zbMATH — the first resource for mathematics

Hill equation and Hatze’s muscle activation dynamics complement each other: enhanced pharmacological and physiological interpretability of modelled activity-pCa curves. (English) Zbl 1388.92015
Summary: In pharmacology, particularly receptor theory, the drug dose-effect relation of bio-active substances is frequently described by a sigmoidal function formulated by A.V. Hill. In biomechanics and muscle physiology then again, H. Hatze had elaborated a mathematical model for the stimulation- and length-dependent dynamics of the calcium-induced activation of mammalian skeletal muscle. Here, we prove that muscular activity-pCa curves described by the Hill equation and the equilibrium state predicted by Hatze’s activation dynamics are equivalent. Thus, the exponent introduced by Hatze can be directly identified with its counterpart in the Hill equation, by which the former model gains further physiological interpretability. Conversely, the Hill constant can now be interpreted as a function of the fibre length, generally allowing for advanced Hill plots based on model ideas. We derive and examine the complementary relation of both model approaches, highlight the benefits of mutually viewing one approach from the perspective of the other, and address the physiology behind sigmoidal curves.

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C40 Biochemistry, molecular biology
Full Text: DOI
[1] Acerenza, L.; Mizraji, E., Cooperativity: a unified view, Biochim. Biophysica Acta, 1339, 1, 155-166, (1997)
[2] Altringham, J. D.; Johnston, I. A., The pca-tension and force-velocity characteristics of skinned fibres isolated from fish fast and slow muscles, J. Physiol., 333, 1, 421-440, (1982)
[3] Atkins, G. L., A simple digital-computer program for estimating the parameters of the Hill equation, Eur. J. Biochem., 33, 1, 175-180, (1973)
[4] Bahler, A. S., Modeling of Mammalian skeletal muscle, IEEE Trans. Bio-Med. Eng., BME-15, 4, 249-257, (1968)
[5] Balnave, C. D.; Allen, D. G., The effect of muscle length on intracellular calcium and force in single fibres from mouse skeletal muscle, J. Physiol., 492, 3, 705-713, (1996)
[6] Barcroft, J.; Roberts, F., The dissociation curve of haemoglobin, J. Physiol., 39, 2, 143-148, (1909)
[7] Bardsley, W. G.; Wright, A. J., A new approach to the measurement of Sigmoid curves with enzyme kinetic and ligand binding data, J. Mol. Biol., 165, 1, 163-182, (1983)
[8] Baskin, R. J.; Paolini, P. J., Volume change and pressure development in muscle during contraction, Am. J. Physiol., 213, 4, 1025-1030, (1967)
[9] Begovic, H.; Zhou, G. Q.; Li, T.; Wang, Y.; Zheng, Y. P., Detection of the electromechanical delay and its components during voluntary isometric contraction of the quadriceps femoris muscle, Front. Physiol., 5, 1, 1-8, (2014)
[10] Bigland, B.; Lippold, O. C.J., Motor unit activity in the voluntary contraction of human muscle, J. Physiol., 125, 2, 322-335, (1954)
[11] Bohr, C., Die sauerstoffaufnahme des genuinen blutfarbstoffes und des aus dem blute dargestellten Hämoglobins, Zentralblatt Physiologie, 23, 1, 688-690, (1904)
[12] Bolitho Donaldson, S. K.; Glenn, W.; Kerrik, L., Characterization of the effects of mg\({}^{2 +}\) on ca\({}^{2 +}\)- and sr\({}^{2 +}\)-activated tension generation of skinned skeletal muscle fibers, J. Gen. Physiol., 66, 4, 427-444, (1975)
[13] Brandt, P. W.; Cox, R. N.; Kawai, M., Can the binding of ca\({}^{2 +}\) to two regulatory sites on troponin C determine the steep pca/tension relationship of skeletal muscle?, Proceedings of the National Academy of Science of the United States of America, 77, 8, 4717-4720, (1980)
