Nonlinear Actomyosin Elasticity in Muscle?

Figures

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  Figure 1. Basic characteristics of model for rigor conditions. (A) A kinetic scheme for model with five actin sites (A−2…A2; separated by 5.5 nm) assumed to be accessible and partly compete for the binding of each myosin head (M) to form rigor (AiM; actomyosin) links. (B) Free-energy diagrams for model states in which free energy of the detached state M is 0 kBT. Full curved lines: free energies of AiM-states in the case of linear (Hookean) cross-bridge elasticity. Dashed curved lines: free energies of AiM-states in the case of nonlinear cross-bridge elasticity (Fig. S1). The gray area indicates the region for averaging and solving of differential equations to calculate force and stiffness. (C) Structural components and their representations in the form of contractile and series elastic elements in the modeling. The series elastic component is limited to myofilament compliance if muscle fiber experiments use sarcomere length control. The myofilament compliance is assumed to be linear and modeled by a separate series elastic element.

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  Figure 2. Model for active contraction. (A) A kinetic scheme. M, myosin; A, actin; D, ADP; T, ATP; P or Pi, inorganic phosphate. The constants given by lower case letters represent rate constants. Some of the constants are strain dependent, as indicated by the argument (x). Constants Ki represent equilibrium constants. The MDP and the AMDP states are in rapid equilibrium. (B) Free-energy diagrams for the different states in the kinetic scheme for the case with linear cross-bridge elasticity (full lines) and nonlinear cross-bridge elasticity (full lines connected to dashed lines) used as standard below (Table S1). The free energy of the weakly attached AMDP state is not shown. It would be represented by a nearly horizontal line (because of very low stiffness, 2.5 kBT below the energy for the MDP state).

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  Figure 3. Tension responses to half-sarcomere length changes imposed on a muscle fiber in rigor. A comparison of model responses to experimental data from a skinned rabbit psoas muscle fiber at sarcomere length (2.37 μm) with maximal overlap between thin and thick filaments (39) is shown. (A) Tension change (light gray; left vertical axis) in response to total half-sarcomere-length changes (black; right vertical axis). Experimental data (filled circles and lines; “Exp”), simulated data assuming linear cross-bridge elasticity (filled straight lines; “Lin”) or simulated data assuming nonlinear cross-bridge elasticity (dashed lines; “Non-lin”) are shown. The cross-bridge force-extension relation for nonlinear case is taken from red line in Fig. S1A. The length change attributed to series elastic element in myofilaments (dark gray; right axis) is also indicated in model simulations. (B) The relationship between force and half-sarcomere length (“T1 curve”) in experiments (full circles) and model simulations (line) assuming nonlinear cross-bridge elasticity. The experimental data were obtained by measurements from Fig. 9 a in (39)). In the model simulations, ∼1/3 of the total sarcomere…

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  Figure 4. Model predictions for force and stiffness changes upon sequence of stretch-stretch-shortening ramps. (A) Time courses with all compliance assumed to reside in cross-bridges. Full lines: linear cross-bridge elasticity. Dashed/dotted lines: nonlinear cross-bridge elasticity. Black: sarcomere length (right vertical axis). Gray full and dashed lines: force (left axis). Gray, dotted lines: stiffness (left axis). Terminology: “prestretch,” “main stretch,” and “main shortening” used regularly in the text. Note nonzero force (“isometric rigor tension”) before any applied length change in the case with nonlinear cross-bridge elasticity. (B) Force-sarcomere-length relationships for data in (A). Full lines: linear cross-bridge elasticity. Dashed lines: nonlinear cross-bridge elasticity. (C) Time courses with 30 and 70% of sarcomere compliance assumed to reside in cross-bridges and myofilaments, respectively. Full lines: linear cross-bridge elasticity. Dashed/dotted lines: nonlinear cross-bridge elasticity. Black: sarcomere length (right vertical axis). Light gray: average length change of cross-bridges (right vertical axis). Dark gray full and dashed lines: force (left axis). Dotted gray…

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  Figure 5. Half-sarcomere stiffness upon stretch ramps (main stretch) of different amplitudes normalized to the stiffness before the stretch. Squares and triangles: simulated data on the assumption of nonlinear cross-bridge elasticity and linear myofilament elasticity. The cross-bridge elasticity was assumed similar to that in (29). Model data are given for main stretches imposed 1 s after a prestretch of amplitude 0 nm (open symbols), 1 nm (black squares), or 2 nm (gray squares). For the case of 0 nm prestretch, data were simulated for two different levels of myofilament stiffness: 150 pN/nm (squares) and 250 pN/nm (triangles). Negative stretch corresponds to shortening. Simulated data are connected by lines for clarity. Velocity of the simulated length changes was 10 nm/s for all cases. A thousandfold increase in velocity did not change the results. The simulated data are superimposed on experimental results from (24) (small full circles; measured from their Fig. 5).

