Activation speed = .88 Swimming speed/mech wavespeed = .65

33. Solver: IBAMR • Grid-refinement monitors locations of immersed boundaries and regions of high vorticity. B.Griffith, et.al J.Comp.Physics 223,10-49, 2007, www.math.nyu.edu/~griffith/

35. Re (body) = 7900 Re (tail) = 127 St = .65 Wave/Activation speed = .88 Swimming speed/mech wavespeed = .65

36. Red: muscle force Black: muscle length

37. Twice as stiff, twice as strong

38. Half as stiff, half as strong

39. Floppy swimmer Negative work -- muscle segment lengthening when muscle is activated.

41. Stiff-bodied swimmer accelerates faster. Floppy-bodied swimmer uses less energy during steady swimming.

42. Faster activation speed--1.5 Hz

43. Tytell and Lauder, J. Exp. Biology, 2004 “The hydrodynamics of eel swimming: wake structure”

44. Hultmark, Leftwich, Smits: Exper. In Fluids, 2007. Tytell, Lauder: J. Exp. Biol., 2004 Tytell, Hsu, Cohen, Williams, Fauci: PNAS, 2010

45. Even faster activation speed—2.0 Hz

46. Swimming in different viscosity Water More viscous than water… Courtesy of Eric Tytell

47. Swimming in different viscosity Water 10X more viscous than water…

48. Turning (muscles on one side stronger – then switched)

49. Observations • Wake structure is a function of model parameters. • Sensory feedback not necessary for phase-lag between activation/curvature.