[14] Brown, I. E.; Loeb, G. E., Measured and modeled properties of Mammalian skeletal muscle. I. the effects of post-activation potentiation on the time course and velocity dependencies of force production, J. Muscle Res. Cell Motil., 20, 5-6, 443-456, (1999)
[15] Carillo, M.; Gonzàlez, J. M., A new approach to modelling sigmoidal curves, Technol. Forecasting Social Change, 69, 3, 233-241, (2002)
[16] Clark, A. J., The reaction between acetyl choline and muscle cells, J. Physiol., 61, 4, 530-546, (1926)
[17] Constantin, L. L.; Podolsky, R. J., Evidence for depolarization of the internal membrane system in activation of frog semitendinosus muscle, Nature, 210, 5035, 483-486, (1966)
[18] Coval, M. L., Analysis of Hill interaction coefficients of the kwon and Brown equation, J. Biol. Chem., 245, 23, 6335-6336, (1970)
[19] De Lean, A.; Munson, P. J.; Rodbard, D., Simultaneous analysis of families of sigmoidal curves: application to bioassay, radioligand assay, and physiological dose-response curves, Am. J. Physiol., 235, 2, E97-E102, (1978)
[20] De Luca, C. J., Control properties of muscle units, J. Exp. Biol., 115, 1, 125-136, (1985)
[21] Del Castillo, J.; Katz, B., Biophysical aspects of neuro-muscular transmission, Prog. Biophys. Biophys. Chem., 6, 1, 121-170, (1956)
[22] Del Pra, P.; Rossini, L.; Segre, G., The kinetics of ouabain uptake in frog heart in relation to the kinetics of inotropic effect and to the activation of transport ATP-ases, Pharmacol. Res. Commun., 3, 2, 177-191, (1970)
[23] Dragomir, C. T., On the nature of forces acting between myofilaments in resting state and under contraction, J. Theor. Biol., 27, 3, 343-356, (1970)
[24] Ebashi, S.; Endo, M., Calcium ion and muscle contraction, Prog. Biophys. Molecular Biol., 18, 1, 123-183, (1968)
[25] Eccles, J. C., The understanding of the brain, (1973), McGraw-Hill Inc., New York
[26] Fabiato, A.; Fabiato, F., Myofilament-generated tension oscillations during partial calcium activation and activation dependence of the sarcomere length-tension relation of skinned cardiac cells, J. Gen. Physiol., 72, 5, 667-699, (1978)
[27] Falk, G., Predicted delays in the activation of the contractile system, Biophys. J., 8, 5, 608-625, (1968)
[28] Fechner, G. T., Elemente der psychophysik, 2, (1860), Breitkopf und Härtel
[29] Fekedulegn, D.; Mac Siutain, M. P.; Colbert, J. J., Parameter estimation of nonlinear growth models in forestry, Silva Fennica, 33, 4, 327-336, (1999)
[30] Filo, R. S.; Bohr, D. F.; Ruegg, J. C., Glycerinated skeletal and smooth muscle: calcium and magnesium dependence, Science, 147, 3665, 1581-1583, (1965)
[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] Frieden, C., Treatment of enzyme kinetic data, J. Biol. Chem., 242, 18, 4045-4052, (1967)
[33] Fuchs, F.; Martyn, D. A., Length-dependent ca\({}^{2 +}\) activation in cardiac muscle: some remaining questions, J. Muscle Res. Cell Motility, 26, 4-5, 199-212, (2005)
[34] Fukuda, N.; Kajiwara, H.; Ishiwata, S.; Kurihara, S., Effects mgadp on length dependence of tension generation in skinned rat cardiac muscle, American Heart Association, 86, 1, E1-E6, (2000)
[35] Gesztelyi, R.; Zsuga, J.; Kemeny-Beke, A.; Varga, B.; Juhasz, B.; Tosaki, A., The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66, 4, 427-438, (2012)
[36] Giuliano, R. A.; Verpooten, G. A.; Verbist, L.; Wedeen, R. P.; De Broe, M. E., In vivo uptake kinetics of aminoglycosides in the kidney cortex of rats, J. Pharmacol. Exp. Ther., 236, 2, 470-475, (1986)