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  Figure 6. Model predictions for steady-state contractile properties on the assumption of linear or non-linear cross-bridge elasticity. (A) Velocity as function of the cross-bridge stiffness (Slow) in the AM/AMD state in the range −75 nm < x < −4 nm. Rate function k2(x) is strain independent (xcrit = 0). Full lines and filled symbols: k2(x) = k2(0) = 1800 s−1. Dashed lines and open symbols: k2(x) = k2(0) = 1600 s−1. Circles: cross-bridge stiffness linear in AMDL and AMDH states. Filled triangles: stiffness linear in AMDL state but nonlinear in AMDH state ((10a), (10b), a and b). Filled squares: stiffness linear in AMDH state but nonlinear in AMDL state ((9a), (9b), a and b). Open squares: nonlinear cross-bridge elasticity in all states. Vertical dotted line: the degree of nonlinearity (Slow = 0.03 pN/nm) observed in AM/AMD state in (29). Horizontal dotted lines: experimental range of velocities (see text for references). (B) The force-velocity relationship. The experimental data (light gray) are from mouse intact muscle at 30°C (62). Black filled circles: simulations assuming linear elasticity and standard parameter values (Tables S1 and S2) without periodic boundary conditions. Black open…

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  Figure 7. Population of different cross-bridge states during isometric contraction (left panels) and during shortening near maximal velocity (right panels). (A) Cross-bridge distributions during isometric contraction, assuming that all attached cross-bridge states have linear (full lines) or nonlinear elasticity (dashed lines). (B) Cross-bridge distributions as in (A) (same grayscale coding) but simulated for steady-state shortening at 14,000 nm/s. (C) Cross-bridge distributions during isometric contraction assuming that all attached cross-bridges have linear elasticity (as in A; full lines) or that the AMD/AM state exhibits nonlinear elasticity, whereas the other states have linear elastic elements (dashed lines). (D) Cross-bridge distributions during shortening at 14,000 nm/s, assuming either linear cross-bridge elasticity of all states (as in B; full lines) or nonlinear elasticity of the AM/AMD state only (dashed lines). The distributions are normalized to the total number of myosin heads at each value of x.

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  Figure 8. The parameter values Vmax and KMv for actin filament sliding velocity versus [MgATP]. Vmax (black, left axis) and KMv (gray, right axis) were obtained in simulations with different combinations of linear and nonlinear (from (29)) cross-bridge elasticity for different states and force dependence of k2(x) ((xcrit = 0 nm; no force dependence) or (xcrit = 0.6 nm; force dependence)). The final group of parameter values (right) was obtained assuming linear cross-bridge elasticity and ATP-induced detachment rate constant without force dependence, but k2(0) increased almost threefold to 6430 s−1 compared to data in Table S2. The box indicates simulations in which both Vmax and KMv values accord best with experimental data.

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  Figure 9. Steady-state ATP turnover rate and thermodynamic efficiency. (A) ATP turnover rate per all available (not only attached) myosin heads as a function of sliding velocity on the assumption of linear (black) or nonlinear (dark gray) cross-bridge elasticity. The nonlinear cross-bridge elasticity was simulated either assuming that the elasticity is nonlinear for all attached states (filled dark gray circles) or only for the AM/AMD state (open gray circles). Experimental data are illustrated by the light gray line and symbols from (69) (circles), (102) (line), and (70) (squares). The velocity data were transformed from the original experimental values by first normalizing velocity to the unloaded velocity in each experiment and then scaling to the high-temperature value of our simulations. The ATP turnover rate was transformed by assuming Q10 = 2.9 for ATP turnover rate at all velocities. For more details, see text. All simulated data represented by circles were obtained without consideration of periodic boundary conditions. Control simulations using periodic boundary conditions are represented by open squares for nonlinear cross-bridge elasticity. (B) Thermodynamic efficiency data…