The equations for change in the fraction of strongly attached cross-bridges with time (\(As\left(t\right))\) for the kinetic schemes in Fig. 1a–c are given below (Eqs. 14, 15) With all cross-bridges initially in the MDP state (assumed at the onset of contraction from zero force and force redevelopment) the following equation applies:

$$As\left(t\right)=\frac_}_}\left(1-^_t}\right)$$

(14)

here, kon is the attachment rate constant and kAS is a function of kon and other parameter values (see Table 3). Because force, in all the schemes in Fig. 1a–c, is proportional to \(As\left(t\right)\) and all other rate constants are assumed to be much greater than (kon + kon-) the rate constant kAS would approximate the rate constant, ktr for the isometric tension rise at full activation such as in force redevelopment.

The alteration in the fraction of strongly attached cross-bridges after a sudden change in [Pi] is also governed by the rate constant kAS in all schemes in Fig. 1a–c according to the following equation:

$$As\left(t\right)=\left(\frac_}_^}-\frac_}_^}\right)^_t}+\frac_}_^}$$

(15)

here, the superscripts S and E denote the value of \(_^\) and \(_^\) before (Start) and after (End) the change in [Pi] in a Pi-transient experiment. The steady state probability of strongly attached states at initial and final [Pi] are given by Asssstart= \(\frac_}_^}\) and Asssend = \(\frac_}_^}\), respectively. The rate constants \(_^\) and \(_^\) are functions of [Pi] (Table 3).

Equations 14, 15 and the first column in Table 3, describe As(t) for all schemes in Fig. 1 with As(t) in Eq. 14 being proportional to level of force redevelopment from zero. Equation 15, on the other hand, describes the time course for the tension changes (represented by rate constant kPi) in response to a sudden change in [Pi] (Pi-transients) only for the scheme in Fig. 1a, i.e. kAS = ktr = kPi in this case (Fig. 5a). For the other schemes (Fig. 1b, c), the theoretical Pi-transient is a double exponential function dominated by a fast phase (Table 3 and Fig. 5b, c). The rate constant of the fast phase is determined by the transitions between different strongly attached states. Because these are assumed to be fast equilibria (KP and KLH), the rate constant is > tenfold higher than kAS. The force redevelopment as well as Pi-transient responses of the different kinetic schemes in Fig. 1 are given in Fig. 5 at varying Pi-concentrations. Composite apparent rate constants, given by 1/half-time (1/t1/2) of the tension change as function of [Pi] are plotted in Fig. 5d for both the rate of force redevelopment and the rate of the Pi-transients for each of the kinetic schemes in Fig. 1. Only the scheme in Fig. 1a predicts similar behavior of ktr and kPi. The differences between ktr and kPi for the other schemes is attributed to the double exponential behavior of the Pi-transient.

Simulations based on kinetic schemes of isometric tension redevelopment and Pi-transients. a Analytical solutions (full lines; Eqs. 14, 15 and Table 3) for the scheme in Fig. 1a with superimposed numerical simulations (dashed lines) based on calculations in Simnon for 5 and 25 mM Pi. b Numerical solutions for the scheme in Fig. 1b clearly depicting differences between Pi-transients (double exponential) and force redevelopment from zero (single exponential). Also note the dominance of the very fast phase, particularly at high [Pi]. c Numerical solutions for the scheme in Fig. 1c. Also in this case, note differences between Pi-transients and force redevelopment as in b. Further, note even greater dominance of the very fast phase than in b. d Apparent rate constant (1/t1/2) for Pi-transient (open squares) and force redevelopment (filled cricles; ktr) vs [Pi] for the schemes in Fig. 1a (black), Fig. 1b (red) and Fig. 1c (blue). Note that ktr = kPi only for the scheme in Fig. 1a. Rate constants in schemes: kon = kon- 100 s−1, KLH = 1 (kLH+  = kLH- 5000 s−1), and kP+ = 3000 s−1, KP = 10 mM

The rate constant ktr predicted by all schemes in Fig. 1 increases with [Pi] according to a rectangular hyperbola from 130 to 150 s−1 at trace [Pi] to 200–260 s−1 at saturating [Pi] (Fig. 5d). Half the maximum Pi-induced increase in rate constant occurs at [Pi] = 10 mM with Kp set to 10 mM. As mentioned above, the [Pi] dependence of kPi predicted by the scheme in Fig. 1a is the same as the [Pi]-dependence of ktr. Also for the schemes in Fig. 1b, c, the apparent kPi increases with [Pi] but it is more than 10 times higher than ktr (Fig. 5d) due to the large fractional contribution of the fast phase.

