1. Force Velocity Describe the force-velocity relationship Explain the "extra heat of shortening" Describe exceptions to the force-velocity relationship – Edman, 1979 – Ford, Huxley & Simmons, 1977 – Rack & Westbury, 1969/74

2. Afterloaded contraction A. V. Hill – Time to move fixed distance against known inertia Magnetic release mechanism Position indicator Counterweights (Inertia) Muscle Safety stops/timer

3. Hyperbolic force-velocity Purely phenomenolocial/empirical – (x-a)(y-b)=c – a, b are coordinates of asymptotes Load (g) Shortening velocity (cm/s)

4. Add some thermodynamics Conservation of energy – Rubber band analogy: Q  W+H – Friction analogy: W+f  Q – Work = Force * Distance; Heat measurable Extra heat of shortening Different distances Same Speed Same distance Different speeds Heat released

5. Extra heat of shortening During whole movement – W = P(  L); H=a(  L) (ie: energy = (P+a)  L) – Time to move varies with P (nonlinear dW/dt, dH/dt) – (P+a)DL/Dt empirically linear (P+a) V = b(P 0 -P) H=a(  L) (P+a)V P0P0 b*P 0 b

6. Interpretation Energetic support for empirical hyperbola – Internal viscosity (a V) – External power (P V) (P+a) V = b(P 0 -P) – P  0; V  V max = bP 0 /a – V = b(P 0 -P)/(P+a); V = V max (1-P/P 0 )/(1+P/a) One ‘material property’ for muscle: a≈P 0 /4 Convenient/accurate estimate for V max – Extrapolate linear relation vs hyperbola

7. Lengthening velocities Work done on muscle – Directly stretch viscous element  greater heat rate – Negative work Sudden yield Heat during overload Heat during shortening Isometric heat Overload heat - work Length during overload (56 g) Catastrophic yield (68 g)

8. Length-tension-velocity Length and velocity are not independent Real motions follow trajectories

9. Troubles Instantaneous behavior from dynamic average – Force to accelerate afterload – Force to move muscle’s own mass – Bath viscosity Whole muscle Heat rate depends on length Lengthening

10. Isotonic and isovelocity experiments Servo control – Feed back some sensor data to match a control signal – Nearly instant change in force, length, velocity without acceleration Largely confirm Hill’s results Lutz & al., 2002

11. K.A. Paul Edman (1979) Single fibers – Sarcomere length control via laser – Simultaneous force measurement Servo length control Force transducer Diffraction screen Laser

12. Tension recovery after shortening “Push” fibers together faster than Vmax – Brief period of 0 tension – Distinct recovery of tension dL-dt slope gives V 0 V0V0 System elasticity

13. V 0 depends on length Isometric tension (●) V 0 (o) Apparent V0 rises sharply with passive tension. Elastic recoil? Lateral compression? Apparent V0 falls below Ls=1.6 um.Thick filament-Z disk resistance?

14. Near-zero loads Discontinuity in slope Loads > P 0 Edman 1988

15. Non-steady state forces Rack & Westbury 1974 Whole muscle (distributed stimulation) Triangular length changes First one is different Decay at submaximal stimulations

16. Yield during dynamic motions During phases of constant velocity, force is not constant Two-stage elasticity

17. Fast tension transients Hill’s viscoelastic system is 1 st order Ford, Huxley & Julian, 1977 – Further refined spot follower – Low-impedance moving coil motor Very small, very fast steps – Crossbridge length – Chemical kinetics

18. Step response Instantaneous, elastic recoil Rapid (2 ms), partial recovery Slow (100 ms), complete recovery 100 ms Tension Step size

19. Fast stages Elastic recoil (short range stiffness) – Linear – 6 nm/hs ~ 0.5% length change Rapid recovery – Complex, up to ~1% length change 100 ms T0T0 T1T1 T2T2

20. Interpretation At least two sources of ‘viscosity’ – Fast & slow – In series “True” viscosity – Velocity dependent process – Contrast: elastic element that relaxes

21. Summary Isotonic shortening: hyperbolic force Isotonic lengthening: catastrophic yield Much of the behavior is viscoelastic – Not P ≥ 0.8 P 0 – Not t < 2 ms