[37] Glenn, W.; Kerrik, L.; Bolitho Donaldson, S. K., The comparative effects of \([\text{Ca}^{2 +}]\) and \([\text{Mg}^{2 +}]\) on tension generation in the fibers of skinned frog skeletal muscle and mechanically disrupted rat ventricular cardiac muscle, Pflügers Archiv, 358, 3, 195-201, (1975)
[38] Godt, R. E.; Lindley, B. D., Influence of temperature upon contractile activation and isometric force production in mechanically skinned muscle fibers of the frog, J. General Physiol., 80, 2, 279-297, (1982)
[39] Gordon, A. M.; Huxley, A. F.; Julian, F. J., The length-tension diagram of single vertebrate striated muscle fibres, Proc. Physiol. Soc., 21, 1, 28P-30P, (1964)
[40] Gordon, A. M.; Huxley, A. F.; Julian, F. J., The variation in isometric tension with sarcomere length in vertebrate muscle fibers, J. Physiol., 184, 1, 170-192, (1966)
[41] Gordon, A. M.; Pollack, G. H., Effects of calcium on the sarcomere length- tension relation in rat cardiac muscle, Circul. Res., 47, 4, 610-619, (1980)
[42] Goutelle, S.; Maurin, M.; Rougier, F.; Barbaut, X.; Bourguignon, L.; Ducher, M.; Maire, P., The Hill equation: a review of its capabilities in pharmacological modelling, Fundam. Clinical Pharmacol., 22, 6, 633-648, (2008)
[43] Günther, M., Computersimulation zur Synthetisierung des muskulär erzeugten menschlichen Gehens unter Verwendung eines biomechanischen Mehrkörpermodells, (1997), Universität Tübingen, Ph.D. thesis
[44] Günther, M.; Schmitt, S.; Wank, V., High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models, Biol. Cybern., 97, 1, 63-79, (2007) · Zbl 1125.92007
[45] 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)
[46] Haeufle, D. F.B.; Günther, M.; Wunner, G.; Schmitt, S., Quantifying control effort of biological and technical movements: an information-entropy-based approach, Physical Review E, 89, 1, (2014)
[47] Hansen, E. A.; Lee, H.; Barrett, K.; Herzog, W., The shape of the force-elbow angle relationship for maximal voluntary contractions and sub-maximal electrically induced contractions in human elbow flexors, J. Biomech., 36, 11, 1713-1718, (2003)
[48] Hatze, H., A myocybernetic control model of skeletal muscle, Biol. Cybern., 25, 2, 103-119, (1977) · Zbl 0346.92011
[49] Hatze, H., A general myocybernetic control model of skeletal muscle, Biol. Cybern., 28, 3, 143-157, (1978) · Zbl 0367.92002
[50] Hatze, H., A teleological explanation of weber’s law and the motor unit size law, Bull. Math. Biol., 41, 3, 407-425, (1979) · Zbl 0409.92001
[51] Hatze, H., Myocybernetic control models of skeletal muscle, (1981), University of South Africa · Zbl 0635.92003
[52] Hellam, D. C.; Podolsky, R. J., The relation between calcium concentration and isometric force in skinned frog muscle fibers, Federation Proceedings, 25, 1, 466, (1966)
[53] Hellam, D. C.; Podolsky, R. J., Force measurements in skinned muscle fibres, J. Physiol., 200, 3, 807-819, (1969)
[54] Herzog, W.; Nigg, B., Biomechanics of the musculo-skeletal system, (2007), John Wiley, Hoboken, New Jersey
[55] Hibberd, M. G.; Jewell, B. R., Calcium- and length-dependent force production in rat ventricular muscle, J. Physiol., 329, 1, 527-540, (1982)
[56] Hill, A., The abrupt transition from rest to activity in muscle, Proc. R. Soc. B, 136, 884, 399-420, (1949)
[57] Hill, A. V., The mode of action of nicotine and curari, determined by the form of the contraction curve and the method of temperature coefficients, J. Physiol., 39, 5, 361-373, (1909)
