Muscle active force-length curve explained by an electrophysical model of interfilament spacing

Abstract

The active isometric force-length relation (FLR) of striated muscle sarcomeres is central to understanding and modeling muscle function. The mechanistic basis of the descending arm of the FLR is well explained by the decreasing thin:thick filament overlap that occurs at long sarcomere lengths. The mechanistic basis of the ascending arm of the FLR (the decrease in force that occurs at short sarcomere lengths), alternatively, has never been well explained. Because muscle is a constant-volume system, interfilament lattice distances must increase as sarcomere length shortens. This increase would decrease thin and thick-filament electrostatic interactions independently of thin:thick filament overlap. To examine this effect, we present here a fundamental, physics-based model of the sarcomere that includes filament molecular properties, calcium binding, sarcomere geometry including both thin:thick filament overlap and interfilament radial distance, and electrostatics. The model gives extremely good fits to existing FLR data from a large number of different muscles across their entire range of measured activity levels, with the optimized parameter values in all cases lying within anatomically and physically reasonable ranges. A local first-order sensitivity analysis (varying individual parameters while holding the values of all others constant) shows that model output is most sensitive to a subset of model parameters, most of which are related to sarcomere geometry, with model output being most sensitive to interfilament radial distance. This conclusion is supported by re-running the fits with only this parameter subset being allowed to vary, which increases fit errors only moderately. These results show that the model well reproduces existing experimental data, and indicate that changes in interfilament spacing play as central a role as changes in filament overlap in determining the FLR, particularly on its ascending arm.

Figure 1

Relative filament overlap versus half-sarcomere length for the case that false cross-bridges produce a marginal amount of (${\tilde{F}}_{\text{false}}=0.1$, solid line), half of (${\tilde{F}}_{\text{false}}=0.5$, dashed line), or the full (${\tilde{F}}_{\text{false}}=1$, dotted line) force of a proper cross-bridge. Full (not half-) sarcomere sketches shown for clarity. Note that actin (red) and myosin (blue-cyan) filament radial spacing decreases at longer lengths (insets). To see this figure in color, go online.

Figure 2

Hill plot: probability of an actin site to be available for myosin head binding (active) (black line) versus relative calcium concentration. At $\tilde{c}={\tilde{c}}_{50}\approx 0.23$, the probability of an actin site to be active was already 50% (black dot). For explanation and parameters, see text.

Figure 3

Cross-sectional view of sarcomere lattice. A double triangular lattice of actin filaments (red-orange circles: red, actin; orange, tropomyosin) and hexagonal lattice of myosin filaments (blue-cyan circle: blue, myosin back bone (mbb); cyan, myosin heads (S1)), with each filament located in the center of six actin filaments. As the sarcomere lengthened from $\approx 1/2\cdot {\ell }_{\text{hs},\text{ref}}$ (left) to twice this length ($\approx {\ell }_{\text{hs},\text{ref}}$, right), cross-sectional area CSA_hsp halved. Black rhombus shows one possible half-sarcomere primitive cell. Every myosin lies on the vertex of a rhombus and is thus shared by four primitive cells, for a total of one myosin per cell. The two interior actin filaments are not shared and hence count fully, resulting in an actin:myosin ratio of 2:1. For parameter meanings see text or Table 2. To see this figure in color, go online.

Figure 4

Change of the potential energy function of the myosin head with sarcomere activity. (Left) Actin inactive state. The actin (red) charges are blocked by tropomyosin (orange) and the potential energy of the myosin head due to the actin electrostatic field is therefore orders of magnitude smaller than the potential energy due to the myosin backbone field (blue). The myosin head (cyan) remains near the backbone at the most negative potential energy value. Black line, the difference between the potential energies of the myosin head due to each cylinder; red line, due to actin; blue line, due to myosin. (Right) Activated state. The tropomyosin has moved aside and the actin charges are no longer blocked. The potential energy of the myosin head due to the activated actin field has become much more negative (its magnitude has increased), and the myosin head therefore orients more toward the actin surface. In both states, myosin surface charge remains unchanged (≈9 zJ potential energy at the surface). A myosin to actin center-to-center distance of approximately 25 nm was chosen, corresponding to the situation at ${\ell }_{\text{hs},\text{ref}}$. Myosin and actin radii were 7.5 nm and 5.5 nm, respectively (92). Myosin and actin charge densities were −12 e₀ and −4 e₀ per nm, respectively (101). To see this figure in color, go online.

Figure 5

Maxwell-Boltzmann distribution: probability density function (PDF) (black) and cumulative distribution function (CDF) (gray).

Figure 6

(Left) Probability of location (f_pos(x)) of the tip of the myosin head (S1) at different half-sarcomere lengths ${\ell }_{\text{hs}}$ and relative calcium ion concentrations $\tilde{c}$. Note how, for short lengths, the calcium concentration did not influence the probability (solid, dashed, and dotted blue lines). The same holds for the change of lengths from short to reference length at low calcium (blue, red dotted lines). Increasing calcium concentration or sarcomere length shifted the probability in favor of the head being arranged toward the actin filament, as increasing calcium increases the visible charges on actin and increasing length decreases the lattice spacing and thus shifts the potential energy function of actin closer toward the head. (Right) The probability of myosin binding, i.e., of a head being “caught” by available actin (P_aas ⋅ P_esa in Eq. (3)), calculated by the integral of the location’s distribution from the minimum of the potential energy function to the tip of the swung-out head (Eq. (14)). The sigmoidal curves shifted right as calcium concentration decreased. To see this figure in color, go online.

