×

zbMATH — the first resource for mathematics

The basic mechanical structure of the skeletal muscle machinery: one model for linking microscopic and macroscopic scales. (English) Zbl 1406.92027
J. Theor. Biol. 456, 137-167 (2018); corrigendum ibid. 488, Article ID 110143, 4 p. (2020).
Summary: Measuring, analysing, and modelling muscle contraction has a long history. In consequence, some signature characteristics of skeletal muscle contraction have been found. On a microscopic level, these are the typical non-steady-state responses of the cross-bridge bindings to steps in force and length. On a macroscopic level, the force-velocity, enthalpy-velocity, and efficiency-velocity relations for concentric steady-state contractions are crucial characteristics. As these characteristics were repeatedly confirmed across animal species and sizes, they are expected to pinpoint basic physical properties of the mechanical structure that embodies the skeletal muscle machinery. The approach presented in this article explains, for the first time, these characteristics at both the microscopic and the macroscopic scale with one model and one set of parameters. According to expectation, this model is solely built on the basic mechanical structure of the muscular, contractile machinery. Its four mechanical elements represent the source of work, the serial elasticity, damping due to mechanical deformation, and damping due to the biochemical ATP hydrolysis in the energy conversion process. For explaining all mentioned non-steady-state and steady-state characteristics at once, the model requires, at maximum, ten parameters of which only three parameters representing damping properties plus one representing muscle-internal steady-state kinematics were free to be chosen. All other parameters were already fixed by literature knowledge of the geometrical structure and force characteristics of one cross-bridge. Amongst other results, we found that (i) the most reduced variant of the model is mathematically equivalent to a former version and (ii) the curvature parameter of the Hill relation can be interpreted as the ratio of strengths of the two modelled damping processes. This model approach not only unifies microscopic and macroscopic experimental findings, but further allows to interpret findings of molecular damping and elasticity and scaling of muscle properties, as discussed in this article.

MSC:
92C10 Biomechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anson, M.; Geeves, M. A.; Kurzawa, S. E.; Manstein, D. J., Myosin motors with artificial lever arms, EMBO J., 15, 22, 6069-6074, (1996)
[2] Baker, J. E.; Thomas, D. D., A thermodynamic muscle model and a chemical basis for a.v. hill’s muscle equation, J. Muscle Res. Cell Motility, 1, 4, 335-344, (2000)
[3] Barclay, C. J., Mechanical efficiency of fast- and slow-twitch muscles of the mouse performing cyclic contractions, J. Exp. Biol., 193, Pt 1, 65-78, (1994)
[4] Barclay, C. J., Mechanical efficiency and fatigue of fast and slow muscles of the mouse, J. Physiol., 497, Pt 3, 781-794, (1996)
[5] Barclay, C. J.; Constable, J. K.; Gibbs, C. L., Energetics of fast- and slow-twitch muscles of the mouse, J. Physiol., 472, 61-80, (1993)
[6] Barclay, C. J.; Lichtwark, G. A., The mechanics of mouse skeletal muscle when shortening during relaxation, J. Biomech., 40, 14, 3121-3129, (2007)
[7] Barclay, C. J.; Woledge, R. C.; Curtin, N. A., Energy turnover for ca\({}^{2 +}\) cycling in skeletal muscle, J. Muscle Res. Cell Motility, 28, 4-5, 259-274, (2007)
