Figure 1.
(A) Isometric (black trace) and stretched (red trace) tension responses in a myofibril consisting of 50 half-sarcomeres in series, with 20% variation in half-sarcomere strength (α = 0.2). The stretch response in a single half-sarcomere (which is equivalent to a multi half-sarcomere system with no variation in half-sarcomere strength) is also shown for comparison (blue trace). (B) Time course of average half-sarcomere length for isometric (black trace) and stretch (red trace) protocols. In each simulation, fibers were activated gradually over a period of one second, and then allowed to stabilize for another second before the stretch was imposed. Tension in the myofibril consisting of a single half-sarcomere decays rapidly back to the isometric value, while tension in the heterogeneous fiber remains elevated for many seconds following stretch, strongly resembling the classical force enhancement response. (C) Time courses of 50 individual half-sarcomeres are shown for a single simulation with an α value of 0.2, and reveal continuing internal motion after the end of stretch. This is the source of enhancement.
Table 1.
Passive length-tension parameter values, obtained by fitting Equation 2 to published data from rabbit psoas muscle [21], [22].
Figure 2.
Velocity dependence of the force enhancement response.
(A) Tension responses in a simulated fiber containing 50 half-sarcomeres in series and an α value of 0.2 at various stretch velocities. The dashed trace represents tension under isometric conditions at the final length (L0). (B) Fiber length as a fraction of final length (L0). (C) Residual force enhancement magnitude (RFE) at each of the simulated stretch velocities. RFE was not strongly dependent on stretch velocity. Colors in each panel link tension, length, and RFE values that have the same stretch velocity.
Figure 3.
Stretch magnitude dependence of the force enhancement response.
(A) Tension responses in a simulated fiber containing 50 half-sarcomeres in series and an α value of 0.2 for various stretch magnitudes. The dashed trace represents tension under isometric conditions at the final length (L0). (B) Fiber length as a fraction of final length (L0). (C) Residual force enhancement magnitude (RFE) at each of the simulated stretch magnitudes. RFE was proportional to stretch magnitude, in agreement with many experimental measurements. Colors in each panel link tension, length, and RFE values that have the same stretch magnitude.
Figure 4.
Force enhancement at different degrees of myofilament overlap.
(A) Residual force enhancement (RFE) as a function of final mean sarcomere length, simulated in a fiber containing 50 half-sarcomeres in series and an α value of 0.2. (B) Enhanced tension after stretch (circles) shown with the steady-state length tension curve produced by the model (solid line). Some enhancement can be seen on the ascending limb, plateau, and descending limb of the length tension curve.
Figure 5.
Sensitivity of residual force enhancement to amount of half-sarcomere heterogeneity.
Residual force enhancement in a fiber composed of 50 half-sarcomeres in series as a function of α, the variation in half-sarcomere strength. Stretch magnitude was 8% of the final length. Note the logarithmic scale on the x-axis. As little as 2% variation in half-sarcomere strength produces 5% residual force enhancement following stretch.
Figure 6.
Heterogeneous passive tension properties also produce enhancement.
The standard stretch protocol was simulated with varying levels of passive (β) and active (α) variation among 50 half-sarcomeres connected in series. Each plot shows tension responses in stretch (red) and isometric conditions (blue). Inset numbers indicate the percentage of residual force enhancement after stretch for the given combination of β and α shown in the margins. Increasing just passive variation among half-sarcomeres (first column) gives rise to residual force enhancement behavior. When α and β are co-varied, it is apparent that the magnitude of enhancement is more sensitive to α.
Figure 7.
Force enhancement in a model with multiple myofibrils.
A model consisting of 6 myofibrils in parallel, each containing 50 half-sarcomeres in series was subjected to a slow ramp stretch (8% L0 over 4 seconds). Z-lines in adjacent myofibrils were mechanically linked so that they stayed in register throughout the protocol, while half-sarcomeres remained unrestricted. α was set to 0.2. (A) Tension in the stretched muscle (red) decayed slowly and remained above the isometric control record (black), giving 12.7% residual force enhancement 6 seconds post-stretch. (B) Average sarcomere length during the two simulations (colors as in panel A). (C) Sarcomere length for each of 25 whole sarcomeres in the simulated muscle fiber (color traces), normalized to the initial average sarcomere length of 2.4 µm. The relative stability and absence of ‘popping’ in sarcomeres is seen by comparison with the time course of half-sarcomere lengths (gray traces, normalized to initial half-sarcomere length of 1.2 µm) appearing in the background.
Figure 8.
Enhancement by dynamic interactions among dissimilar half-sarcomeres.
Residual force enhancement is produced here in a model of two dissimilar half-sarcomeres connected in series. (A) The length-tension coordinates of the two half-sarcomeres are shown here one second after the end of stretch (circles) and at steady-state (triangles, >30 seconds post stretch). The steady-state length-tension curves, which include passive and active tension, are shown for both half-sarcomeres, along with horizontal lines indicating levels of isometric and enhanced tension for the system. (B) Tension response to stretch (solid trace). The circle marks the time point corresponding to the length-tension coordinates marked by circles in panel A. (C) Time courses of average (black trace) and individual half-sarcomere lengths (red and blue traces). Circles correspond with those plotted in panel A.