Crossbridge model
• Crossbridge biophysics
• Force generation
• Energetics
Crossbridge Cycle
ATP
ADP
Shape change
Pi
Shape change
Animation: Graham Johnson & Ron Vale
Myosin physics
S-1 Fragment
• Globular head
Actin Binding
– Actin binding
– ATP binding
• Filamentous neck
Hinge
ATP cleft
– Flexible
– Light chain binding
• Filamentous tail
– Dimerization
– Oligomerization
Native Myosin
Neck
Laser Trap
• Photon momentum = E/c
• Refraction changes momentum
• 3D Position control
Measuring myosin steps
• Compliant traps
• Low ATP
• Record position
“Step”
Position data:
Brownian
Motion
Many steps:
Actin-myosin chemical scheme
•
•
•
•
•
State/compartment model
Actin-myosin bound/unbound
ATP bound/unbound
ATP/ADP+Pi
Hidden states
Crossbridge Cycle
• Actin catalyzes Pi release
• ATP catalyzes A release
P
AMDP
First cycle:
M
MT
D
AMD
AM
A
M
T
D
P
MDP
T
Repeatable: T
P
Actin
Myosin
ATP
ADP
Pi
D
AM
AMT
AMDP
AMD
AM
MT
MT
MDP
MD
M
T
Shape Changes P
D
Lymn & Taylor 1971
Quenched-flow chemistry
• Reactions in moving medium
– Steady-state relation btw time
and distance
– Measure very fast reactions
Reagent 1
Reagent 2
Mix
ATPPi by
o Actin-myosin
• Myosin alone
Quench
o AM + ATPAMADP + Pi
• M+ATP  MADP + Pi
After an initial burst, actin accelerates reaction
Initial ATP hydrolysis independent of actin,
sustained Rx catalyzed by actin
Actin-myosin dissociated by ATP
• Stopped-flow measurements
• Light scattering by A-M filaments
Reagent 1
– ie, turbidity
Mix
Turbidity
T
AM
AMT
MT
MT
Reagent 2
Detector
AM + ATP  AQuench
+ M●ATP
Lymn & Taylor (1971)
Phosphate release catalyzed by actin
Add actin
• Pi release by fluorescence
• More actinfaster release
75 s-1
T
AM
P
AMT
AMDP
AMD
MT
MDP
MD
1-2 s-1 P
Heeley & al (2002)
Chemical summary
• Myosin is an ATPase with large shape differences
– M-MATP
– MATP-MADP
– MADP-M
• Filamentous actin facilitates Pi release
• ATP facilitates f-actin release
Relate chemistry to force
• AF Huxley 1957 Crossbridge model
• Two states: myosin attached or myosin not
attached
• Force results from elasticity of individual
crossbridges
• Myosin interacts with actin at discrete sites
• Attachment and detachment rates are
position dependent
Cartoon: capture the minimal process
• Modeling crossbridge attachment
– Imagine Pi release & power stroke instantaneous
– A + M  AM + Force with rate constant f
– AM  A + M●ATP with rate constant g
• Think about behavior of single crossbridge
• Imagine many crossbridges spanning all configs
Thick filament
Rigor
State
x=0
Thin filament
Max
Attachment
length
x=h
Mathematics
• Two states: myosin attached (n) or myosin not
attached (1-n)
dn
 (1  n ) f ( x )  ng ( x )
dt
dn
 f ( x)  n f ( x)  g ( x)
dt
• Force results from elasticity of individual
crossbridges
– Individual: Fb=kx
– All: F  k  x  n  dx
Mathematical features
• First order: exponential
• Steady state
– dn/dt  0
dn
– dt  f ( x )  n f ( x )  g ( x ) 
– n(x) = f/(f+g)
Crossbridge attachment rate
• Relate crossbridge physics to x
• Energy released by binding
• Energy required for deformation
“Energy”2.0
1.5
f
Binding
1.0
0.5
Deformation
0.0
-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0
-0.5
Position
(X)
-1.0
-1.5
-2.0
Prohibit
attachment
x>h
An unbound myosin is positioned just at “x=1”
and can drop onto actin without any bending
f1
0.0
0
h
Position (X)
Crossbridge detachment rate
• Release deformation energy
• Release conformation energy
– Discrete change x<0
Binding
2.0 “Energy”
Deformation
g
g3
1.5
1.0
0.5
0.0
-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0
-0.5
Position
g1
(X)
-1.0
-1.5
-2.0
A bound myosin is positioned just at “x=0”
and any displacement requires bending
0.0
0
h
Position (X)
Steady state crossbridge attachment
• n(x) = f/(f+g)
• x<0 ; x>h n=0
– 0<x<h  n=f1/(f1+g1)
• Force=∫k∙n∙xdx
– k(f1/(f1+g1))(h2/2)
– Crossbridge stiffness
– Ratio of f:g
f1/(f1+g1)
g3
f1
g1
0.0
0
h
Position (X)
Crossbridge behavior during shortening
• Since n=n(x), dn/dt depends on dx/dt
dn n x
n

 v
dt x t
x
v
n
 f ( x )   f ( x )  g ( x ) n
x
• Crossbridge moving in from x>>h
– No chance to attach until x=h
– High probability to attach, but limited time
– Probability to attach decreases to x=0, but time rises
– Rapid detachment x<0
Crossbridge distribution
• V=0
• V= Vmax/3
– Uniform attachment
– Mean x = h/2
– No saturation
– Mean x
1
0.8
0.8
0.6
0.6
n
0.4
0.4
0.2
0.2
0
x>0force > 0
n
1
0
-0.01
0
x
0.01
-0.01
0
x
0.01
x>0force < 0
These crossbridges resist shortening
Dynamic response
Transition to lengthening
• Fully attached crossbridges get over-stretched
• Unattached crossbridges dragged in from left
Faster lengthening
Transition to shortening
• Fully attached crossbridged get compressed
• Unbound crossbridges dragged in from right
Faster shortening
Damping without viscosity
• Qualitative (and quantitative) results of
crossbridge and Hill models similar
– Even the math: dL/dt = F/b - k/b L
–
dn/dt = f - (f+g)n
• Mechanisms behind the models are very
different
– Crossbridge predicts/validated by biochemistry
Energy prediction
• Energy liberation
– Power from P*v
– Heat from dn/dt: increased by shortening
Shortening Vopt
1
0.8
Accelerated release
n
0.6
Accelerated binding
0.4
0.2
0
-0.01
0
x
0.01
Total energy rate
– Hill’s data
o Huxley’s model
Issues
• Fast length changes
– < 2 ms (500 s-1)
– Violates “one process”
assumption
T0
T2
100 ms
T1
• Lengthening
– Too many very long x-bridges
• Residual force enhancement
Model
• Double-hyperbolic F-V
Data
Summary
• Crossbridge cycle:
– AM+TA+MTA+MDPAMDPAMDAM
• Attachment of elastic crossbridges explains
force-velocity relationship
– Reduced attachment during shortening
– Shorter length of attachment
• Higher state models fit better