[8] Behrmann, E.; Müller, M.; Penczek, P. A.; Mannherz, H. G.; Manstein, D. J.; Raunser, S., Structure of the rigor actin-tropomyosin-myosin complex, Cell, 150, 2, 327-338, (2012)
[9] Bennett, A. F., Thermal dependence of muscle function, Am. J. Physiol., 247, 2, R217-R229, (1984)
[10] Bennett, A. F., Temperature and muscle, J. Exp. Biol., 115, 333-344, (1985)
[11] Bickham, D. C.; West, T. G.; Webb, M. R.; Woledge, R. C.; Curtin, N. A.; Ferenczi, M. A., Millisecond-scale biochemical response to change in strain, Biophys. J., 101, 10, 2445-2454, (2011)
[12] Bonifasi-Lista, C.; Lake, S. P.; Small, M. S.; Weiss, J. A., Viscoelastic properties of the human medial collateral ligament under longitudinal, transverse and shear loading, J. Orthopaedic Res., 23, 1, 67-76, (2005)
[13] Bormuth, V.; Varga, V.; Howard, J.; Schäffer, E., Protein friction limits diffusive and directed movements of kinesin motors on microtubules, Science, 325, 5942, 870-873, (2009)
[14] Brandt, P. W.; Lopez, E.; Reuben, J. P.; Grundfest, H., The relationship between myofilament packing density and sarcomere length in frog striated muscle, J. Cell Biol., 33, 2, 255-263, (1967)
[15] Carter, N. J.; Cross, R. A., Mechanics of the kinesin step, Nature, 435, 7040, 308-312, (2005)
[16] Civan, M. M.; Podolsky, R. J., Contraction kinetics of striated muscle fibres following quick changes in load, J. Physiol., 184, 511-534, (1966)
[17] Close, R.; Hoh, J. F.Y., Influence of temperature on isometric contractions of rat skeletal muscles, Nature, 217, 5134, 1179-1180, (1968)
[18] Close, R. I., Dynamic properties of Mammalian skeletal muscles, Physiol. Rev., 52, 1, 129-197, (1972)
[19] Cooke, R., The sliding filament model, J. Gen. Physiol., 123, 6, 643-656, (2004)
[20] Curtin, N. A.; Woledge, R. C., Efficiency of energy conversion during sinusoidal movement of white muscle fibres from the dogfish scyliorhinus canicula, J. Exp. Biol., 183, Pt 1, 137-147, (1993)
[21] Decostre, V.; Bianco, P.; Lombardi, V.; Piazzesi, G., Effect of temperature on the working stroke of muscle myosin, Proc. Natl. Acad. Sci. USA, 102, 39, 13927-13932, (2005)
[22] Dobbie, I.; Linari, M.; Piazzesi, G.; Reconditi, M.; Koubassova, N.; Ferenczi, M. A.; Lombardi, V.; Irving, M., Elastic bending and active tilting of myosin heads during muscle contraction, Nature, 396, 6709, 383-387, (1998)
[23] Dominguez, R.; Freyzon, Y.; Trybus, K. M.; Cohen, C., Crystal structure of a vertebrate smooth muscle myosin motor domain and its complex with the essential light chain: visualization of the pre-power stroke state, Cell, 94, 5, 559-571, (1998)
[24] Duke, T. A.J., Molecular model of muscle contraction, Proc. Natl. Acad. Sci. USA, 99, 6, 2770-2775, (1999)
[25] EbashiS and Endo, M., Calcium ion and muscle contraction, Progr. Biophys. Mol. Biol., 18, 123-183, (1968)
[26] Edman, K. A.P., The velocity of unloaded shortening and its relation to sarcomere length and isometric force in vertebrate muscle fibres, J. Physiol., 291, 143-159, (1979)
[27] Edman, K. A.P.; Josephson, R. K., Determinants of force rise time during isometric contraction of frog muscle fibres, J. Physiol., 580, 3, 1007-1019, (2007)
[28] Einstein, A., Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen, Annalen der Physik, 322, 8, 549-560, (1905) · JFM 36.0975.01
[29] Fenn, W. O.; Marsh, B. S., Muscular force at different speeds of shortening, J. Physiol., 85, 277-297, (1935)
[30] Fisher, M. E.; Kolomeisky, A. B., The force exerted by a molecular motor, Proc. Natl. Acad. Sci. USA, 96, 12, 6597-6602, (1999)
