Cross-bridge model for predicting muscle short-range stiffness during movement

Figures

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  Fig 1. Muscle thixotropy is a history-dependent reduction in muscle short-range stiffness. A. Experimental set-up used for characterizing single fiber muscle mechanics. A single permeabilized muscle fiber is attached to a length controller on one end and a force transducer at the other. Muscle activation is controlled using a bathing solution with a fixed calcium (Ca2+) concentration. B. Typical example data from a muscle mechanics experiment in which triangular length changes were imposed on an isolated muscle fiber. When the fiber has been isometric for a while, force rapidly increases following onset of the first stretch (ΔF). But when the fiber is shortened and stretched again, the force increase ΔF is smaller for the same length increase ΔL. C. The history dependence of short-range stiffness may be explained by a cross-bridge mechanism. Molecular cross-bridges are thought to act as springs when attached (1) that get strained during stretch (2), and either unstrained or detached during shortening (3). When another stretch occurs shortly after shortening, some detached cross-bridges will not have had the time to reattach. This mechanism would explain why force increases less rapidly…

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  Fig 2. Modeled and measured forces for fitting trials: Hill-type models and a regular 2-state cross-bridge (XB) model. A. Entire protocol. Top row shows measured fiber length changes for a condition with a conditioning stretch (darker solid lines) and without a conditioning stretch (lighter dotted lines). Bottom row shows the corresponding empirical forces (“Data”, grey) at three different calcium concentrations (i.e. pCa 4.5, pCa 6.1 and pCa 9.0), alongside forces of a Hill-type model without a series element (“Hill (no SE)”, light blue), a Hill-type model with a series element (“Hill (with SE)", darker blue), and a regular 2-state cross-bridge model (“2-state XB”, red). B. Forces during the conditioning stretch (Cond.) and test stretch (Test) at submaximal activation levels.

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  Fig 3. Modeled and measured forces for fitting trials: cross-bridge (XB) models. A. Entire protocol. Top row shows measured fiber length changes for a condition with a conditioning stretch (darker solid lines) and without a conditioning stretch (lighter dotted lines). Bottom row shows the corresponding empirical forces (“Data”, grey) at three different calcium concentrations (i.e. pCa 4.5, pCa 6.1 and pCa 9.0), alongside forces of a regular 2-state cross-bridge model (“2-state XB”, red), a cross-bridge model with cooperative thin filament activation (“2-state XB coop”, yellow), and a cross-bridge model with both cooperative thin filament activation and cooperative thick filament activation (“3-state XB coop”, purple). B. Forces during the conditioning stretch (Cond.) and test stretch (Test) at submaximal activation levels.

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  Fig 4. Modeled and measured forces for testing trials: cross-bridge (XB) models. A. Entire protocol. Top row shows measured fiber length changes for a condition with a short recovery of 0.001 s (darker solid lines) and with a longer recovery of 0.316 s (lighter dotted lines). Bottom row shows the corresponding empirical forces (“Data”, grey) at three different calcium concentrations (i.e. pCa 4.5, pCa 6.1 and pCa 9.0), alongside forces of a Hill-type model with a series element (“Hill (with SE)”, blue), a cross-bridge model with cooperative thin- and thick filament activation (“3-state XB coop”, purple), and a cross-bridge model with cooperative thin- and thick filament activation, and a forcibly-detached state (“4-state XB coop”, green). B. Forces during the conditioning stretch (Cond.) and test stretch (Test) at submaximal activation levels.

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  Fig 5. Root-mean-squared deviations (RMSDs) between model and experimental forces, normalized with respect to the maximal isometric force F0. Force RMSDs σF are shown for Hill-type models without and with series elasticity (“Hill”, light and dark blue), 2-state cross-bridge model (“2-state XB”, red), 2-state cross-bridge model with cooperative dynamics (“2-state coop”, yellow), 3-state cross-bridge model with cooperative dynamics (“3-state coop”, purple) and 4-state cross-bridge model with cooperative dynamics (“4-state coop”, green). RMSDs are shown for entire stretch-shorten protocol (top row), conditioning stretch (middle row) and test stretch (bottom row). A. Averaged over all trials. Asterisks indicate significance at p < 10-4. B. Effect of calcium activation level at constant amplitude (3.83%L0) and recovery time (0.001 s). C. Effect of stretch amplitude at constant activation level (pCa 6.3) and recovery time (0.001 s). D. Effect of recovery time at constant amplitude (3.83%L0) and activation level (pCa 6.3). In panels B, C and D, darker shades indicate lower pCa, larger amplitude, and longer recovery, respectively. In these panels, bars with a white surface area indicate the…

