Mechanokinetics of rapid tension recovery in muscle: the Myosin working stroke is followed by a slower release of phosphate

Abstract

Crystallographic and biochemical evidence suggests that the myosin working stroke that generates force in muscle is accompanied by the release of inorganic phosphate (Pi), but the order and relative speed of these transitions is not firmly established. To address this problem, the theory of A. F. Huxley and R. M. Simmons for the length-step response is averaged over elastic strains imposed by filament structure and extended to include a Pi-release transition. Models of this kind are applied to existing tension-recovery data from length steps at different phosphate concentrations, and from phosphate jumps upon release of caged phosphate. This body of data is simulated by the model in which the force-generating event is followed by Pi release. A version in which the Pi-release transition is slow provides a better fit than a version with rapid Pi release and a slow transition preceding force generation. If Pi is released before force generation, the predicted rate of slow recovery increases with the size of the step, which is not observed. Some implications for theories of muscle contraction are discussed.

Scheme 1

FIGURE 1

Hypothetical Gibbs-energy landscape seen by a bound myosin head as a function of angle θ between lever arm and filament, and elastic strain x in the prestroke state (Eisenberg et al., 1980; Wood and Mann, 1981). The prestroke state (2) and poststroke state (3) are defined by sharp angular potential wells. At intermediate angles, the lever arm is assumed free to rotate unless tethered to the thick filament, when rotation occurs against the elastic strain energy for a lever-arm of length R. The working stroke to state 3 is here set at 10 nm. Note that strain creates an additional energy barrier to the force-generating transition 2→3 for whereas more negative strains create an additional barrier to the reverse transition. The resulting kinetics is described in the main text.

FIGURE 2

(A) Phase-2 tension transients, normalized to unity, calculated from the two-state model with strain averaging for various length steps. With constant and all strain-dependent kinetics in the forward rate Eq. 4 predicts universal curves as functions of scaled time and scaled length steps = −10, −9,…, −1, 1,…, 9, 10, where E₁(x) is the exponential integral of order 1 (Press et al., 1992). (B) Step dependence of the rate of recovery, to 1/e of the total change, from the above curves (solid line). The function (dotted line) is predicted in the absence of sarcomere averaging (Huxley and Simmons, 1971). The solid curve approximates data of Ford et al. (1977), (♦) for the frog at 4°C, scaled with and (kh = 4.8 pN), except for large releases where recovery is very fast. Rate constants determined experimentally as inverse half-times have been multiplied by 0.693. The dashed curve shows the effect of using the bounded form of (see main text) with K₂ = 20; this curve falls below the solid curve for

Scheme 3

FIGURE 3

Rates of the faster and slower phases (the eigenvalues of Eqs. 8 and 11) versus reduced strain in versions A and B of the three-state model with (Version A) and or for 1 mM or 25 mM [Pi], respectively. Both rates always increase with phosphate and decrease with step size. (Version B) and and 0.4 at low and high Pi as above. The slower rate increases with step size when This condition is met at high phosphate [Pi] > 6.7 mM).

Scheme 4

Scheme 5

FIGURE 4

Biphasic tension recovery from a length step in the two versions of the three-state model, showing, from top to bottom: i), tension recovery at low Pi for 10 different stretches and releases as in Fig. 2 A; ii), rates of the faster phase (logarithmic left-hand vertical scale) and slower phase (right-hand vertical scale) versus reduced step size at low and high Pi; and iii), the fractional amplitude of the fast phase versus step size.

FIGURE 5

Phosphate dependence of the rate constants shown in Fig. 4, for a large stretch and large release Phosphate concentration is represented by the rate of binding (say 20 s⁻¹ for 1 mM Pi) in version A, or the binding constant (say 0.1 at 1 mM Pi) in version B. Values of kinetic parameters are given under Fig. 3.

FIGURE 6

Measured rates of tension change after photo release of caged Pi as a function of final Pi concentration (Dantzig et al., 1992; coordinates and standard errors supplied by courtesy of J. Dantzig), fitted to hyperbolic functions passing through the origin (Eq. 12). Unweighted least-squares fitting gives at 10°C, and at 20°C. For the 10°C data, error-weighted fitting gives similar results. Residuals as shown are positive at low Pi.

FIGURE 7

Predictions of versions A and B for caged-Pi experiments, for 10 upward phosphate jumps from nominally 1 mM Pi, generated by increasing the rate of Pi binding (A) or Pi affinity (B) by successive powers of Values of all other rate constants are given under Fig. 3. (Upper panels) Tension transients, normalized to the total change after equilibration. The slower phase is predominant (see main text). (Lower panels) Empirical rates of change (to 1/e = 0.37) versus final [Pi] as measured by or with various values of Computed coordinates are shown as crosses, and the lines are fitted hyperbolic functions (Eqs. 13A or 13B). For version A, and ; and in increasing order of For version B, and similarly, whereas throughout.