[31] Ford, L. E.; Huxley, A. F.; Simmons, R. M., Tension responses to sudden length change in stimulated frog muscle fibres near slack length, J. Physiol., 269, 2, 441-515, (1977)
[32] Ford, L. E.; Huxley, A. F.; Simmons, R. M., The relation between stiffness and filament overlap in stimulated frog muscle fibres, J. Physiol., 311, 219-249, (1981)
[33] Fusi, L.; Brunello, E.; Reconditi, M.; Piazzesi, G.; Lombardi, V., The non-linear elasticity of the muscle sarcomere and the compliance of myosin motors, J. Physiol., 592, Pt 5, 1109-1118, (2014)
[34] Geeves, M. A.; Holmes, K. C., Structural mechanism of muscle contraction, Ann. Rev. Biochem., 68, 687-728, (1999)
[35] Gerritsen, K. G.M.; van den Bogert, A. J.; Nigg, B. M., Direct dynamics simulation of the impact phase in heel-toe running, J. Biomech., 28, 6, 661-668, (1995)
[36] Greene, L. E.; Eisenberg, E., Cooperative binding of myosin subfragment-1 to the actin-troponin-tropomyosin complex, Proc. Natl. Acad. Sci. USA, 77, 5, 2616-2620, (1980)
[37] Günther, M.; Ruder, H., Synthesis of two-dimensional human walking: a test of the λ-model, Biol. Cybernet., 89, 2, 89-106, (2003) · Zbl 1084.92004
[38] Günther, M.; Schmitt, S., A macroscopic ansatz to deduce the Hill relation, J. Theor. Biol., 263, 4, 407-418, (2010)
[39] Günther, M.; Schmitt, S.; Wank, V., High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models, Biol. Cybernet., 97, 1, 63-79, (2007) · Zbl 1125.92007
[40] Haeufle, D. F.B.; Günther, M.; Bayer, A.; Schmitt, S., Hill-type muscle model with serial damping and eccentric force-velocity relation, J. Biomech., 47, 6, 1531-1536, (2014)
[41] Haeufle, D. F.B.; Günther, M.; Blickhan, R.; Schmitt, S., Can quick release experiments reveal the muscle structure? a bionic approach, J. Bionic Eng., 9, 2, 211-223, (2012)
[42] Haeufle, D. F.B.; Günther, M.; Wunner, G.; Schmitt, S., Quantifying control effort of biological and technical movements: an information-entropy-based approach, Phys. Rev. E, 89, 012716, (2014)
[43] Hatze, H., A theory of contraction and a mathematical model of striated muscle, J. Theor. Biol., 40, 219-246, (1973)
[44] Hatze, H., The complete optimization of human motion, Math. Biosci., 28, 1-2, 99-135, (1976) · Zbl 0331.92003
[45] Hatze, H., A myocybernetic control model of skeletal muscle, Biol. Cybernet., 25, 2, 103-119, (1977) · Zbl 0346.92011
[46] Hill, A. V., The heat of shortening and the dynamic constants of muscle, Proc. R. Soc. Lond. B, 126, 136-195, (1938)
[47] Hill, A. V., The abrupt transition from rest to activity in muscle, Proc. R. Soc. B, 136, 884, 399-420, (1949)
[48] Hill, A. V., The series elastic component of muscle, Proc. R. Soc. Lond. B, 137, 273-280, (1950)
[49] Hill, A. V., The effect of load on the heat of shortening of muscle, Proc. R. Soc. Lond. B, 159, 297-318, (1964)
[50] Holmes, K. C., Muscle proteins - their actions and interactions, Curr. Opin. Struct. Biol., 6, 6, 781-789, (1996)
[51] Holmes, K. C., The swinging lever-arm hypothesis of muscle contraction, Curr. Biol., 7, 2, R112-118, (1997)
[52] Houdijk, H.; Bobbert, M. F.; de Haan, A., Evaluation of a Hill based muscle model for the energy cost and efficiency of muscular contraction, J. Biomech., 39, 3, 536-543, (2006)
[53] Houdusse, A.; Kalabokis, V. N.; Himmel, D.; Szent-Györgyi, A. G.; Cohen, C., Atomic structure of scallop myosin subfragment S1 complexed with mgadp: a novel conformation of the myosin head, Cell, 97, 4, 459-470, (1999)
