Crossbridge model

• Crossbridge biophysics• Force generation• Energetics

Crossbridge CycleATP

Pi

ADP

Shape change

Shape change

Animation: Graham Johnson & Ron Vale

Myosin physics

• Globular head– Actin binding– ATP binding

• Filamentous neck– Flexible– Light chain binding

• Filamentous tail– Dimerization– Oligomerization

Actin Binding

ATP cleftHinge

Neck

S-1 Fragment

Native Myosin

Laser Trap• Photon momentum = E/c• Refraction changes momentum• 3D Position control

Measuring myosin steps• Compliant traps• Low ATP• Record position

Position data:

Many steps:

BrownianMotion

“Step”

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

AM AMT AMDP AMD AM

MT MDPMT MD M

T

T P

P D

D

AMDP AMD AM

MT MDPM

T

PP DD

A ActinM MyosinT ATPD ADPP Pi

Shape ChangesLymn & Taylor 1971

First cycle:

Repeatable:

Quenched-flow chemistry

• Reactions in moving medium– Steady-state relation btw time

and distance– Measure very fast reactions

Reagent 1 Reagent 2

Mix

ATPPi byo Actin-myosin• Myosin alone

o AM + ATPAMADP + Pi• M+ATP MADP + Pi

Quench

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

– ie, turbidity

AM + ATP A + M●ATP

Turbidity

Reagent 1 Reagent 2

Mix

Quench

Detector

Lymn & Taylor (1971)

AM AMT

MTMT

T

Phosphate release catalyzed by actin

• Pi release by fluorescence• More actinfaster release

Heeley & al (2002)

AM AMT AMDP AMD

MT MDP MD

T P

P

Add actin

75 s-1

1-2 s-1

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

Thin filament

Rigor State

x=0

Max Attachment length

x=h

Mathematics

• Two states: myosin attached (n) or myosin not attached (1-n)

• Force results from elasticity of individual crossbridges– Individual: Fb=kx

– All:

)()()1( xngxfndt

dn

)()()( xgxfnxfdt

dn

dxnxkF

Mathematical features

• First order: exponential• Steady state

– dn/dt 0– – n(x) = f/(f+g)

)()()( xgxfnxfdt

dn

Crossbridge attachment rate

• Relate crossbridge physics to x• Energy released by binding• Energy required for deformation

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Position (X)

Binding

Deformation

“Energy”

An unbound myosin is positioned just at “x=1” and can drop onto actin without any bending

0 h0.0

f1

Position (X)

f

Prohibit attachment x>h

Crossbridge detachment rate

• Release deformation energy• Release conformation energy

– Discrete change x<0

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Position (X)

BindingDeformation“Energy”

0 h0.0

g1

g3

Position (X)

g

A bound myosin is positioned just at “x=0” and any displacement requires bending

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 0 h

0.0

g1

g3

Position (X)0 h

0.0

f1

Position (X)

f1/(f1+g1)

Crossbridge behavior during shortening

• Since n=n(x), dn/dt depends on dx/dt

• 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

nxgxfxfx

nv )()()(

x

nv

t

x

x

n

dt

dn

Crossbridge distribution

• V=0– Uniform attachment– Mean x = h/2

• V= Vmax/3– No saturation– Mean x

-0.01 0 0.010

0.2

0.4

0.6

0.8

1

x

n

-0.01 0 0.010

0.2

0.4

0.6

0.8

1

xn

x>0force > 0

x>0force < 0These crossbridges resist shortening

Dynamic response

Transition to lengthening

• Fully attached crossbridges get over-stretched• Unattached crossbridges dragged in from left

Faster lengthening

• Fully attached crossbridged get compressed• Unbound crossbridges dragged in from right

Transition to shortening

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

-0.01 0 0.010

0.2

0.4

0.6

0.8

1

x

n

Shortening Vopt

Accelerated binding

Accelerated release

Total energy rate

– Hill’s datao Huxley’s model

Issues

• Fast length changes– < 2 ms (500 s-1)– Violates “one process”

assumption

• Lengthening– Too many very long x-bridges

• Residual force enhancement• Double-hyperbolic F-V

100 ms

T0

T1

T2

ModelData

Summary

• Crossbridge cycle:– AM+TA+MTA+MDPAMDPAMDAM

• Attachment of elastic crossbridges explains force-velocity relationship– Reduced attachment during shortening– Shorter length of attachment

• Higher state models fit better