A Re-examination of Calcium Activation in the Huxley Cross-Bridge Model

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Abstract
This paper investigates mathematical relations between models of calcium activation kinetics and Huxley-type models of cross-bridge dynamics in muscle. It is found that different calcium-activation schemes lead to the same form of generalized Huxley rate equation with calcium activation (∂n/∂t)−v(∂n/∂x)=rf(α−n)−gn if it is assumed that calcium–troponin interaction rates are fast compared to the rates of transition associated with force-generating cross-bridge states. Calcium affects cross-bridge dynamics by modifying the bonding rate f, but does not affect the number of interacting cross bridges α or the unbonding rate g; this occurs through the appearance in the equation of an activation factor, r, which is a pure function of sarcoplasmic free calcium concentration. In particular, it is shown that both the “tight-coupling” and “loose-coupling” calcium-activation schemes introduced by Zahalak and Ma [1] lead to the same rate equation with the same activation factor; the difference between them appears in the calcium mass-balance equation. While both of these activation models can be made to fit simple twitch and force-velocity data equally well, experimentally observed load-dependent shifts in the free calcium concentration are compatible with the tight-coupling scheme, but not with loose coupling.