[54] Houdusse, A.; Sweeney, H. L., Myosin motors: missing structures and hidden springs, Curr. Opin. Struct. Biol., 11, 2, 182-194, (2001)
[55] Houdusse, A.; Szent-Györgyi, A. G.; Cohen, C., Three conformational states of scallop myosin S1, Proc. Natl. Acad. Sci. USA, 97, 21, 11238-11243, (2000)
[56] Huxley, A. F., Muscle structure and theories of contraction, Progr. Biophys. Biophys. Chem., 7, 255-318, (1957)
[57] Huxley, A. F., A note suggesting that the cross-bridge attachment during muscle contraction may take place in two stages, Proc. R. Soc. Lond. B, 183, 83-86, (1973)
[58] Huxley, A. F.; Niedergerke, R., Structural changes in muscle during contraction. interference microscopy of living muscle fibres, Nature, 173, 4412, 971-973, (1953)
[59] Huxley, A. F.; Simmons, R. M., Proposed mechanism of force generation in striated muscle, Nature, 233, 5321, 533-538, (1971)
[60] Huxley, H. E.; Hanson, J., Changes in the cross-striations of muscle during contraction and stretch and their structural interpretation, Nature, 173, 4412, 973-976, (1954)
[61] Huxley, H. E.; Stewart, A.; Sosa, H.; Irving, T. C., X-ray diffraction measurements of the extensibility of actin and myosin filaments in contracting muscle, Biophys. J., 67, 6, 2411-2421, (1994)
[62] Irving, M.; Lombardi, V.; Piazzesi, G.; Ferenczi, M. A., Erratum: myosin head movements are synchronous with the elementary force-generating process in muscle, Nature, 357, 6380, 704, (1992)
[63] Irving, M.; Lombardi, V.; Piazzesi, G.; Ferenczi, M. A., Myosin head movements are synchronous with the elementary force-generating process in muscle, Nature, 357, 6374, 156-158, (1992)
[64] Irving, M.; Piazzesi, G.; Lucii, L.; Sun, Y. B.; Harford, J. J.; Dobbie, I. M.; Ferenczi, M. A.; Reconditi, M.; Lombardi, V., Conformation of the myosin motor during force generation in skeletal muscle, Nat. Struct. Biol., 7, 6, 482-485, (2000)
[65] Julian, F. J., The effect of calcium on the force-velocity relation of briefly glycerinated frog muscle fibres, J. Physiol., 218, 1, 117-145, (1971)
[66] Lampinen, M. J.; Noponen, T., Electric dipole theory and thermodynamics of actomyosin molecular motor in muscle contraction, J. Theor. Biol., 236, 4, 397-421, (2005)
[67] Levin, A.; Wyman, J., The viscous elastic properties of muscle, Proc. R. Soc. Lond. B, 101, 218-243, (1927)
[68] Linari, M.; Dobbie, I.; Reconditi, M.; Koubassova, N.; Irving, M.; Piazzesi, G.; Lombardi, V., The stiffness of skeletal muscle in isometric contraction and rigor: the fraction of myosin heads bound to actin, Biophys. J., 74, 5, 2459-2473, (1998)
[69] Lombardi, V.; Piazzesi, G.; Ferenczi, M. A.; Thirlwell, H.; Dobbie, I.; Irving, M., Elastic distortion of myosin heads and repriming of the working stroke in muscle, Nature, 374, 6522, 553-555, (1995)
[70] Lombardi, V.; Piazzesi, G.; Linari, M., Rapid regeneration of the actin-myosin power stroke in contracting muscle, Nature, 355, 6361, 638-641, (1992)
[71] Lombardi, V.; Piazzesi, G.; Reconditi, M.; Linari, M.; Lucii, L.; Stewart, A.; Sun, Y. B.; Boesecke, P.; Narayanan, T.; Irving, T. C.; Irving, M., X-ray diffraction studies of the contractile mechanism in single muscle fibres, Philosoph. Trans. R. Soc. B, 359, 1452, 1883-1893, (2004)
[72] Lymn, R. W.; Taylor, E. W., Mechanism of adenosine triphosphate hydrolysis by actomyosin, Biochemistry, 10, 25, 4617-4624, (1971)
[73] McMahon, T. A., Muscles, reflexes, and locomotion, (1984), Princeton University Press Princeton, NJ
