The Two-state Cross-bridge Model of Muscle is an Asymptotic Limit of Multi-state Models

The relationship between the two-state model of muscle contraction and multi-state models is examined from the perspective of matched asymptotic expansions, under the assumption that transition rates between attached states are fast compared to those between detached and attached states. A detailed formal analysis of a three-state model reveals that the classic Huxley (1957. Prog. Biophys. Biophys. Chem.7, 225–318) rate equation, as modified for thermodynamic self-consistency by Hill et al. (1975. Biophys. J.15, 335–372), governs the “outer” solution of the three-state equations. Thus, the two-state model remains a valid description of muscle dynamics on physiologically relevant time scales, which are slow compared to millisecond-scale transitions between attached states. But the asymptotic analysis reveals also that the cross-bridge force must be considered to be a nonlinear function of the cross-bridge strain, in contrast to the usual assumption of two-state models. This apparent, or effective, force is determined by both the intrinsic stiffness of the cross-bridge and the equilibrium distribution of cross-bridges among attached states. Further, the asymptotic analysis yields an expression for the energy liberation rate that implies a reduced rate in stretch vs. shortening. Some behaviors of multi-state models that are suggested by the three-state analysis are discussed in qualitative terms.