A re-interpretation of the rate of tension redevelopment (k(TR)) in active muscle

Abstract

A slackening to zero tension by large length release (~20%) and a restretch of active muscle fibres cause a fall and a redevelopment in tension. According to the model of Brenner (Proc Natl Acad Sci USA 85(9):3265-3269, 1988), the rate constant of tension redevelopment (k TR) is the sum of attachment and detachment rate constants, hence is limited by the fast reaction. Here we propose a model in which, after restretch, cross-bridges cycle many times by stretching series elastic elements, hence k(TR) is limited by a slow reaction. To set up this model, we made an assumption that the stepping rate (v) decreases linearly with tension (F), which is consistent with the Fenn effect. The distance traveled by a cross-bridge stretches series elastic elements with stiffness σ. With these assumptions, we set up a first order differential equation, which results in an exponential time course with the rate constant k(TR) = ση(0)ν(0)(1 - λ)/F(1), where λ = ν(1)/ν(0), η = step size, the subscript 0 indicates unloaded condition, and the subscript 1 indicate isometric condition. We demonstrate that the ATP hydrolysis rate (=[myosin head]/ν(0)) is proportionate to k(TR) as the ambient temperature is changed, and that the published data fit to this relationship well if λ = 0.28. We conclude that k(TR) is limited by the cross-bridge turnover rate; hence it represents the rate constant of the slowest reaction of the cross-bridge cycle, i.e. the ADP isomerization step before ADP is released.

Fig. 1

A graphic representation of the proposed cross-bridge model, in which cross-bridges cycle many times in the tension recovery period by stretching series elastic elements. Z-line is to the right, and M-line to the left

Fig. 2

Length (in a) and tension (in b and c) time courses. A rabbit psoas single muscle fibre was activated in the standard activating solution at 20 °C. When the tension plateau was reached, the length was suddenly released by 20 % L₀ and restretched to L₀ after 50 ms. The solid lines are the actual record, and the dotted lines are exponential fits (one exponential fit in b, and two exponential fit in c) with the following parameters: b k_TR = 8 s^–1, A = 0.93 mN; c k_TR(slow) = 5.7 s^–1, A_slow = 0.58 mN, k_TR(fast) = 16.3 s^–1, A_fast = 0.42 mN; where k_TR is the rate constant and A is its amplitude

Fig. 3

A graphic representation of the Fenn effect (Eq. 4 in the text), i.e. the rate of stepping (v) becomes increasingly less as force develops

Fig. 4

The effect of temperature on fast and slow rate constants of k_TR in rabbit psoas fibres

Fig. 5

Correlation between the ATP hydrolysis rate (ordinate) and the slow component of k_TR (abscissa) in rabbit psoas fibres in the standard activating solution as the temperature is changed between 10 °C and 25 °C. The ATPase data were taken from our earlier publication (Zhao and Kawai, 1994), and normalized to that at 20 °C

Fig. 6

Correlation between the apparent rate constant 2πa (sinusoidal analysis) and k_TR as temperature is changed between 10 °C and 25 °C in rabbit psoas fibres in a, or different muscle fibres (psoas open circle, EDL open square, TA filled triangle) are compared at 20 °C in b. The points for EDL and TA overlap. From the regression line in a, Eq. 20 was deduced as the empirical relationship. The same line is drawn in b