[74] Mobley, B. A.; Eisenberg, B. R., Sizes of components in frog skeletal muscle measured by methods of stereology, J. Gen. Physiol., 66, 1, 31-45, (1975)
[75] Mörl, F.; Siebert, T.; Schmitt, S.; Blickhan, R.; Günther, M., Electro-mechanical delay in Hill-type muscle models, J. Mech. Med. Biol., 12, 5, 85-102, (2012)
[76] Parmeggiani, A.; Jülicher, F.; Ajdari, A.; Prost, J., Energy transduction of isothermal ratchets: generic aspects and specific examples close to and far from equilibrium, Phys. Rev. E, 60, 2 Pt B, 2127-2140, (1999)
[77] Pate, E.; White, H.; Cooke, R., Determination of the myosin step size from mechanical and kinetic data, Proc. Natl. Acad. Sci. USA, 90, 6, 2451-2455, (1993)
[78] Piazzesi, G.; Dolfi, M.; Brunello, E.; Fusi, L.; Reconditi, M.; Bianco, P.; Linari, M.; Lombardi, V., The myofilament elasticity and its effect on kinetics of force generation by the myosin motor, Arch. Biochem. Biophys., 552-553, 108-116, (2014)
[79] Piazzesi, G.; Linari, M.; Reconditi, M.; Vanzi, F.; Lombardi, V., Cross-bridge detachment and attachment following a step stretch imposed on active single frog muscle fibers, J. Physiol., 489, 3-15, (1997)
[80] Piazzesi, G.; Lombardi, V., A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle, Biophys. J., 68, 5, 1966-1979, (1995)
[81] Piazzesi, G.; Lucii, L.; Lombardi, V., The size and the speed of the working stroke of muscle myosin and its dependence on the force, J. Physiol., 545, Pt 1, 145-151, (2002)
[82] Piazzesi, G.; Reconditi, S.; Linari, M.; Lucii, L.; Bianco, P.; Brunello, E.; Decostre, V.; Stewart, A.; Gore, D. B.; Irving, T. C.; Irving, M.; Lombardi, V., Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size, Cell, 131, 4, 784-795, (2007)
[83] Podolsky, R. J., Kinetics of muscular contraction: the approach to the steady state, Nature, 188, 4751, 666-668, (1960)
[84] Rayment, I.; Holden, H. M.; Whittaker, M.; Yohn, C. B.; Lorenz, M.; Holmes, K. C.; Milligan, R. A., Structure of the actin-myosin complex and its implications for muscle contraction, Science, 261, 5117, 58-65, (1993)
[85] Rayment, I.; Rypniewski, W. R.; Schmidt-Bäse, K.; Smith, R.; Tomchick, D. R.; Benning, M. M.; Winkelmann, D. A.; Wesenberg, G.; Holden, H. M., Three-dimensional structure of myosin subfragment-1: a molecular motor, Science, 261, 5117, 50-58, (1993)
[86] Reconditi, M., Recent improvements in small angle X-ray diffraction for the study of muscle physiology, Rep. Progr. Phys., 69, 10, 2709-2759, (2006)
[87] Reconditi, M.; Linari, M.; Lucii, L.; Stewart, A.; Sun, Y. B.; Boesecke, P.; Narayanan, T.; Fischetti, R. F.; Irving, T. C.; Piazzesi, G.; Irving, M.; Lombardi, V., The myosin motor in muscle generates a smaller and slower working stroke at higher load, Nature, 428, 6982, 578-581, (2004)
[88] Reedy, M. K.; Holmes, K. C.; Tregear, R. T., Induced changes in orientation of the cross-bridges of glycerinated insect flight muscle, Nature, 207, 5003, 1276-1280, (1965)
[89] Rode, C.; Siebert, T.; Blickhan, R., Titin-induced force enhancement and force depression: a ’sticky-spring’ mechanism in muscle contractions?, J. Theor. Biol., 259, 2, 350-360, (2009) · Zbl 1402.92042
[90] Rosenfeld, E. V., The interrelation between mechanical characteristics of contracting muscle, cross-bridge internal structure, and the mechanism of chemomechanical energy transduction, Eur. Biophys. J., 41, 9, 733-753, (2012)
[91] Rosenfeld, E. V., The influence of filament elasticity on transients after sudden alteration of length of muscle or load, Eur. Biophys. J., 43, 8-9, 367-376, (2014)
