Transcript Force Velocity - School of Applied Physiology

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
Afterloaded contraction
• A. V. Hill
– Time to move fixed distance against known inertia
Magnetic release
mechanism
Counterweights
(Inertia)
Position
indicator
Safety stops/timer
Muscle
Hyperbolic force-velocity
• Purely phenomenolocial/empirical
Shortening velocity (cm/s)
– (x-a)(y-b)=c
– a, b are coordinates of asymptotes
Load (g)
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
Heat released
Different distances
Same Speed
Same distance
Different speeds
Extra heat of shortening
• During whole movement
– W = P(DL); H=a(DL) (ie: energy = (P+a) DL)
– Time to move varies with P (nonlinear dW/dt, dH/dt)
– (P+a)DL/Dt empirically linear
• (P+a) V = b(P0-P)
b*P0
(P+a)V
H=a(DL)
b
P0
Interpretation
• Energetic support for empirical hyperbola
– Internal viscosity (a V)
– External power (P V)
• (P+a) V = b(P0-P)
– P0; VVmax = bP0/a
– V = b(P0-P)/(P+a); V = Vmax (1-P/P0)/(1+P/a)
• One ‘material property’ for muscle: a≈P0/4
• Convenient/accurate estimate for Vmax
– Extrapolate linear relation vs hyperbola
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
Catastrophic yield (68 g)
Length during overload
(56 g)
Length-tension-velocity
• Length and velocity are not independent
• Real motions follow trajectories
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
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
K.A. Paul Edman (1979)
• Single fibers
– Sarcomere length control via laser
– Simultaneous force measurement
Diffraction
screen
Force
transducer
Servo length
control
Laser
Tension recovery after shortening
• “Push” fibers together faster than Vmax
– Brief period of 0 tension
– Distinct recovery of tension
• dL-dt slope gives V0
V0
System elasticity
V0 depends on length
Apparent V0
falls below Ls=1.6
um.Thick filament-Z
disk resistance?
V0 (o)
Isometric tension (●)
Apparent V0 rises
sharply with passive
tension. Elastic recoil?
Lateral compression?
Near-zero loads
• Discontinuity in slope
• Loads > P0
Edman 1988
Non-steady state forces
• Rack & Westbury 1974
• Whole muscle (distributed stimulation)
• Triangular length changes
First one is different
Decay at submaximal stimulations
Yield during dynamic motions
• During phases of constant velocity, force is not
constant
• Two-stage elasticity
Fast tension transients
• Hill’s viscoelastic system is 1st order
• Ford, Huxley & Julian, 1977
– Further refined spot follower
– Low-impedance moving coil motor
• Very small, very fast steps
– Crossbridge length
– Chemical kinetics
Step response
Tension
• Instantaneous, elastic recoil
• Rapid (2 ms), partial recovery
• Slow (100 ms), complete recovery
Step size
100 ms
Fast stages
• Elastic recoil (short range stiffness)
– Linear
– 6 nm/hs ~ 0.5% length change
• Rapid recovery
– Complex, up to ~1% length change
T0
T2
100 ms
T1
Interpretation
• At least two sources of ‘viscosity’
– Fast & slow
– In series
• “True” viscosity
– Velocity dependent process
– Contrast: elastic element that relaxes
Summary
• Isotonic shortening: hyperbolic force
• Isotonic lengthening: catastrophic yield
• Much of the behavior is viscoelastic
– Not P ≥ 0.8 P0
– Not t < 2 ms