# Sliding filament theory Andrew F. Huxley's 1957 model

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Sliding filament theory: Andrew F. Huxley’s 1957 model

A theoretical model should include or predict the
following empirical observations.

When active muscle shortens, the
relationship between force T
and velocity v is given by Hill’s
equation:
(T + a )(v + b) = constant
where
a   b
=     =c
T0 vmax
and c is in the range 0.15 to 0.25.
Fenn’s effect: an active muscle that shortens and does work
produces more heat than when it contracts isometrically.

Hill (1938) approximated this relationship by

Rate of heat production        v
=c +c
2
,
T0 vmax              vmax
where c is the same parameter as in the force-velocity
relationship.
There is a discontinuity in
the slope of the force-
velocity relationship at zero
velocity. Slope for
lengthening is about 6 times
the slope for shortening.

Active muscle “yields” when
the force exceeds about
1.8T0.
Restrictions of the Huxley model.

Only tension produced by active contraction is considered.

The number of potential
actin and myosin binding
sites are assumed to be
fixed. Effectively, this
means that sarcomere
length is within the plateau
range of the force-length
curve.

The muscle is assumed to
be fully activated at all
times.
Hypotheses of the Huxley model.
• Myosin has side pieces M that can slide along the main
backbone of the filament in either direction from an
equilibrium position.

• Actin has fixed binding sites A.
• A and M can bind to form AM. When bound, a force kx is
generated, where spacing distance x is defined as
x = (position of A) − (equilibrium position of M ).
• The rate constant for binding, f(x), and the rate constant for
unbinding, g(x), both depend on x.

• Breaking the bond between A and M requires energy
(splitting of ATP).
• There are many A and M pairs that have the potential to
bind. The x values for these pairs are uniformly distributed
(over some large range of values).
Rate equations

Consider all A and M pairs whose spacing distance is between
x and x + dx. Let n(x,t) be the fraction of such pairs that are
bound at time t.

If muscle length is constant, then x is constant for each A and
M pair, so that
∂n( x, t )
= f ( x)[1 − n( x, t )] − g ( x)n( x, t ) (Huxley's equation)
∂t
In general, if muscle velocity is v(t) (v > 0 for shortening), then
∂n( x, t )         ∂n( x, t )
− v(t )            = f ( x)[1 − n( x, t )] − g ( x)n( x, t )
∂t                 ∂x
(Huxley - Zahalak equation)
Huxley (1957) assumed constant force and velocity,
corresponding to the equilibrium solution n(x,t)=n(x):
∂n( x)
−v        = f ( x)[1 − n( x)] − g ( x)n( x).
∂x
The total rate of energy liberation per unit volume is
me ∞                        me ∞
E=
l ∫−∞ f ( x)[1 − n( x)]dx = l ∫−∞ g ( x)n( x)dx,
where e is energy liberated per M site per cycle, m is the
number of M sites per unit volume, and l is the spacing
between A sites.
The tension per cross-sectional area is

msk       ∞
P=
2l   ∫−∞
n( x) xdx,

where s is the sarcomere length.

The rate of doing mechanical work per unit volume is PV,
where V = 2v/s is normalized velocity in terms of half-
sarcomere lengths per second.

The rate of liberation of heat per unit volume is E – PV.
Choices for rate functions:
 f1 x / h if 0 < x < h,             g2       if x < 0,
f ( x) =                           g ( x) = 
 0         otherwise.               g1 x / h if x > 0.

It is critical that these function are not symmetric around x=0.

g1    3
= ,
f1 + g1 16
g2
= 3.919.
f1 + g1
Solutions for n(x)
For v = 0 and 0 < x < h :
f1   13
n( x ) =        = .
f1 + g1 16

For v > 0 and x < 0 :
∂n( x)
−v        = − g 2 n( x )
∂x
 g2 x 
⇒ n( x) = n(0) exp         .
 v 

For v = vmax :
msk         ∞
P=
2l     ∫−∞
n( x) xdx = 0.
Comparison with Hill’s equations
Force-velocity relationship:
a   b
(T + a)(v + b) = constant     and           =     =c
T0 vmax
⇒ ( P / P0 + c)(V / Vmax + c) = c(1 + c)
Normalized rate of heat liberation per unit volume:
E − PV        V
=c +c
2
.
P0Vmax       Vmax
Huxley fit Hill’s equations with c=1/4 by setting
g1    3      g2                w 3
= ,            = 3.919,     = ,
f1 + g1 16    f1 + g1             e 4
where w=kh2/2 is the maximum work done in 1 cycle at 1 site.
Force-velocity relationship     Rate of heat liberation

P     V                     E − PV        V
 + c      + c  = c(1 + c)          =c +c
2
.
P     V                     P0Vmax       Vmax
 0    max     
Other forms of Hill’s equations follow from the force-velocity
and heat liberation equations.

Rate of heat liberation        Total rate of energy liberation
E − PV        V                                E
=c +c
2
.
P0Vmax       Vmax                            P0Vmax
Model predictions for lengthening muscle

When active muscle
lengthens (V < 0), there are
bound sites with x > h,
leading to increased tension.
Force-velocity curve shows qualitative agreement with data:

Slope of force-velocity curve is discontinuous at 0 velocity.
Slope for lengthening is f1/g1 = 4.33 times slope for shortening.
Muscle yields when force is
(f1 + g1)/g1 = 5.33 times P0.
What does Huxley’s model suggest about the relationship
between the amount of actin-myosin overlap and Vmax?
Further development of Huxley-like cross-bridge models

Huxley and Simmons (1971): Change in tension following
sudden changes in length.
A model with two attached
states was used to
account for the data.

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