Next we moved on from the simple kinetic schemes to instead simulate Pi-transients and the force redevelopment using the mechanokinetic model. We started our analysis by obtaining steady-state solutions of the distributions of the cross-bridge populations and force as a function of x (Fig. 3). In accordance with previous modelling results (Månsson 2021) this simulation suggests that increased [Pi] particularly reduces the occupancy probability of the pre-power-stroke states (e.g. AMDL) and to only minor degree that of the post-power-stroke AMDH state. The steady-state population of the pre-power-stroke AMDP state as well as the detached MDP states both increase (not shown) by the increase in [Pi] during isometric contraction. The reason for only a minor reduction in the AMDH state population upon increased [Pi] is that the strong binding-energy (low free energy) in the AMDH state for low values of x limits population of the AMDL state in which Pi-binding is possible.

By observing the force integrals along x (Fig. 3a) at both 0.5 and 25 mM Pi (with integration starting at x = 14 nm and progressing in the negative x-direction) along with their difference at given x-values (orange symbols in Fig. 3a) it is clear that the decrease in steady-state isometric force between 0.5 to 25 mM Pi has two main components that contribute to the Pi-transient. The major component (ΔFPslow) is due to loss of cross-bridges from the AMDL and AMDP states to the MDP state upon Pi-binding and reversal of the cross-bridge attachment step. The second, smaller component of force decrease (ΔFPfast) is due to reversal of the Pi-release. By mass action it leads to force-reduction due to depletion of the AMDH state for x close to x ≈ x1 where the free energy of the AMDH and AMDL states are similar. Notably, only for this minor component is reversal of the power stroke (AMDH—> AMDL state) involved.

The two components are apparent (Fig. 3a) from the inflection point in the difference plot (orange dots and lines) between the two integrals of force over x for 0.5 (black) and 25 mM Pi (red) and from the double peaks of the difference in cross-bridge force vs x between 0.5 and 25 mM Pi (Fig. 3b, c). The latter differences are superimposed on the rate constant for the slow (Fig. 3b) and fast (Fig. 3c) component, respectively. The results in Fig. 3 with two distinct components accounting for the difference in steady-state tension between low and high [Pi] suggest a complex response of a mechanokinetic model to changes in [Pi]. During the Pi-transient, one thus expects one slow dominating single exponential component of the force change. This component, ΔFPslow, behaves similarly to the kinetic scheme in Fig. 1a but with a distribution of rate constants and force levels. It is due to the Pi-induced detachment from the AMDL state via the AMDP state to the MDP state at high x-values. A smaller fast component, ΔFPfast, in the tension response is dominated by a first rapid shift of the equilibrium out of the AMDH to the AMDL, and particularly, the AMDP state. This may be followed by slow subsequent detachment into the MDP state. The ΔFPfast component is analogous of the kinetic scheme in Fig. 1c. However, with the rate functions used in the mechanokinetic model, the amplitude of the slow phase of the ΔFPfast component is generally negligible. Thus, to summarize, the tension response of the present mechanokinetic model to a sudden change in [Pi] can be approximated by the sum of two components. The dominant component ΔFPslow is fitted by a single exponential function analogous to the kinetic scheme in Fig. 1a (detachment from the AMDL state at high strain). The other smaller fast component ΔFPfast is analogous to the kinetic scheme in Fig. 1c but with a negligible slow phase. Finally, for the force redevelopment from zero force (rate constant ktr) the mechanokinetic model suggests the existence of one additional Pi-independent component (Fig. 3a). This is attributed to the virtually irreversible transition from the MDP over the AMDP and AMDL states to the AMDD state at x < < x1 where KLH > > 1. For these x-values, the cross-bridge behaviour according to the mechanokinetic model is similar to that of the kinetic scheme in Fig. 1c if cross-bridge detachment by ADP-release and ATP binding is neglected.

