Active Force Generation in Cardiac Muscle Cells: Mathematical Modeling and Numerical Simulation of the Actin-Myosin Interaction

Figures

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  Fig. 1. Representation of a sarcomere. Inside sarcomeres, thin and thick filaments are arranged with a regular structure. M-lines, located at the center of the sarcomere, have the function of connecting thick filaments together. Z-discs link adjacent sarcomeres to each other and to the extracellular matrix and are connected to thick filaments through a huge cytoskeletal protein named titin

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  Fig. 2. Representation of the Lymn-Taylor cycle

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  Fig. 3. Representation of the force-velocity curve (a) and tension-elongation curves after a fast transient (b) that is typically obtained in experiments

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  Fig. 4. Typical length and tension traces obtained during the force clamp (left) and length-clamp (right) experiments. In the force clamp case, the slope of the length trace gives the steady-state shortening velocity v (represented in the picture). The phases 1,2 and 4 (we recall that phase 3 is absent in the cardiac case) are represented by circled numbers

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  Fig. 5. Sketch of the phenomenological model of [31]. A contractile element, following the law (2), is coupled in series with an elastic element, to which a quadratic energy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{e}(x) = \frac {1}{2}k x^{2}$\end{document}ue(x)=12kx2 is associated

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  Fig. 6. Scheme of the H57 model. The attachment-detachment rates of MHs (denoted respectively by f and g) depend on the XB distortion x (i.e., to which amount the myosin arm is stretched). The myosin arm is modeled as a linear elastic element with stiffness kXB

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  Fig. 7. Scheme of the CMC19 model. The MH is described by two degrees of freedom (z and y). When the MH is attached, the degree of freedom z coincides with the variable x. The attachment-detachment rates of MHs (f and g) depend on the XB distortion x. The myosin arm is modeled as a linear elastic element with stiffness kXB, whereas the degree of freedom y is associated with a bistable energy, which depends on the XB attachment state

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  Fig. 8. The force-velocity relationship (a) is characterized by the maximum shortening velocity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v^{\max \limits }$\end{document}vmax (the intercept of the curve with the axis Ta = 0) and by the inverse sensitivity of the force to velocity in isometric conditions v0, which can be interpreted as the intercept with the axis Ta = 0 of the tangent to the curve in isometric conditions. On the other hand, the response to fast transients is characterized by the normalized stiffness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde {k}_{2}$\end{document}k~2, where the subscript 2 reflects the fact that this value characterizes the T2-L2 response

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  Fig. 9. Force-velocity relationship obtained with the model (17), compared with experimental data from [9]

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  Fig. 10. Normalized force after the application of a fast length step ΔL. The fast steps reported by the blue lines (model result) and the blue circles (T2-L2 experimental data from [9]) are applied within a time interval of Δt = 200μs, whereas the red lines refer to fast steps applied with a shortening velocity of v = 0.5s− 1. Finally, the black dashed lines refer to the asymptotical response for Δt → 0