Exhaustion of Skeletal Muscle Fibers Within Seconds: Incorporating Phosphate Kinetics Into a Hill-Type Model

Figures

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  Figure 1. Force-time courses of GAS (left column) and PLA (right column) at various muscle lengths (see section 2). Models were fit to data, where different lengths are indicated by different colors as labeled in the top row. Parameter values are summarized in Table 2. (A,B) Classical model (GAS−, PLA−) without phosphate dynamics. (C,D) New model (GAS+, PLA+) including phosphate dynamics. (E,F) New model with switched off ATP hydrolysis (GAS+, 0, PLA+, 0, i.e., k^hyd=0).

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  Figure 2. Force-time course of an additional isometric experiment on GAS (dots) as well as the model output of GAS+ (gray line), utilizing parameter values from Table 2. Stimulation duration is 1.4 s, GAS length ℓGAS = 126mm. Inlay shows the concentration-time courses of the most relevant bio-chemical reactants (blue: ATP; red: Pi; green PCR) as well as the relative chemical potential of ATP (black: μ~ATP). For more details see Figure A1 in Appendix B.

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  Figure 3. Example sensitivity-time matrices for each model parameter at a long length GAS (A, ℓGAS = 132mm, cf. dark blue curve in Figure 1A) and a short length PLA (B, ℓPLA = 116mm, cf. yellow curve in Figure 1B). Large absolute sensitivities indicate the local importance of parameter value accuracy. For further explanation see text. For scaling purpose, sensitivity values had been truncated at absolute values >1. For example for ℓSEE, 0, sensitivities go up to absolute values in the magnitude of 104.

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  Figure 4. (A) Experimental force rates (time derivatives) of GAS at different MTU lengths, normalized to maximum force. Color encoding is the same as in Figure 1. Data were smoothed by a moving average filter with 40 ms width for clarity of depiction, but not for calculations. Inlay shows the mean force rates (solid lines) in the 0.1 s interval between 0.58 and 0.68 s, where near steady-state conditions were assumed. (B) Experimental force decay rates vs. CE length at different MTU lengths (colored circles) as extracted from (A) in comparison to theoretical modified CE force-length relation (Equation 8, black line). An additional data point (diamond) was taken from another GAS experiment (see Figure 2). The best-fit parameter κGAS = 0.163 s−1 was found by an optimization routine, all other parameters were taken from Table 2. The unmodified and scaled force-length relation [κGAS·F~isom(ℓCE), gray line] is given for reference. In (C,D) the procedure is mirrored for PLA, where κPLA = 0.165 s−1.

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  Figure 5. Schematic cross-bridge (Lymn-Taylor) cycle (1-2-3-4-5-6-7-1). New abbreviations are used as follows: Troponin C (TnC), actin (A), open and closed myosin switch-2-region (MO and MC), light chain domain (LCD), catalytic domain (CD). For their explanations, see text. Although most processes are easily reversible, for the sake of clarity only one direction is depicted.

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  Figure 6. Simulated time courses for CE length (A) and MTU force (B) for a GAS muscle during isometric contractions, utilizing fixed parameters from Table 2. CE length are displayed relative to ℓCE, opt ≈ 0.0198m and forces relative to Fmax ≈ 133N. Muscle length varies about −10, …, +15mm, in steps of 5 mm, around the reference length ℓMTU = 122mm. Colors and line-styles in both sub-figures are coherent, indicating MTU lengths [as specified in (A)] and stimulation levels [as specified in (B)]. Two observations should be highlighted. First, the force level in (B) at which (theoretically) infinite exhaustion time occurs settles at around 0.16, corresponding to 16%Fmax, which is in perfect agreement with the 15%Fmax from in vivo experimental data (Rohmert, 1960). Second, the force at the longest muscle length [blue line in (B)] settles at significantly higher values, which is due to passive (PEE) forces that were herein not modeled to show any exhaustion.

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  Figure 7. (A) Simulated force-length-exhaustion time diagram for variations in MTU lengths and stimulation levels as given in Figure 6. (B–D) Contour plots through the three different planes in (A). In (B), the force-length diagram for various exhaustion times is shown. The active and passive force-length characteristic of the CE is clearly visible. In (C), the force-exhaustion time courses for various CE length are displayed. The longer the CE, the longer the exhaustion time. The typical exponential (or hyperbolic) characteristic (Rohmert, 1960; Frey-Law and Avin, 2010) is visible. (D) Shows CE length-exhaustion time curves for various force levels. The higher the force, the shorter the exhaustion time at the same length. For longer muscles, the passive force determines the exhaustion time.

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  Figure A1. Time courses of [ATP], [Pi], [ADP], and [PCr] (left ordinate) as well as μ~ATP (right ordinate) for GAS (A) and PLA (B). Solid lines represent the time courses for the shortest muscle examined in our experiments (cf. Figure 1). Dashed lines represent the respective longest muscle. Time courses for [Cr] are not shown due to a near congruence with [Pi](t). Note that the ATP affinity decreases substantially, due to increasing Pi, while the [ATP] itself remains almost constant.