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

Calcium-driven dynamics of undulatoryswimmers: a tale of two Reynolds numbers.

(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

Today’s cast of characters

• Mammalian spermatozoa (Re 10 -2)

• Lamprey (Re 103)

Neuroscience: How does the nervous system create a complete behavior

(i.e. swimming)?

How is force generated to produceswimming?

Free swimming of lamprey

University of Chicago 3/27/07

Sensory system

Fluid Dynamics

Musculature

Neuronal network

University of Chicago 3/27/07

Fluid Dynamics

Musculature

Neuronal network

Neural activation and fictive swimming in lamprey

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]

From Fish and Wildlife.

Neural activation and fictive swimming in lamprey

From Fish and Wildlife.

R7

L7R17

L17

Methods

Avis Cohen andEric Tytell, UMD

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

• 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

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.

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.

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.

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.

Output of CPG

Ultrastructure of vertebrate skeletal muscle

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.

Mass action equations

[c] : unbound calcium[cs]: calcium bound SR sites[s]: unbound SR sites[f]: unbound filament sites[cf]: calcium bound filament sites

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]

Lamprey anatomy: spinal cord and muscles

[Peters & MacKay, J. Anatomy 95 (4), 575-585, 1961.]

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).

Illustration of the simplified lamprey model structure

• Some muscle segments on either side are shown in red.

McMillen and Holmes, Mathematical Biology,53:843-886, 2006

McMillen, Williams and Holmes, PLOS Computational Biology,1-16,2008

Can we build a ‘compu-lamprey’?

Can we build a ‘compu-lamprey’?

1. Takes a wave of action potential as input and…

Can we build a ‘compu-lamprey’?

1. Takes a wave of action potential as input and…

2. Generates realistic muscle forces from this input and…

Can we build a ‘compu-lamprey’?

1. Takes a wave of action potential as input and…

2. Generates realistic muscle forces from this input and…

3. Reflects passive elastic properties of real lamprey and...

Can we build a ‘compu-lamprey’?

1. Takes a wave of action potential as input and…

2. Generates realistic muscle forces from this input and…

3. Reflects passive elastic properties of real lamprey and...

4. Swims?

Body plan.

Lamprey built out of three filaments, with points connected by springs. Each spring has a particularrestlength.

How does this translate to macroscopic bend modulus?

Lim and Peskin, 2004

Energy= .5 A κ2 L

What are the forces exerted by model lamprey?

• Passive elastic forces (linear springs).

• Active muscle contraction forces.

• May include muscle damping forces.

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 calciumdynamics and muscle forces are solved on each muscle segment!!!

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

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/

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

Red: muscle forceBlack: muscle length

Twice as stiff, twice as strong

Half as stiff, half as strong

Floppy swimmer

Negative work -- muscle segment lengthening when

muscle is activated.

Stiff-bodied swimmer accelerates faster.

Floppy-bodied swimmer uses less energyduring steady swimming.

Faster activation speed--1.5 Hz

Tytell and Lauder,J. Exp. Biology,2004“The hydrodynamics ofeel swimming: wakestructure”

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

Even faster activation speed—2.0 Hz

Swimming in different viscosity

Water

More viscous than water…

Courtesy of Eric Tytell

Swimming in different viscosity

Water

10X more viscous than water…

Turning (muscles on one side stronger – then switched)

Observations

• Wake structure is a function of model parameters.

• Sensory feedback not necessary for phase-lag between activation/curvature.

Olson, Suarez, Fauci: J. Theor. Biol. 2011

Olson, Suarez, Fauci, J. Theor. Biology, 283(2011)203–216

CASA – computer-aided sperm analysis.VAP - average path velocity