Thermodynamics and Kinetics of a Binary Mechanical System: Mechanisms of Muscle Contraction

Figure 1. Binary mechanical model system. (A) Two-state (M and AM) scheme in which a single motor (blue ovals) undergoes a discrete conformational change upon binding to a track (black rectangle), generating a displacement, d, at a rate f+ (M to AM). The reverse transition occurs at a rate f– (AM to M). Motors can be irreversibly detached (AM to M) through an active (ATP-dependent) process that occurs at a rate v. (B) If a single system spring with stiffness κsys equilibrates with the surroundings, F determines the distribution of states between M and AM (eq 4). If force generation either occurs against a fixed length (no movement, x, of the blue bar) or occurs much more rapidly than the spring equilibrates with the surroundings, F is mechanistically determined by working steps (eq 3) that displace the system spring an effective distance, deff = d/(a·N). (C) Equation 4 (colored curves) is plotted at ΔG° increments of 1 kT, and eq 3 is plotted (black lines) at NM° increments of 5.

Figure 2. Simulation of a thermodynamic work loop. (A) Four binding pathways of a thermodynamic work loop were simulated (eqs 5 and 6, parameters in Table 1) and plotted against Figure 1C. Isothermal stretch (pathway 1) was simulated both as a slow continuous stretch (dashed red line) and as small discrete stretches that generate force (vertical red lines) and then relax through the reverse adiabatic force generation (red lines parallel to black lines) equilibrating at the ΔG° = −2 kT isotherm. Adiabatic force generation (pathway 2) was simulated following an increase in ΔG° from −2 kT to −6 kT (red line parallel to black lines). Isothermal shortening (pathway 3) was simulated both as a slow continuous shortening (dashed red line) and as small discrete shortening steps (vertical red lines) re-stretched by adiabatic force generation (red lines parallel to black lines) that equilibrates with the ΔG° = −6 kT isotherm. Reversal of adiabatic force generation (pathway 4) was simulated following a decrease in ΔG° from −6 kT to −2 kT (red line parallel to black lines). All simulations in panel (A) were obtained from simulated time courses of force transients using eqs 5 and 6. Examples of time…

Figure 3. Simulations of force transients. (A) Three phases of a force transient following a rapid increase (phase 1, red) and decrease (phase 1, blue) in force are simulated (eqs 5 and 6, Table 1 parameters) and plotted against Figure 1C. Adiabatic force generation (phase 2, blue) overshoots the ΔG° = −6 kT isotherm by (1 – a)ΔG° if system forces are not equilibrated at the end of phase 2. Equilibration of forces occurs through isothermal loss (phase 3, blue) or gain (phase 3, red) of (1 – a)ΔG° as heat to the surroundings at a rate of 4 s–1. (B) Time course of the simulations in panel (A).

Figure 4. Simulations of active (v > 0) isometric force generation. (A) Entropic beating starts with F = 0 and NM = 30. Simulated adiabatic force generation (red line) occurs rapidly along an adiabatic binding curve (black lines) equilibrating with the ΔG° = −6 kT isotherm. Entropic force generation then occurs through the reaction cycle (v). Both discrete and continuous simulations of entropic force generation are simulated. Continuous simulations are performed using eqs 5 and 6 with a rate, v, of 1 s–1. Discrete simulations are irreversibly detachment (ΔNM = 5, horizontal red line segments) followed by simulations of adiabatic force generation that equilibrates with the ΔG° = −6 kT. Entropic force generation continues until no motors are bound, at which point the system spring catastrophically shortens returning to initial conditions (F = 0, NM = 30). (B) Force recovery starts with F = 0 and NM = 0. Force generation occurs through the reaction cycle like entropic force generation in panel (A).

Figure 5. Simulations of single isotonic reaction cycles at different forces, F. Starting from NM = 15 and F set to a value that satisfies equilibrium conditions (eq 1, ΔrG = 0), one motor is irreversibly transferred from AM to M, increasing NM to 16 and adding energy, δE = ΔrG = ΔG° + F·deff to the system (horizontal line not shown). Simulated (eqs 5 and 6, Table 1 parameters) adiabatic force generation equilibrates with the a·ΔG° isotherm. Loss of ΔrG from the system by shortening of the system spring (power stroke) is simulated at a rate of 0.8 s–1.