We next evaluated (without in depth consideration of how realistic this is in other resepcts) if changes in parameter values in the mechanokinetic model could completely eliminate the fast component (ΔFPfast) of the Pi-transient. To that end, we varied key parameter values in the model with the aim to modify the relative population of the AMDL and AMDH states in isometric steady-state contractions. First we noted (Fig. 6) that, for all parameter values tested, the fractional amplitude of ΔFPslow decreases with increased [Pi] with corresponding increase of ΔFPfast, as also reflected in [Pi]-induced changes in Fig. 3b, c. Second, changes in parameter values also have marked effects on the fractional amplitudes of ΔFPfast and ΔFPslow. As one example, an increase in x1 and a reduction of x2 lead to an increased ΔFPslow component and a corresponding decrease of ΔFPfast. Thus, strikingly, if we reduced x2 from 2 to 1 nm (with only 1 out of a 9 nm total power-stroke due to the second sub-stroke) and increased x1 to 9 nm, ΔFPfast becomes smaller than 20% of the total force change for [Pi] up to 50 mM. We now denote the standard condition defined in Tables 1, 2 as C(7.7, 2) (× 1 = 7.7 nm and x2 = 2 nm) whereas the altered condition is denoted C(9,1) (see Fig. 6). Under the latter condition (C(9,1)), ΔFPfast is less than 10% for [Pi] < 25 mM, usually studied in Pi-transients and related experiments (Dantzig et al. 1992; Tesi et al. 2000; Stehle 2024). In summary, a mechanokinetic model with Pi-release before the power stroke (the AMDL-AMDH transition) predicts a complex Pi-transient dominated by a slow component (< 100 s−1 in the present model). However, the model also predicts the existence of a fast component (> 1000 s−1) with an amplitude > 5–10%. This is in contrast to the experimental results which suggest a single exponential Pi-transient and similar amplitudes and [Pi]-dependence of ktr and kPi.

The fraction of slow (ΔFPslow) [Pi]-dependent components of Pi-transients (filled symbols) and the fraction of Pi-independent component of the force redevelopment (open symbols) in mechanokinetic model. All parameter values are as in Tables 1, 2 with exceptions as follows: Circles C(7.7, 2), no exception. Squares, ΔGon = 2 kBT and kon′ = 89 s−1. Triangles, x1 = 9 nm and x2 = 1.5 nm (peak down kon′ = 20 s−1 peak up kon′ = 40 s−1; C9,1.5)). Diamonds, x1 = 9 nm and x2 = 1 nm (C(9,1))

The C(9,1)-condition gives a Pi-transient closest to a single exponential function, i.e. it has a quite small ΔFPfast component. However, before evaluating this condition in greater detail, We investigated if it would reasonably well predict the force–velocity relationship of striated muscle. Unfortunately it does not (Fig. 7), giving a shape of the relationship far from the hyperbolic equation of Hill (1938). We therefore also considered a condition intermediate between C(9,1) and the standard C(7.7, 2) condition, namely (C(9,1.5) with x1 = 9 nm and x2 = 1.5 nm but otherwise the same parameter values as in Tables 1, 2. The comparision in Fig. 7 suggests that the latter choice of parameter values gives the most reasonable prediction of the force–velocity relationship, particularly if the attachment rate constant is doubled to better account for the maximum power output. We next moved on to compare the conditions C(9,1), C(9,1.5) and C(7.7,2) in different respects to better elucidate the key factors that determine the Pi-transient and initial force-redevelopment kinetics in mechanokinetic models with Pi-release before the power stroke. Despite the poor prediction of the force–velocity relationship with x1 = 9 nm and x2 = 1 nm we did not exclude this condition from additional evaluation. The reason is that it is of interest to see if any mechanokinetic model may be able to account for the Pi-transients to similar extent as the kinetic scheme in Fig. 1a. This is reasonable because the scheme in Fig. 1a cannot be at all evaluated against the force–velocity relationship.

Simulated force–velocity relationships compared to experimental values for living mammalian muscle at 30 °C (purple; (Månsson et al. 1989)). Force data normalized to maximum isometric force. Simulated data based on parameter values in Tables 1, 2 (black) or as in Tables 1, 2 with modifications as follows: x1 = 9 nm and x2 = 1 nm (red), x1 = 9 nm and x2 = 1.5 nm (blue) or x1 = 9 nm and x2 = 1.5 nm and kon′ = 40 s−1 (orange)