Figure 7

Final result of Eq. (3). Filament overlap (P_ovl, solid gray line) times the product of the amount of cleared active sites and myosin binding (P_aas ⋅ P_esa, black dotted lines) is shown for several relative calcium concentrations (colored dashed lines). The scaled version of this amount ($\varsigma \cdot P_{\text{ovl}}\cdot P_{\text{aas}}\cdot P_{\text{esa}}$) is shown by the correspondingly colored solid lines and represents the relative active force-length relation (FLAR) of the half-sarcomere. The dashed vertical lines show the edges of the Gordon et al. (1966) plateau region (see text). The dotted vertical line at 0.8 nm is the initial (half-)myosin filament length. To see this figure in color, go online.

Figure 8

Optimal fit of model $\mathfrak{F}$ to the available datasets, see section “muscle datasets.” The dashed vertical lines in each panel show the edges of the Gordon et al. (1966) plateau region. The dotted vertical line in each panel is myosin filament length, at which the ends of the thick filament would reach the Z disk.

Figure 9

Optimal fit of model ${\mathfrak{F}}_{\text{simple}}$ to the available datasets, see section “muscle datasets.” Note that the color bar differs from Fig. 8 due to inherently different Hill exponents. The dashed vertical lines show the edges of the Gordon et al. (1966) plateau region. The dotted vertical line is myosin filament length, the sarcomere length at which the ends of the thick filament would reach the Z disk.

Figure A. 10

Conceptual explanation of FLAR rightward shift with decreasing [Ca²⁺]. Some changes (rightward translocation, column A; squaring, column C) in the myosin force generation curve (${\tilde{F}}_{\text{asc}}$, top row) result in rightward shifts of the FLAR peak (middle row). Changing ${\tilde{F}}_{\text{asc}}$ slope, alternatively, does not (column B). An understanding of these different responses of the FLAR peak to changes in ${\tilde{F}}_{\text{asc}}$ can be obtained by considering how these changes affect the equation that gives rise to the FLAR, $\tilde{F}\left({\ell }_{\text{hs}}\right)={\tilde{F}}_{\text{asc}}\cdot {\tilde{F}}_{\text{des}}$ (bottom row). See text for detailed explanation. Keys in (A2) and (A3) apply to all other panels in same row. To see this figure in color, go online.

Figure B. 11

Sensitivity analysis. The factors in front of the parameters indicate the lower (dotted) and upper (dash-dotted) bound at which the mean deviation D_r from the three solid curves is exactly 10%. The metric, i.e., distance, of a function g(x) from the solid curves r(x) was therefore defined by $D_r:=\underset{{\ell }_{\text{hs}}}{\int }|r\left(x\right)-g\left(x\right)|\text{d}x/\underset{{\ell }_{\text{hs}}}{\int }r\left(x\right)\text{d}x$.

Figure B. 12

Range over which each parameter in Fig. B11 varied, arranged by increasing magnitude. Two clear breaks are apparent, one between r_act and ${\mathcal{Z}}_{\text{act}}$ and the another between ϵ_r and ${\mathcal{Z}}_{\text{S}1}$. We identified the first eight parameters as being sensitive parameters (see Table 3). Note that the bar for ${\tilde{F}}_{\text{false}}$ is truncated.

Figure B. 13

Best (Stephenson, upper row) and worst (Guschlbauer (a), lower row) examples of fixing non-sensitive parameters. Left column shows results from the parameter fit with bounds from Table 4. Right column shows the optimal fit after fixing the nine most insensitive parameters. Note that for Guschlbauer (a), the fits to the two highest activation FLAR datasets were identical (both found $\tilde{c}=1$). For residuals, see Table B1. To see this figure in color, go online.

Figure C. 14

Electrostatic potential (left) and force (right) resulting from different model approaches (Table C1; black, blue, lilac, green) under different conditions (solid, dashed, dotted). Experimental conditions as in (98), ϵ_r = 80 and I = 0.05 mol/L, are used as reference against increased permittivity and increased ionic strength, respectively. The solid blue lines represent the corresponding reference case Debye-Hückel proposed by Nakajima et al. (1997). Reference values of the potential (gray dotted line) and force (gray dotted and dash-dotted line) taken from (98) given for orientation. The actin radius r_act (red dotted line) marks the start of the cylinder and sphere formulations. To see this figure in color, go online.

Figure D15

Correlation coefficients between each pair of optimized model parameters, see Table 4. White closed and open circles are α = 0.01 and α = 0.05 deviations of the correlation coefficient ρ from zero, respectively. The test statistic was calculated as $t=\frac{\rho \cdot \sqrt{n-2}}{\sqrt{1-{\rho }^2}}$ (153) and compared against the corresponding quantile of the Student’s t-distribution with n − 2 degrees of freedom (two-sided test), where n = 12 is the number of available datasets.