5.4 Mechanical Properties of Contractile Fibers
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At the molecular level, this behavior has been explained by the degree of overlap
between actin and myosin filaments, with more overlap allowing the formation of
more cross-bridges. As a muscle is stretched, the overlap increases to a peak at an
optimal stretch and then begins to decrease (Fig. 5.7b).
The second type of experiment, called a quick-release test, is used to study
how tension affects contraction velocity. For skeletal muscle, shortening velocity
is measured by first tetanizing a muscle held isometrically, then suddenly releasing
one end (by removing a clamp), allowing the muscle to contract against a known
force. If the applied force f is smaller than the maximum contractile force f max ,
the muscle begins to shorten immediately, and the initial shortening velocity v is
measured. After conducting this test for several loads, plotting v versus f yields a
hyperbolic curve (Fig. 5.7c) that can be described by Hill’s equation (Fung 1993)
(v + b)(f + a) = b(f max + a),
(5.20)
where a and b are constants. Note that the maximum force f = f max occurs when
v = 0, corresponding to isometric contraction.
For cardiac muscle, which cannot tetanize, the end is released at various times
during a twitch. Plotting v as a function of the force at the time of release yields
similarly shaped curves. Notably, experiments also reveal that the same lengthtension and force-velocity relationships are reasonable approximations for smooth
muscle (Fung 1997).
5.4.2 Time-Varying Elasticity
Common experience tells us that muscles stiffen when they contract, and experiments suggest that this behavior applies to CFs in general. During isometric
contraction, for example, the tension increases at a fixed strain, reflecting an increase
in effective stiffness. The reason for this behavior lies in the formation of crossbridges connecting actin and myosin filaments. When an active CE is stretched,
cross-bridge elasticity resists the tension that pulls the filaments apart. As the
number of active cross-bridges increases, the force needed to stretch the CE a given
amount increases and, therefore, CE stiffness increases.
The intrinsic change in CE stiffness during contraction represents a timedependent change in material properties, which we call time-varying elasticity.
The material properties depend on the value of the contraction ratio K, which is a
time-dependent measure of the degree of contraction.
To lay the groundwork for developing constitutive relations that capture the
fundamental active behavior of a CE, we first synthesize the above experimental
results into a consistent picture. For convenience, the results are described in terms
of the active first Piola-Kirchhoff stress P a0 (active force divided by undeformed
cross-sectional area in the passive state), as well as the total stretch ratio λ relative
to the passive ZSS. Subscript zero on the stress indicates a state of isometric
221
At the molecular level, this behavior has been explained by the degree of overlap
between actin and myosin filaments, with more overlap allowing the formation of
more cross-bridges. As a muscle is stretched, the overlap increases to a peak at an
optimal stretch and then begins to decrease (Fig. 5.7b).
The second type of experiment, called a quick-release test, is used to study
how tension affects contraction velocity. For skeletal muscle, shortening velocity
is measured by first tetanizing a muscle held isometrically, then suddenly releasing
one end (by removing a clamp), allowing the muscle to contract against a known
force. If the applied force f is smaller than the maximum contractile force f max ,
the muscle begins to shorten immediately, and the initial shortening velocity v is
measured. After conducting this test for several loads, plotting v versus f yields a
hyperbolic curve (Fig. 5.7c) that can be described by Hill’s equation (Fung 1993)
(v + b)(f + a) = b(f max + a),
(5.20)
where a and b are constants. Note that the maximum force f = f max occurs when
v = 0, corresponding to isometric contraction.
For cardiac muscle, which cannot tetanize, the end is released at various times
during a twitch. Plotting v as a function of the force at the time of release yields
similarly shaped curves. Notably, experiments also reveal that the same lengthtension and force-velocity relationships are reasonable approximations for smooth
muscle (Fung 1997).
5.4.2 Time-Varying Elasticity
Common experience tells us that muscles stiffen when they contract, and experiments suggest that this behavior applies to CFs in general. During isometric
contraction, for example, the tension increases at a fixed strain, reflecting an increase
in effective stiffness. The reason for this behavior lies in the formation of crossbridges connecting actin and myosin filaments. When an active CE is stretched,
cross-bridge elasticity resists the tension that pulls the filaments apart. As the
number of active cross-bridges increases, the force needed to stretch the CE a given
amount increases and, therefore, CE stiffness increases.
The intrinsic change in CE stiffness during contraction represents a timedependent change in material properties, which we call time-varying elasticity.
The material properties depend on the value of the contraction ratio K, which is a
time-dependent measure of the degree of contraction.
To lay the groundwork for developing constitutive relations that capture the
fundamental active behavior of a CE, we first synthesize the above experimental
results into a consistent picture. For convenience, the results are described in terms
of the active first Piola-Kirchhoff stress P a0 (active force divided by undeformed
cross-sectional area in the passive state), as well as the total stretch ratio λ relative
to the passive ZSS. Subscript zero on the stress indicates a state of isometric