The effects of inorganic phosphate on muscle force development and energetics: challenges in modelling related to experimental uncertainties

Fig. 1. Models used. a Schematic illustration of model geometry for target zones (T) on actin with 1 (10:1 model) and 3 (9:3 model) sites. b Kinetic scheme for 1 site in the 3-site model with 9 states (9:3 model). The myosin head (M) is either attached to actin (A) or not. Except for in the “rigor” state (AM), the heads have ATP (T), ADP (D) and/or inorganic phosphate (P, Pi) at the active site. The AMDPT state is a transient intermediate under physiological conditions. Rate constants are written generically as ki and equilibrium constants as Ki. An argument (x) indicates strain dependence of the “constant”. The green and grey arrows refer to transitions to the states in green and grey font, respectively at the neighbouring sites along actin (superscript “p” and “m” towards the plus end and minus end of actin, respectively). c Free energy diagrams for the 9:3 model plotted versus variable x which is the distance of the myosin head from the position where the free energy in the AM state at the central site attains its minimum. The full and dashed coloured lines represent linear and non-linear cross-bridge elasticity, respectively. d Kinetic scheme for the 1-site model with 10 states (10:1…

Fig. 2. Effects of increase from 0.5 to 25 mM Pi on free energy diagrams. a Central site in the 9:3 model. Comparison of free energies of different strongly attached states at 25 mM Pi (dashed coloured lines) and 0.5 mM Pi (full coloured lines). b 10:1 Model. Comparison of free energies of different strongly attached states at 25 mM Pi (dashed coloured lines) and 0.5 mM Pi (full coloured lines). Both in a and b the absolute values of the free energies at 25 mM Pi are shifted upwards (to compensate for reduced free energy of ATP at increased [Pi]) so that the free energies of the MTP, MDP and the AMDP state (black lines) coincide at 0.5 mM and 25 mM Pi in both plots and also those of the AMDPPP and the AMDPPiR in b. (Color figure online)

Fig. 3. Force–velocity relationships at different [Pi]. Model with 9 states and 3 sites (9:3-model). a Model data for different [Pi] assuming linear cross-bridge elasticity. Comparison to experimental data from living fast mouse muscle (Månsson et al. 1989) and from actin–myosin ensembles in vitro (Pertici et al. 2018) under non-fatiguing conditions with expected low [Pi] (< 1 mM). b Details of the high-force, low-velocity region in a. Horizontal dotted lines “e”–“g” refer to cross-bridge distributions in e–g. Curved full line: a fit used in analysis below (see Fig. 5). c Similar analysis as in a but assuming non-linear cross-bridge elasticity. d Details of the high-force, low-velocity region in c. For ease of comparison the experimental force–velocity data are scaled to the maximum velocity predicted by the model. Furthermore as motivated in the text the experimental maximum isometric force is scaled to the maximum isometric force predicted by the model for [Pi] = 0.5 mM Pi. e Population of low-force (AMDL) and high-force (AMDH) states as function of strain-parameter x in isometric contraction (labelled “e” in b) at 0.5 mM (black) and 25 mM (grey) Pi. f Similar data as in e but at a…

Fig. 4. Force–velocity relationship at different [Pi] using model with 10 states and 1 site (10:1-model) and the assumption of linear cross-bridge elasticity throughout. a Original model of Rahman et al. (2018) with minor modification (Table S2). b Same data as in a, but in the high-force, low-velocity range only. c Model of Rahman et al. (2018) but with shift of x-positions for free energy minima of states AMDPPP and AMDPPiR from 8.7 to 7.2 nm and from 7.7 to 6.9 nm, respectively compared to original model. d Same data as in c, but in high-force, low-velocity range only. Experimental data from Månsson et al. (1989) same as in Fig. 3

Fig. 5. Anomalous force–velocity relationship in relation to sarcomere-non-uniformities. The grey squares and the full curved grey line are reproduced from the model in Fig. 3b. The black squares and the dashed black line represent the same data shifted towards lower maximum force to simulate lower force-producing capacity without changes in the shape of the force–velocity relationship. Black and grey sloped straight lines at negative velocity: tentative force velocity relationship during lengthening with higher absolute value of the derivative of force against velocity than during the shortening part of the force–velocity relationship (Edman 1988). Dashed dark grey line: average force–velocity relationship from two half-sarcomeres in series, derived as described in the text. Dashed vertical red line: a given load (force) level illustrating that three different constant velocities are possible for the strong half-sarcomere at this load. See text for details of the analysis. (Color figure online)

Fig. 6. Isometric force, and isometric ATP turnover rate as function of [Pi] for the 10:1 model. Data normalized to value at 0.5 mM Pi. a The version of the model most similar to the original one (Rahman et al. 2018, cf. Table S2 for difference). Blue: simulated ATP turnover rate. Black: simulated force. Purple: experimental data for force (circles, Coupland et al. 2001) and ATP turnover rate (open squares, 15 °C, Potma et al. 1995). Filled circles and full lines: 30 °C. Open circles and dashed-dotted lines: 5 °C. Absolute value of simulated ATP turnover rate at 0.5 mM Pi: 2.4 s−1 at 30 °C and 0.12 s−1 at 5 °C. b Model similar to that in a but parameter x11 and x1 reduced to 7.2 and 6.9 nm, respectively. Colour coding as in a. Absolute value of simulated ATP turnover rate at 0.5 mM Pi: 5.5 s−1 at 30 °C and 0.36 s−1 at 5 °C. (Color figure online)