Constitutive equations of skeletal muscle based on cross-bridge mechanism

The statistical mechanics of cross-bridge action is considered in order to develop constitutive equations that express fiber tension as a function of degree of activation and time history of speed of contraction. The kinetic equation of A.F. Huxley (1) is generalized to apply to the partially activated state. The rate parameters of attachment and detachment, and cross-bridge compliance are assumed to be step functions of extension, x, with a finite number of discontinuities. This assumption enables integration of the kinetic equation and its moments with respect to x resulting in analytic equations from which x has been eliminated. When the constants in the rate parameters and the force function are chosen so that Hill's force-velocity relation and features of the transient kinetic and tension data can be fitted, the resulting cross-bridge mechanism is quite similar to the one proposed by Podolsky et al. (2). Because the derived constitutive equations simplify mathematical analysis, the influence of various cross-bridge parameters on the mechanical behavior of muscle fibers may be evaluated. For example (a) instantaneous elastic response (T0-T1) and the magnitude of rapid recovery (T2-T1) after a step length change can be adequately explained when the rate of attachment is assumed high for positive x. In that case T2 corresponds to the force generated by cross-bridges in the region of negative x. (b) Kinetic transients occur as a result of the jumps that exist in the distribution of attached cross-bridges during the isometric state. Because of the hyperbolic nature of the kinetic equation, these jumps propagate in the--x direction causing rapid changes in the speed of contraction. (c) When the number of actin sites available for attachment is assumed to depend on the degree of activation, computational results indicate that the speed of shortening is insensitive to the degree of activation at each relative load. (d) It is shown that during sinusoidal oscillation, the mean and second-order harmonics of the experimental force-time curve are strongly dependent on cross-bridge parameters. Therefore, significant information may be lost when the data is expanded into Fourier series and only the first term is considered.