[92] Rosenfeld, E. V.; Günther, M., An enhanced model of cross-bridge operation with internal elasticity, Eur. Biophys. J., 43, 4-5, 131-141, (2014)
[93] Rupp, T. K.; Ehlers, W.; Karajan, N.; Günther, M.; Schmitt, S., A forward dynamics simulation of human lumbar spine flexion predicting the load sharing of intervertebral discs, ligaments, and muscles, Biomech. Model. Mechanobiol., 14, 5, 1081-1105, (2015)
[94] Schmitt, S.; Haeufle, D. F.B.; Blickhan, R.; Günther, M., Nature as an engineer: one simple concept of a bio-inspired functional artificial muscle, Bioinspir. Biomimet., 7, (2012)
[95] Scott, S. H.; Winter, D. A., Biomechanical model of the human foot: kinematics and kinetics during the stance phase of walking, J. Biomech., 26, 9, 1091-1104, (1993)
[96] Seow, C. Y., Hill’S equation of muscle performance and its hidden insight on molecular mechanisms, J. Gen. Physiol., 142, 6, 561-573, (2013)
[97] Siebert, T.; Weihmann, T.; Rode, C.; Blickhan, R., cupiennius salei: biomechanical properties of the tibia-metatarsus joint and its flexing muscles, J. Compar. Physiol., 180, 2, 199-209, (2010)
[98] Sliasas, A.; Tullis, S., Modelling the effect of oar shaft bending during the rowing stroke, J. Sports Eng. Technol., 225, 4, 265-270, (2011)
[99] Suda, H., Origin of friction derived from rupture dynamics, Langmuir, 17, 6045-6047, (2001)
[100] Suzuki, Y.; Yasunaga, T.; Ohkura, R.; Wakabayashi, T.; Sutoh, K., Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps, Nature, 396, 6709, 380-383, (1998)
[101] Sweeney, H. L.; Houdusse, A., Structural and functional insights into the myosin motor mechanism, Ann. Rev. Biophys., 39, 539-557, (2010)
[102] Tawada, K.; Sekimoto, K., A physical model of ATP-induced actin-myosin movement in vitro, Biophys. J., 59, 2, 343-356, (1991)
[103] Tawada, K.; Sekimoto, K., Protein friction exerted by motor enzymes through a weak-binding interaction, J. Theor. Biol., 150, 2, 193-200, (1991)
[104] Uyeda, T. Q.; Abramson, P. D.; Spudich, J. A., The neck region of the myosin motor domain acts as a lever arm to generate movement, Proceedings of the National Academy of Sciences of the USA, 93, 9, 4459-4464, (1996)
[105] Vale, R. D.; Milligan, R. A., The way things move: looking under the Hood of molecular motor proteins, Science, 288, 5463, 88-95, (2000)
[106] Valiant, G. A., A Determination of the Mechanical Characteristics of the Human Heel Pad in Vivo, (1984), Pennsylvania State University USA, Ph.D. thesis
[107] Valiant, G. A., Transmission and attenuation of heelstrike accelerations, (1990), Human Kinetics Publishers Champaign, Ill.
[108] van Soest, A. J.; Bobbert, M. F., The contribution of muscle properties in the control of explosive movements, Biol. Cybernet., 69, 3, 195-204, (1993)
[109] Veigel, C.; Schmidt, C. F., Friction in motor proteins, Science, 325, 5942, 826-827, (2009)
[110] von Smoluchowski, M., Zur kinetischen theorie der brownschen molekularbewegung und der suspensionen, Annalen der Physik, 326, 14, 756-780, (1906) · JFM 37.0814.03
[111] Wakabayashi, K.; Sugimoto, Y.; Tanaka, H.; Ueno, Y.; Takezawa, Y.; Amemiya, Y., X-ray diffraction evidence for the extensibility of actin and myosin filaments during muscle contraction, Biophys. J., 67, 6, 2422-2435, (1994)
[112] Woledge, R. C., The energetics of tortoise muscle, J. Physiol., 197, 3, 685-707, (1968)
[113] Woledge, R. C.; Curtin, N. A.; Homsher, E., Energetic aspects of muscle contraction, Monographs of the Physiological Society 41, 1-357, (1985), Academic Press London
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.