A kinetic model that explains the effect of inorganic phosphate on the mechanics and energetics of isometric contraction of fast skeletal muscle

Abstract

A conventional five-step chemo-mechanical cycle of the myosin-actin ATPase reaction, which implies myosin detachment from actin upon release of hydrolysis products (ADP and phosphate, Pi) and binding of a new ATP molecule, is able to fit the [Pi] dependence of the force and number of myosin motors during isometric contraction of skeletal muscle. However, this scheme is not able to explain why the isometric ATPase rate of fast skeletal muscle is decreased by an increase in [Pi] much less than the number of motors. The question can be solved assuming the presence of a branch in the cycle: in isometric contraction, when the force generation process by the myosin motor is biased at the start of the working stroke, the motor can detach at an early stage of the ATPase cycle, with Pi still bound to its catalytic site, and then rapidly release the hydrolysis products and bind another ATP. In this way, the model predicts that in fast skeletal muscle the energetic cost of isometric contraction increases with [Pi]. The large dissociation constant of the product release in the branched pathway allows the isometric myosin-actin reaction to fit the equilibrium constant of the ATPase.

Scheme 1.

Conventional chemo-mechanical cycle.

Figure 1.

Responses of the simulation. (a) Effect of [Pi] on the occupancy of the cross-bridge states (as defined by the colours) during isometric contraction, calculated with scheme 1. The attached AM state is not significantly populated and is omitted. Pink line, M·ATP; green line, A-M·ADP·Pi; blue line, AM′·ADP·Pi; red line, AM′·ADP. (b) Pi-dependence of isometric force (relative to the force in control solution, 1 mM Pi). Observed relation, filled circles (data from figs 1f and 6a of Caremani et al. 2008). Simulated relations: blue line (scheme 1) and red line (scheme 2). (c) Pi-dependence of ATPase rate: observed relation, open symbols (circles, Bowater & Sleep 1988; triangles, Potma et al. 1995; diamonds, Potma & Stienen 1996). Simulated relations: blue line (scheme 1) and red line (scheme 2). Values relative to those in control solution. (d) Time course (calculated with scheme 1) of the number of attached cross-bridges (the sum of the occupancy of AM′·ADP·Pi and AM′·ADP) following the end of the unloaded shortening in control solution ([Pi], 1 mM) and at different [Pi] (3.5, 6, 11, 16 and 21 mM). Values on the ordinate are normalized for the steady isometric value at the respective [Pi]. (e) Superposition of the time courses (calculated with scheme 1) of the number of attached cross-bridges following either a step of [Pi] from 0 to 11 mM (dashed line) or a period of unloaded shortening in 11 mM Pi (continuous line). For facilitating the comparison, in the inset, the data are plotted after making them relative to their respective maximum change and applying the operation (1 − y) to data (y) for the rise in number of cross-bridges following unloaded shortening. (f) Pi dependences of r_F and r_Pi. r_F relation: observed (filled circles) from fig. 6c of Caremani et al. (2008); simulated: scheme 1 (blue continuous line) and scheme 2 (red continuous line). r_Pi relation: observed (open squares) from fig. 6 of Dantzig et al. 1992; simulated: scheme 1 (blue-dashed line) and scheme 2 (red-dashed line). The simulated r_Pi relations lie slightly below the observed relation probably because of the different procedure to estimate the rate constant of the force transient. While in the experimental records of Dantzig et al. (1992), the initial lag does not contribute to the estimate of the rate constant, in our simulation the lag is incorporated in the estimate.

Scheme 2.

Unconventional chemo-mechanical cycle.

Figure 2.

Responses of the simulation with scheme 2. (a) Effect of [Pi] on the occupancy of the cross-bridge states (as defined by the colours: pink line, M·ATP; green line, A-M·ADP·Pi; dark blue line, AM′·ADP·Pi; red line, AM′·ADP; light blue line, M′·ADP·Pi) during isometric contraction. (b) Pi dependence of fluxes through the conventional pathway (dot-dashed line) and the branched pathway (dashed line) and simulated ATPase rate (continuous line, given by the sum of the two fluxes). (c) Pi-dependence of the ATPase rate per myosin head (dashed line) and per attached myosin head (continuous line) calculated as described in the text.