Elastic Energy Storage and Radial Forces in the Myofilament Lattice Depend on Sarcomere Length

<div><p>We most often consider muscle as a motor generating force in the direction of shortening, but less often consider its roles as a spring or a brake. Here we develop a fully three-dimensional spatially explicit model of muscle to isolate the locations of forces and energies that are difficult to separate experimentally. We show the strain energy in the thick and thin filaments is less than one third the strain energy in attached cross-bridges. This result suggests the cross-bridges act as springs, storing energy within muscle in addition to generating the force which powers muscle. Comparing model estimates of energy consumed to elastic energy stored, we show that the ratio of these two properties changes with sarcomere length. The model predicts storage of a greater fraction of energy at short sarcomere lengths, suggesting a mechanism by which muscle function shifts as force production declines, from motor to spring. Additionally, we investigate the force that muscle produces in the radial or transverse direction, orthogonal to the direction of shortening. We confirm prior experimental estimates that place radial forces on the same order of magnitude as axial forces, although we find that radial forces and axial forces vary differently with changes in sarcomere length.</p> </div

Example axial and radial forces.

<p>The mean (lines) and standard deviations (shaded regions) of axial and radial forces as they develop at a sarcomere length of over the course of 10 runs. Each run consists of 400 time steps, each 1 ms long. Maximum forces are calculated from the mean of the last 50 ms of such runs.</p

Models produce radial and axial forces.

<p>The one-dimensional cross-bridge model shown in (A) produces force and exists only in the axial direction. The two-dimensional cross-bridge model shown in (B) produces both axial and radial forces, and responds to changes in lattice spacing. A multi-filament model using one-dimensional cross-bridge, shown in (C), is diagrammed as a three-dimensional system but is insensitive to changes in lattice spacing and unable to explore radial force produced during contraction. Using two-dimensional cross-bridges in the same model geometry, in (D), allows the recording of radial forces and altered force dynamics with altered lattice spacing.</p

Radial force is of the same order of magnitude as axial force.

<p>Asymptotic maxima of 10 runs at each sarcomere length with standard deviation. Radial and axial forces obey similar scaling trends across the sarcomere lengths and lattice spacings of a classic length-tension curve. The level of radial force varies from 2.4 times the level of axial force at extremely short sarcomere lengths to 0.9 times the axial force at the longest sarcomere lengths. The radial force plateau ends at a shorter sarcomere length than does axial force plateau.</p

Energy stored varies with sarcomere length as well as energy input.

<p>All energy present in the isometrically contracting half-sarcomere derives from the hydrolysis of ATP. This permits a direct comparison of the energy input to the system, as measured by the consumption of ATP, to the energy stored across all filaments and cross-bridges. The fraction of energy stored is shown to change as sarcomere length drops below . A contractile lattice with a stored energy dependent only on the rate at which ATP is consumed would not exhibit the hysteresis present as sarcomere length changes. This suggests that of the energy released by ATP, the fraction which is stored instead of being dissipated is partially determined by sarcomere length.</p

Mechanical properties of the myofilament lattice influence cooperative force production and maximal XB turnover.

<p>Simulation results for steady-state magnitude and rate of force development (<i>k_dev</i>) are plotted against pCa as (A–B) XB stiffness (<i>k_xb</i>) varied, (D–E) thick or thin filament stiffness (<i>k_m</i> or <i>k_a</i>) varied independently, or (G–H) both filament stiffness values (<i>k_fil</i>) varied simultaneously. Parameter values from 3 parameter Hill fits to these force-pCa relationships demonstrate that mechanical properties (<i>i.e.</i> stiffness) of they myofilament lattice can influence cooperativity (<i>n_H</i>; panel C) and Ca²⁺ sensitivity (<i>pCa₅₀</i>; panel F) of force production. The rates of XB cycling or turnover directly correlate with ATPase values (I). All relative values were normalized to results for the standard parameter values (solid black circle) of <i>k_xb</i> = 3 pN nm⁻¹, <i>k_fil</i> = <i>k_m</i> = <i>k_a</i> = 1X, <i>RU_span</i> = 9 actins, and all possible forms of cooperativity were implemented for all simulations. Symbols represent mean±SE in all panels except C and F, and error bars reside within the symbol when not visible. Symbols in panels C and D depict predicted parameter values with error bars representing 95% confidence intervals.</p

Cross-bridge (XB) binding stabilizes and augments thin filament activation.

<p>(A) Steady-state thin filament activation is plotted against pCa for simulations where standard parameter values were applied (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi-1002506-g003" target="_blank">Figure 3</a>), in the presence and absence of kinetic forms of cooperativity. XBs produce larger increases in fractional thin filament activation when cooperative kinetics were present, evidenced by the increase in activation in the presence versus absence of XB binding. These XB-dependent increase in thin filament activation are greatest at submaximal pCa levels, shown by the differences between the two curves when kinetic forms of cooperativity were implemented. Symbols represent mean±SE, where error bars reside within the symbol when not visible. (B) Average thin filament activation is plotted against time for simulations near the <i>pCa₅₀</i> value of the steady-state force-pCa response (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi-1002506-g003" target="_blank">Figure 3</a>), with (pCa = 5.9) and without (pCa 4.25) kinetic forms of cooperativity. These time series activation traces demonstrate the representative slowing of thin filament activation kinetics and increased magnitude of thin filament activation due to XBs in the presence of cooperativity. All simulations used standard parameter values: <i>ρ_Tn</i> = 1, <i>k_xb</i> = 3 pN nm⁻¹, <i>k_fil</i> = <i>k_m</i> = <i>k_a</i> = 1X, and <i>RU_span</i> = 9 actins.</p

Multiple forms of cooperativity combine to simulate the physiological force-pCa relationship.

<p>Various combinations of cooperative thin filament activation kinetics affect the steady-state force-pCa response differently, shown for all possible source combinations when <i>r_t,12</i> was targeted (A) versus both <i>r_t,12</i> and <i>r_t,23</i> being targeted in combination (B). Each line depicts the 3-parameter Hill fit to the simulated force-pCa response, while symbols show <i>n_H</i> (C) and <i>pCa₅₀</i> (D) values for these fits, with error bars representing 95% confidence intervals. All simulations used standard parameter values: <i>ρ_Tn</i> = 1, <i>k_xb</i> = 3 pN nm⁻¹, <i>k_fil</i> = <i>k_m</i> = <i>k_a</i> = 1X, and a <i>RU_span</i> of 9 actins. The dashed lines illustrate an example simulation in the absence of cooperative thin filament activation kinetics, which compares well with prior studies <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi.1002506-Chase1" target="_blank">[25]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi.1002506-Tanner1" target="_blank">[27]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi.1002506-Tanner2" target="_blank">[43]</a> after adjusting for <i>K′₁</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi-1002506-t003" target="_blank">Table 3</a>). The shaded underlay in panels A and B represents the measured physiological range from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002506#pcbi-1002506-g002" target="_blank">Figure 2</a>.</p

Hill-fit parameters to steady-state force and thin filament activation responses versus pCa.

<p>Hill parameter values are listed as mean±SD.</p

Values of <i>RU_span</i> compared to physiological thin filament structures.

<p>Values of <i>RU_span</i> compared to physiological thin filament structures.</p