The in depth analysis of the three conditions C(7.7, 2), C(9,1) and C(9,1.5) (the latter with attachment rate constant kon′ increased from 20 to 40 s−1) is depicted in Fig. 8. The analysis first clarifies the origin of the main differcnces in the fractional contribution of ΔFPslow and ΔFPfast (Fig. 6) based on the [Pi]-induced changes in cross-bridge distributions. Moreover, it shows that, in terms of the present mechanokinetic model, the reduction in the number of strongly attached cross-bridges with increased [Pi] is primarily due to a reduced occupation probability of the AMDL and AMDP cross-bridge states (at x > 11 nm). The analysis in Fig. 8 also reveals that, in addition to [Pi]-dependent components associated with ΔFPslow and ΔFPfast, there is a component of the total isometric force (at x < 6–7 nm) that is negligibly affected by [Pi] for all the parameter values tested (see also Figs. 3 and 6). The latter component does not contribute to the Pi-transients but it contributes to the redevelopment of isometric force.

Cross-bridge distributions, force, number of attached cross-bridges and effective rate constants for Pi-induced reversal of attachment. a–c Cross-bridge distributions giving fraction of cross-bridges in biochemical states AMDP, AMDL and AMDH at different x-values either at 0.5 mM (black) or 25 mM Pi (red). d–f Force integrated over x (starting from x = 14 nm and progressing to x = 4 nm; left vertical axis) corresponding to cross-bridge distributions in top row for 0.5 mM (black) and 25 mM Pi (red). The difference in the integrated force (right vertical axis) between 0.5 and 25 mM Pi is depicted in orange, showing the two main components (cf. Figure 3) of the force-decrease during a Pi-transient upon change of [Pi] from 0.5 mM to 25 mM. Blue symbols depict difference in the integrated number of attached cross-bridges (right vertical axis) between 0.5 and 25 mM Pi showing that the drop in number of attached cross-bridges upon increase in [Pi] is particularly attributed to cross-bridges at x > 11 nm. g–i Rate constants (left vertical axis) and force difference (right vertical axis) between 0.5 mM Pi and high [Pi] vs x: Black, 5 mM Pi; blue, 10 mM Pi; red, 25 mM Pi. Left column a, d, g All parameter values as in Tables 1, 2 (C(7.7, 2)). Middle column b, e, h Parameter values as in left column but x1 = 9 nm, x2 = 1.5 nm (C(9, 1.5)) and kon′ = 40 s−1. Right column c, f, i Parameter values as in left column but x1 = 9 nm and x2 = 1 nm (C(9, 1))

In view of the above arguments, a mechanokinetic model is not expected to predict that ktr = kPi. In addition to having the [Pi] independent component, the force redevelopment would also lack a fast component related to ΔFPfast that contributes to the Pi-transients. The [Pi]-independent fraction of the force redevelopment corresponds to myosin heads in the region below x = 6.5 nm (Fig. 3) and we thererfore obtained a quantitative estimate of the [Pi]-independent component of force (FintPind) as the force contribution of myosin heads in this range (for further details, see Model section). This quantitative estimate is given as a function of [Pi] for different model conditions in Fig. 6. In accordance with the definition, the [Pi]-dependence is negligible. If we neglect effects due to myosin head detachment via ADP-dissociation and ATP-binding, the rate constant of the [Pi]-independent process is given by kon(x) because the transition is made essentially irreversible by the high value of KLH at x < 6.5 nm.

Taking the rate constants and force-data in Fig. 8 as starting points, we simulated Pi-transients and force redevelopment (Fig. 9) using the mechanokinetic model as described in the Model section (revolving around Eqs. 7–13). The results in Fig. 9 show that the Pi-transients, but not the force redevelopment from zero, deviate from a single exponential function with a fast initial phase. All conditions tested, C(7.7, 2) (Fig. 9a), C(9, 1.5) (Fig. 9b) and C(9,1) (Fig. 9c) predict a reduction of the isometric force with increased [Pi] that is reasonable (Coupland et al. 2001) for the preparation and temperature (25–30 °C).