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  Fig 6. Empirical and model relative short-range stiffness across muscle activation levels, stretch amplitudes and recovery times. Fiber-averaged (n = 7) model predictions and errors are shown for Hill-type model without series compliance (“Hill (no SE)”, light blue), the Hill-type model with series compliance (“Hill (with SE)”, dark blue), the 2-state cross-bridge model (“2-state XB”, red), 2-state cross-bridge model with cooperative dynamics (“2-state coop”, yellow), 3-state cross-bridge model with cooperative dynamics (“3-state coop”, purple), and 4-state cross-bridge model with cooperative dynamics (“4-state coop”, green). Data are shown both averaged across all fibers (n = 11, grey circles), and for individual fibers (black dots). Four (out of 11) fibers were not modeled because they did not include stretch-shortening-stretch trials at Intermediate and/or Maximal activation levels, which was key to parameter fitting. Filled symbols and error bars indicate mean ± s.d. across trials with considerable short-range stiffness reductions, open symbols indicate trials with little to no short-range stiffness reductions. Squares indicate the condition that panels A-C have in common. A. Effect…

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  Fig 7. Empirical and model relative short-range stiffness across conditions. Model predictions are averaged over all fibers in the fitting set (n = 7), data are averaged over all data in the dataset (n = 11). A. Relative short-range stiffness at submaximal activation levels across amplitudes and recovery times. Relative short-range stiffness is indicated by both the vertical location and surface color. Fitting trials are indicated with stems, connecting to the corresponding dots. The bottom surface indicates the experimental grid of stretch amplitudes and recovery times, distinguishing between conditions measured in all fibers (dark grey lines, n = 11) and conditions measured in a subset of fibers (light grey lines, n = 3), conditions with history-dependent stiffness reductions (grey surface) and conditions without such stiffness reductions (white surface). At submaximal activation levels, empirical short-range stiffness (grey dots) decreases with larger stretch amplitudes and shorter recovery times. This is not captured by the Hill-type model without series compliance (light blue, left), partially captured by the Hill-type model with series compliance (dark blue, middle), and mostly…

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  Fig 8. Computational cost of discretized and approximated cross-bridge models. The processing time (left column) and the force error (middle column) of solution methods based on discretization depends on the strain vector used. The strain vector can be varied in two ways: (1) fixing the strain range and increasing the number of bins (top row), or (2) fixing the bin width and increasing the number of bins (bottom row). In both cases, processing time and force error increase and decrease with the number of strain bins. Consequently, the force error depends on the processing time (right column). Two types of discretization methods are shown: the traditional method (yellow) and the method of characteristics (brown). In addition, a Gaussian approximated solution method is shown (black dotted line and black dot).

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  Fig 9. Experimental data. Data were obtained using stretch-shortening-stretch protocols that differed in conditioning stretch amplitude, recovery time, and calcium concentration. A. Muscle fiber length and force trajectories for stretch-shorten-stretch protocols at constant stretch amplitude and recovery time but variable calcium concentration (pCa). B. Short-range stiffness was defined as the slope of the force versus length data during the first 10 ms after stretch onset bSRS. Red and blue dots indicate data points used to determine stiffness during the conditioning stretch and test stretch, respectively. C. Trajectories for protocols at constant calcium concentration and recovery time but variable conditioning stretch amplitude (AMP). D. Trajectories for protocols at constant amplitude and calcium concentration but variable recovery time (RT). E. Grid of tested amplitudes and recovery times. Models were fitted on data from two trials per calcium concentration (red) and evaluated on data from the remaining trials (blue). Most fibers (n = 8) had grids with 4 AMPs and 5 RTs. A subset of fibers (n = 3) had grids with 7 AMPs and 7 RTs. Empirical and modeled short-range stiffness during…

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  Fig 10. Cross-bridge muscle model dynamics and interface with elastic elements. A. Cross-bridge muscle models distinguish between dynamics for thin filament activation, thick filament activation, and cross-bridge cycling. Thin filament activation depends on myoplasmic calcium (Ca2+), and is optionally mediated by the fraction of active thin filament binding sites and the fraction of formed cross-bridges (dashed lines). Thick filament activation is mediated by cross-bridge force FCB (dashed lines), which depends on cross-bridge strain x. B. Cross-bridge muscle model dynamics describe the contractile element (CE), which is interfaced with series and parallel elastic elements (SE and PE).