Meccanica

pp 1–13 | Cite as

A chemo-mechanical model for the single myofibril in striated muscle contraction

Active Behavior in Soft Matter and Mechanobiology

Abstract

Based on the framework of sliding-filament theory and on the cross-bridges dynamics, a mathematical model for the simulation of the force response and length change of individual myofibril is presented. The myofibril is modeled as a group of segments placed in series, each segment represents a half-sarcomere with active and elastic properties. A multiple-state cross-bridge formalism relates the half Sarcomere force to the chemical kinetics of ATP hydrolysis. The corresponding system of nonlinear nonlocal partial differential equations of the model is analyzed. A numerical approach is introduced and some numerical tests are performed. The proposed in-silico model enables the study of biologically relevant process in the muscle contraction process, also in the case of muscular diseases, with reasonable computational effort.

Keywords

Muscle contraction Multiscale chemo-mechanical model Nonlinear nonlocal equations 

References

  1. 1.
    Blinks JR, Rudel R, Taylor SR (1978) Calcium transients in isolated amphibian skeletal muscle fibers: detection with aequorin. J Physiol 277:291–323CrossRefGoogle Scholar
  2. 2.
    Campbell KS (2009) Interactions between connected half-sarcomeres produce emergent mechanical behavior in a mathematical model of muscle. PLoS Comput Biol 5(11):e1000560ADSCrossRefGoogle Scholar
  3. 3.
    Campbell SG, Hatfield PC, Campbell KS (2011) A mathematical model of muscle containing heterogeneous half-sarcomeres exhibits residual force enhancement. PLoS Comput Biol 7(9):e1002156ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Capelo A, Comincioli V, Minelli R, Poggesi C, Reggiani C, Ricciardi L (1981) Study and parameters identification of a rheological model for excised quiescent cardiac muscle. J Biomech 14:1–11CrossRefMathSciNetGoogle Scholar
  5. 5.
    Colli PL (1986) A mathematical model of heterogeneous behavior of single muscle fibres. J Math Biol 24:103–118CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Colli P, Comincioli V, Naldi G, Torelli A (1988) Some mathematical and computational aspects of muscle contraction. In: Eisenfeld J, Levine DS (eds) IMACS Transactions on Scientific Computing, 12th IMACS World Congress, Biomedical Modelling and Simulation. J.C. Baltzer AG, Scientific Publishing Co. Basel, pp 159–161Google Scholar
  7. 7.
    Colli P, Comincioli V, Naldi G, Reggiani C (1988) Mathematical modelling for contracting muscle. In: Ricciardi LM (ed) Biomathematics and related computational problems. Kluwer Academic Publishers, Dordrecht, pp 603–613CrossRefGoogle Scholar
  8. 8.
    Colli P, Comincioli V, Naldi G, Torelli A (1990) A mathematical study of the plasticity effects in muscle contraction. Appl Math Optim 22(1):1–26CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Comincioli V, Naldi G (1990) Mathematical models in muscle contraction: parallelism in the numerical approach. Math Comput Model 13(1):109–115CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Comincioli V, Torelli A (1983) Mathematical aspects of the cross-bridge mechanism in muscle contraction. Nonlinear Anal 7(6):661–683CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Comincioli V, Torelli A (1988) A mathematical model of contracting muscle with viscoelastic elements. SIAM J Math Anal 19:593–612CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Comincioli V, Torelli A (1988) A mathematical study of a continuum-state cross-bridge model of muscle contraction. Math Methods Appl Sci 10(3):289–302CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Comincioli V, Torelli A, Poggesi C, Reggiani C (1984) A four-state cross bridge model for muscle contraction. Mathematical study and validation. J Math Biol 20(3):277–304CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Edman KAP, Reggiani C (1984) Redistribution of sarcomere length during isometric contraction of frog muscle fibres and its relation to tension creep. J Physiol 351:169–198CrossRefGoogle Scholar
  15. 15.
    Edman KAP, Reggiani C (1985) Differences in maximum velocity of shortening along single muscle fibres of the frog. J Physiol 365:147–163CrossRefGoogle Scholar
  16. 16.
    Edman KAP, Reggiani C (1987) The sarcomere length tension relation determined in short segment of intact muscle fibres of the frog. J Physiol 385:709–732CrossRefGoogle Scholar
  17. 17.
    Eisenberg E, Hill TL (1978) A cross-bridge model of muscle contraction. Prog Biophys Mol Biol 33:55–82CrossRefGoogle Scholar
  18. 18.
    Eisenberg E, Hill TL, Chen YD (1980) Cross-bridge model of muscle contraction: quantitative analysis. Biophys J 29:195–227CrossRefGoogle Scholar
  19. 19.
    Givli S (2010) Towards multi-scale modeling of muscle fibers with sarcomere non-uniformities. J Theor Biol 264:882–892CrossRefGoogle Scholar
  20. 20.
    Glantz AS (1977) A three-element description for muscle with viscoelastic passive elements. J Miomech 10:5–20Google Scholar
  21. 21.
    Hill TL (1974) Theoretical formalism for the sliding filament model of contraction of striated muscle Part I. Prog Biophys Mol Biol 28:267–340CrossRefGoogle Scholar
  22. 22.
    Hill TL (1975) Theoretical formalism for the sliding filament model of contraction of striated muscle Part II. Prog Biophys Mol Biol 29:105–159CrossRefGoogle Scholar
  23. 23.
    Huxley AF (1957) Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7:255–318Google Scholar
  24. 24.
    Huxley AF, Simmons RM (1971) Proposed mechanism of force generation in striated muscle. Nature 233:533–538ADSCrossRefGoogle Scholar
  25. 25.
    Julian FJ, Sollins KR, Sollins MR (1974) A model for the transient and steady-state mechanical behaviour of contracting muscle. Biophys J 14:546–562CrossRefGoogle Scholar
  26. 26.
    Mirsky I, Parmley WW (1974) Evaluation of passive elastic stiffness for left ventricle and isoIated heart muscle. In: Mirsky I, Ghista DM, Sandier H (eds) Mechanics cardiac. Wiley, New YorkGoogle Scholar
  27. 27.
    Naldi G (1988) Multiple actin sites models of muscle contraction. Mathematical study. Pub. IAN N. 600, PaviaGoogle Scholar
  28. 28.
    Robertson SP, Johnson JD, Potter JD (1981) The time-course of \(\text{ Ca}^{2+}\) exchange with calmodulin, troponin, parvalbumin and myosin in response to transient increases in \(\text{ Ca}^{2+}\). Biophys J 34:559–569CrossRefGoogle Scholar
  29. 29.
    Schappacher-Tilp G, Leonard T, Desch G, Herzog W (2015) A novel three-filament model of force generation in eccentric contraction of skeletal muscles. PLoS ONE 10(3):e0117634CrossRefGoogle Scholar
  30. 30.
    Silverthorn DU, Johnson BR, Ober WC, Ober CE, Silverthorn AC (2016) Human physiology: an integrated approach. Pearson, San FranciscoGoogle Scholar
  31. 31.
    Smart DR (1974) Fixed point theorems. Cambridge University Press, CambridgeMATHGoogle Scholar
  32. 32.
    Torelli A (1997) Study of a mathematical model for muscle contraction with deformable elements. Rend Sem Mat Univ Poi Torino 55(3):241–271MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli studi di MilanoMilanItaly

Personalised recommendations