Fig 1.
Schematic of a half-sarcomere.
(A) Half-sarcomere model showing the geometry of a system containing two actin nodes and two myosin nodes. Cross-bridges are bound, connecting the thick and thin filaments. (B) Schematic showing the three potential mechanical states of a cross-bridge. State 1 shows a cross-bridge unbound from the thin filament. State 2 shows a bound, pre-power stroke cross-bridge in a low force bearing state. State 3 shows a bound, post-power stroke cross-bridge in a high force bearing state. In state 3, the cross-bridge has also undergone a conformational change where the cross-bridge rest length (b0) has shortened by the length of a power stroke (dps). The model is one dimensional, but this figure illustrates the model in two dimensions for clarity. All forces in this model are assumed to be one-dimensional, parallel to the filaments.
Fig 2.
Schematic of the three-state system.
Shorthand for the biochemical states of the actomyosin complex are displayed in black frames: A-actin, M-myosin, ATP-adenosine triphophate, ADP-adenosine diphosphate, Pi-inorganic phosphate. All elements in each black box are bound. Between state 3 and state 1, ATP binds the actomyosin complex and is hydrolyzed. Rates for the transitions between each state are labeled such that kij represents the transition rate from state i to j. Association of ATP to the actomyosin complex and the dissociation of ADP and Pi from the actomyosin complex are indicated at the appropriate transition.
Fig 3.
Schematic of free energy landscape for one full cycle.
Shorthand of the biochemical states of the actomyosin complex are framed in black (key in Fig 2 caption). Transition states are denoted by dashed lines and dashed frames. ΔGhyd = ΔGT assoc. + ΔGT hydr. + ΔGD,Pi rel.. Free energy of association of ATP is ΔGT assoc.. Free energy of ATP hydrolysis is ΔGT hydr.. Free energy of of ADP and Pi release from actomyosin is ΔGD,Pi rel.. GADP,Pi = kBT ln([ADP][Pi]). GATP = kBT ln([ATP]). Note: The free energies of each state depicted in this free energy landscape assume there is no cross-bridge deformation, and therefore do not include the elastic potential energy contributions of such deformations. The complete free energies of each state, including elastic potential energies, are fully defined in Eqs 2–5.
Table 1.
Variables and the corresponding units or values used in the sarcomere model. Values were selected after a parameter exploration was performed on a range of values pulled from literature from both models and experiments (details in Section D of S1 Text).
Fig 4.
Single half-sarcomere consisting of 16 myosin and 24 actin nodes under (A-B) normal ([ATP] = 5 mM) and (C-D) reduced ([ATP] = 0.5μM) ATP conditions. (A,C) Force output profile denoted by black circles. Average force denoted by the dashed line. Force averaged over a sliding window of (A) 12 ms and (C) 14 ms denoted by blue lines with blue diamonds. (B,D) ATP consumption rate (molecules/s) denoted by black circles. ATP consumption rate averaged over a sliding window of 50 ms denoted by green diamonds.
Fig 5.
One-myosin system simulations.
Comparison of free energy schema in terms of ATP consumption (A) Ratios of [ATP] to [ADP][Pi]. (B) Plateau ATP consumption rate for a one-myosin system where attachment of ATP and detachment of ADP and Pi effectively happen simultaneously, allowing for no time delay between these events. For such a model, the rate kinetics, and thus ATP consumption, depend only the ratio of [ATP]/[ADP][Pi]. (C) Our model’s plateau ATP consumption rate simulation results for a one-myosin system. Note the plot’s asymmetry compared to (B) and the proximity of the standard physiological conditions (bold red lines) to the crossover between regimes. One-myosin simulation results of (D) Duty ratio and (E) Average force for various ([ATP], [ADP][Pi]) combinations. (F) Changes in duty ratio (black), average force output (purple), and plateau ATP consumption rate (blue) at standard physiological [ADP][Pi] and varying [ATP] concentrations. (B-F) Standard physiological conditions are denoted by bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM2). Dashed white/gray lines denote the the [ATP] concentration associated with the onset of rigor ([ATP] = 0.5 mM) [75, 79]. (C-F) n = 10.
Fig 6.
Results for 16-myosin simulations (n = 10) showing (A,D) Duty ratio, (B,E) Average force, and (C, D) Plateau ATP consumption rate. Average force and plateau ATP consumption rate are reported as per whole sarcomere. (D) Duty ratio and ATP consumption rate and (E) Average force at standard physiological [ADP][Pi] and varying [ATP] concentrations. (A-E) Standard physiological concentrations for [ATP] and [ADP][Pi] conditions are denoted by the bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM2). Dashed white/gray lines represent literature values of [ATP] concentrations where the onset of rigor has been observed experimentally ([ATP] = 0.5 mM) [75, 79]. (F) Percent of total elastic potential energy in the spring elements internal to the sarcomere (i.e. actin, myosin, titin, and cross-bridges) and external to the sarcomere (i.e. external spring).
Fig 7.
Analytical results for a one-myosin system and varying parameters.
(A) Analytically calculated duty ratio (Eq 16) where the power stroke’s effective sliding distance (ESD) = 6 nm. (B) Analytically calculated average force (Eq 17) where ESD = 6 nm. (C) Analytically calculated ATP consumption rate (Eq 18) where ESD = 6 nm. (D) Analytically calculated duty ratio where ESD = dps = 7 nm, where 7 nm is the power stroke distance against no resistance. This causes an upward and rightward shift in the plot. (E) Analytically calculated duty ratio where ESD = 3 nm. This causes a downward and leftward shift in the plot. (F) Analytically calculated duty ratio where ESD = 6 nm and kATP,0 = 7 × 10−5 s−1. This causes a rightward shift in the plot. (A-I) Standard physiological concentrations for [ATP] and [ADP][Pi] conditions are denoted by the bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM2). Dashed white/gray lines denote the [ATP] concentration associated with the onset of rigor ([ATP] = 0.5 mM) [75, 79]. Effect of reducing kATP,0 on one-myosin analytical and 16-myosin simulation (n = 10) values for (G) Duty ratio, (H) Average force, and (I) Plateau ATP consumption rate at standard physiological [ADP][Pi] or increased [ADP][Pi] and varying [ATP] concentrations. Reducing kATP,0 causes a rightward shift in all of the plots. Increasing environmental [ADP][Pi] by a factor of 103 alters 16 myosin system behavior. Average force and plateau ATP consumption rate are reported as per whole sarcomere. (H, inset) The one-myosin analytical system’s predictions of average force compared to muscle strip data (red circles) adapted from White [79]. Data are normalized to force under complete rigor, at [ATP] = 0.1 μM for the analytical system and [ATP] = 0mM for experimental data.