The time course of Pi-transients and force redevelopment (related to ktr) simulated by mechanokinetic model with different parameter values. a C(7.7, 2), all as in Tables 1 and 2. Best single exponential fits to the data superimposed. b C(9, 1.5) and kon′ = 40 s−1 otherwise as in Tables 1 and 2. Best single exponential fits to the data superimposed. Force redevelopment shown also for 0.5 mM Pi (top black). c C(9,1) otherwise as in Tables 1 and 2. Best single exponential fits to the data superimposed. Force redevelopment shown also for 0.5 mM Pi (top black). Area in dashed box shown in greater detail as an inset in f. d Steady-state isometric force vs [Pi] for conditions C(9,1.5) and C(9,1) from b and c. e Rate constants (corresponding to ktr and kPi) derived in single exponential fits for conditions C(9,1.5) and C(9,1) shown in b and c. f Fractional contribution to tension change during Pi-transient attributed to fast component for all conditions simulated in a–c. Inset: Early part of tension decay for the C(9,1) condition (c) with the very fast initial component of the tension decay indicated. Best single exponential fit as in c superimposed on the data

This effect is summarized in Fig. 9d for the C((9, 1) and C(9,1.5) conditions, the latter with the attachment rate constant doubled (kon´= 40 s-1). It is also shown for these conditions in Fig. 9e that the effects on isometric force are coupled to reciprocal effects of increased [Pi] on ktr and kPi. The coupling between the change in isometric force and ktr does not have exactly the same functional relationship as in a recent experimental study of guinea pig cardiac myofibrils at 10 °C (Stehle 2024). In the latter study, the force shows a clear saturation behaviour at high [Pi] whereas the dependence of ktr on [Pi] is nearly linear. In our model both ktr and isometric force show similar saturation behaviour. We do not investigate the basis for this difference. However, we note that the [Pi]-effects vary greatly with temperature (e.g. (Coupland et al. 2001)). Moreover, other studies using fast rabbit skeletal muscle myofibrils (Tesi et al. 2000) have indicated more similar saturation behaviour between the variation with [Pi] of ktr and isometric force.

Recent work (Moretto et al. 2022) suggests that the assumption of multistep Pi-binding would mitigate the differences from experimental results of the mechanokinetic model by removing the fast phase of tension decay from the Pi-transients following increases in [Pi].

We tested this idea by a simplified treatment described in the Methods and in Fig. 4. For the Pi-transients we obtained the amplitude of the fast (lower scheme in Fig. 4) and slow (upper scheme in Fig. 4) component from ΔFPfast and ΔFPslow, respectively where the latter values were obtained from the mechanokinetic modelling. The amplitudes for redevelopment of force from zero, on the other hand, were obtained from the integral force at the given [Pi] below (fast component; lower scheme) and above (slow component; upper scheme) the x-value of the force-integral′s inflection point. We focused the analysis on the conditions C(7.7, 2) and C(9,1.5) that give most faithful reproduction of the force–velocity relationship. The results are depicted in Fig. 10. As expected, both the Pi-transients and the force redevelopment are well approximated by single exponentials. However, ktr is consistently higher than kPi, particularly at low [Pi]. Whereas this difference can be reduced by altered parameter values i, e.g. higher ratio, kPs/kon (Fig. 10e vs 10f) it seems difficult to completely eliminate the differences. The faster average rates compared to Fig. 9 are due to choice of rate constants kon and kon- as for the analysis using kinetic schemes (Fig. 5). These values are faster than the corresponding rates in the mechanokinetic model from which we, however, take the amplitudes of the fast and slow component (details in Methods) corresponding to the relative contribution of the two parallel schemes in Fig. 4.

Simulation of Pi-transients and rate of force redevelopment vs [Pi] for simplified representation of multistep (two-step) Pi-release. a Time courses with contribution of top (slow) and bottom (fast) scheme in Fig. 4 with respective amplitudes determined from the mechanokinetic model for C(7.7, 2) condition as described in the Model section. Pi-transients from 0.5 mM to higher final [Pi]. Final Pi-concentration either 5 mM (black), 10 mM (blue) or 25 mM (red). Single exponential fits superimposed on the data. b Simulations using mechanokinetic model as in a but for the C(9, 1.5) condition with the Pi-release rate (kPs) equal to kon, i.e. kps = kon = 100 s−1. Single exponential fits superimposed on the data. c Simulations using mechanokinetic model as in a but for the C(9, 1.5) condition with kPs = 200 s−1 and kon = 100 s−1. Single exponential fits superimposed on the data. d Estimates of kPi (open symbols) and ktr (filled symbols) from single exponential fits and their ratio for the condition in a. e Estimates of kPi (open symbols) and ktr (filled symbols) from single exponential fits and their ratio for the condition in b. f Estimates of kPi (open symbols) and ktr (filled symbols) from single exponential fits and their ratio for the condition in c