Sarcomeric determinants of striated muscle relaxation kinetics

Abstract

Ca²⁺ is the primary regulator of force generation by cross-bridges in striated muscle activation and relaxation. Relaxation is as necessary as contraction and, while the kinetics of Ca²⁺-induced force development have been investigated extensively, those of force relaxation have been both studied and understood less well. Knowledge of the molecular mechanisms underlying relaxation kinetics is of special importance for understanding diastolic function and dysfunction of the heart. A number of experimental models, from whole muscle organs and intact muscle fibres down to single myofibrils, have been used to explore the cascade of kinetic events leading to mechanical relaxation. By using isolated myofibrils and fast solution switching techniques we can distinguish the sarcomeric mechanisms of relaxation from those of myoplasmic Ca²⁺ removal. There is strong evidence that cross-bridge mechanics and kinetics are major determinants of the time course of striated muscle relaxation whilst thin filament inactivation kinetics and cooperative activation of thin filament by cycling, force-generating cross-bridges do not significantly limit the relaxation rate. Results in myofibrils can be explained well by a simple two-state model of the cross-bridge cycle in which the apparent rate of the force generating transition is modulated by fast, Ca²⁺-dependent equilibration between off- and on-states of actin. Inter-sarcomere dynamics during the final rapid phase of full force relaxation are responsible for deviations from this simple model.

Appendix. Relaxation kinetics: predictions from models

Models predict that force decays more slowly during relaxation than it develops during activation. This can be most simply explained in terms of a two-state model of the cross-bridge cycle [26] in which Ca²⁺ regulates the formation of force-generating cross-bridges (via a Ca²⁺-dependent apparent rate constant f_app), whereas cross-bridge detachment occurs with apparent rate constant g_app that is independent of [Ca²⁺] [8]. As can be shown both by numerical simulation and mathematical analysis (not shown here), this two-state model gives equivalent relations for the dependence of force kinetics on steady-state force as the three-state model shown in Fig. 1 (described by one Ca²⁺-dependent rate constant for thin filament switch on (k_on) and three Ca²⁺-independent rate constants (k_off, f and g), provided that the switch of the thin filament in the three-state model is a rapid equilibrium (k_on+k_off>>f+g) (see also [9, 24]).

Any perturbation in the equilibrium between force- and non-force-generating states of the cross-bridges is predicted to induce mono-exponential force kinetics with rate constant k_obs=f_app+g_app towards the new steady-state force level F, which is proportional to f_app/(f_app+g_app). Regardless of whether force kinetics are induced by a sudden increase (k_obs=k_ACT) or decrease (k_obs=k_REL) in [Ca²⁺] or by mechanical perturbations during steady-state Ca²⁺ activation (k_obs=k_TR), k_obs is related directly to the final isometric steady-state force (F) approached at the end of the force transient. At zero final force (F=0), f_app by definition equals 0 and hence k_obs=g_app. At maximum Ca²⁺-activated force (F_MAX), f_app reaches its maximum value (f_app^MAX), as does k_obs=f_app^MAX+g_app. After preselection of the values of g_app and f_app^MAX+g_app to match the observed kinetics at very low F and at F_MAX, respectively, the relation (see line in Fig. 7A; see also Fig. 4D) between k_obs and the relative force F_REL (=F/F_MAX) is predefined by $k_{{{\text{obs}}}} = g_{{{\text{app}}}} /{\left( {1 - F_{{{\text{REL}}}} f_{{{\text{app}}}} ^{{{\text{MAX}}}} /{\left( {f_{{{\text{app}}}} ^{{{\text{MAX}}}} + g_{{{\text{app}}}} } \right)}} \right)}$ .

Fig. 7A–D

Force kinetics during activation-relaxation cycles predicted by the two-state cross-bridge model. A Dependence of the observed rate constant of exponential force activation and relaxation kinetics on final Ca²⁺-activated force. The relation (line) is calculated from the equation given in the text using g_app=1.5 s⁻¹ and f_app^MAX=7.5 s⁻¹. The different coloured points superimposed on the relation indicate the k_obs used for calculation of the force transients in the subsequent panels. B Simulation of Ca²⁺-induced force transients ([Ca²⁺] is suddenly increased at time 0 s) to different final Ca²⁺-activated force levels. Each activation is followed by complete relaxation (Ca²⁺ is abruptly removed at time 2 s). C As in B but force relaxation transients are simulated for partial relaxations initiated from maximum Ca²⁺ activation (at 2 s [Ca²⁺] is suddenly reduced to different levels). D Initial parts of the relaxation transients shown in C. Numbers indicate the final steady-state force for each transient

Figure 7B illustrates how relaxation kinetics will relate to activation kinetics in the case Ca²⁺ removal leads to complete relaxation (final force=0). Force will decay with a rate constant g_app irrespective of the initial force F_ACT during Ca²⁺ activation (Fig. 7B). The initial rate of force decay will be given by g_app/F_ACT. Hence, for a physiological contraction-relaxation cycle, force can always be expected to decay with a slower rate constant than that with which it developed during the preceding contraction. Only at low Ca²⁺ activation, when f_app becomes low compared with g_app, will force development kinetics approach the kinetics of full relaxation. The rate constant of force decay can be made faster than the rate of force development, as is the case in myofibrils during the final fast phase of full relaxation, only if g_app increases. Such an increase can be due to changes in the working conditions imposed on sarcomeres and to increased importance of pathways for cross-bridge detachment other than those associated with forward turn-over kinetics.

It might be argued that the two-state model is too simple to describe the real situation in a myofibril. However, even if thin filament switch kinetics take part in rate-limiting force kinetics, this will not change the argument that relaxation should be slower than activation. As thin filament switch kinetics following changes in [Ca²⁺] are determined by k_on+k_off, the switch-off following the fall in [Ca²⁺] should be slower due to the Ca²⁺-dependent reduction of k_on, again resulting in slower force kinetics during relaxation than during activation.

A further prediction of the two-state model (and of any cross-bridge models that do not involve feedback mechanisms of force-generating cross-bridges on thin filament activation) is that, upon partial Ca²⁺ removal, force will decay monoexponentially to a new steady state with the same kinetics as those of the Ca²⁺-induced force development leading to the same final steady-state force (Fig. 7C; see also Fig. 4). Both k_REL and k_ACT will increase equally with the final force level according to k_obs=f_app+g_app in Fig. 7A. In other words, force kinetics only depend on the final [Ca²⁺] but are independent of the initial [Ca²⁺]. The rate constant of force decay will increase in partial relaxations as compared to full relaxation (Fig. 7C) simply because the redistribution between cross-bridge states determined by f_app+g_app is faster with the contribution of f_app than when the regulatory system is completely switched off and f_app is zero.

However, the absolute rate of the force decay is lower in partial relaxations to higher force levels than in full relaxation (Fig. 7D). The reason is that the rate of the force decay reflects the net flux of cross-bridges out of force generating states, i.e. detachment minus attachment. As long as the regulatory system remains partially activated, formation of force-generating cross-bridges by f_app counteract the cross-bridge detachment by g_app. The apparent rate by which cross-bridges leave force-generating states reflected by g_app/F_ACT (refer to the initial slope of the red line in Fig. 7D) is therefore the maximum absolute rate of force decay following a decrease in [Ca²⁺].