[58] Hill, A. V., The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves, Proc. Physiol. Soc., 1, 1, iv-vii, (1910)
[59] Hill, A. V., The combinations of haemoglobin with oxygen and with carbon monoxide, Biochem. J., 7, 5, 471-480, (1913)
[60] Hill, A. V., The heat of shortening and the dynamic constants of muscle, Proc. R. Soc. London B, 126, 843, 136-195, (1938)
[61] Hill, A. V., First and last experiments in muscles mechanics, (1970), Cambridge University Press
[62] Hill, G. W., On the part of the motion of lunar perigee which is a function of the mean motions of the Sun and Moon, Acta Math., 8, 1, 1-36, (1886) · JFM 18.1106.01
[63] Hodgkin, A. L.; Horowicz, P., Potassium contractures in single muscle fibres, J. Physiol., 153, 2, 386-403, (1960)
[64] Huang, W.; Wilson, G. J.; Brown, L. J.; Lam, H.; Hambly, B. D., EPR and CD spectroscopy of fast myosin light chain conformation during binding of trifluoperazine, Eur. J. Biochem., 257, 2, 457-465, (1998)
[65] Huxley, A. F., Muscle structure and theories of contraction, Prog. Biophys. Biophys. Chem., 7, 1, 255-318, (1957)
[66] Huxley, A. F.; Peachey, L. D., Local activation of crab muscle, J. Cell Biol., 23, 1, 107a, (1964)
[67] Huxley, A. F.; Simmons, R. M., Proposed mechanism of force generation in striated muscle, Nature, 233, 5321, 533-538, (1971)
[68] 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)
[69] 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)
[70] Irving, M.; St Claire Allen, T.; Sabido-David, C.; Craik, J. S.; Brandmeier, B.; Kendrick-Jones, J.; Corrie, J. E.; Trentham, D. R.; Goldman, Y. E., Tilting of the light chain region of myosin during step length changes and active force generation in skeletal muscle, Nature, 375, 6533, 688-691, (1995)
[71] Jewell, B. R.; Wilkie, D. R., The mechanical properties of relaxing muscle, J. Physiol., 152, 1, 30-47, (1966)
[72] Jöbsis, F. F.; O’Connor, M. J., Calcium release and reabsorption in the sartorius muscle of the toad, Biochem. Biophys. Res. Commun., 25, 2, 246-252, (1966)
[73] Joumaa, V.; Herzog, W., Calcium sensitivity of residual force enhancement in rabbit skinned fibers, Am. J. Physiol., 307, 4, C395-C401, (2014)
[74] 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)
[75] Julian, F. J.; Moss, R. L., Sarcomere length-tension relations of frog skinned muscle fibres at lengths above the optimum, J. Physiol., 304, 1, 529-539, (1980)
[76] Kardel, T., Niels stensen’s geometrical theory of muscle contraction (1667): a reappraisal, J. Biomech., 23, 10, 953-965, (1990)
[77] Katz, B., The relation between force and speed in muscular contraction, J. Physiol., 96, 1, 45-64, (1939)
[78] Katz, B.; Miledi, R., The measurement of synaptic delay, and the time course of acetylcholine release at the neuromuscular junction, Proc. R. Soc. B, 161, 985, 483-495, (1965)
[79] Kenakin, T., New concepts in pharmacological efficacy at 7TM receptors: IUPHAR review 2, British J.Pharmacol., 168, 3, 554-575, (2013)
[80] Kentish, J. C.; ter Keurs, H. E.D. J.; Ricciardi, L.; Bucx, J. J.J.; Noble, M. I.M., Comparison between the sarcomere length-force relations of intact and skinned trabeculae from rat right ventricle, Circ. Res., 58, 6, 755-768, (1986)
[81] Kistemaker, D. A.; van Soest, A. J.; Bobbert, M. F., Length-dependent \([\text{Ca}^{2 +}]\) sensitivity adds stiffness to muscle, J. Biomech., 38, 9, 1816-1821, (2005)
[82] Konhilas, J. P.; Irving, T. C.; de Tombe, P. P., Length-dependent activation in three striated muscle types of the rat, J. Physiol., 544, 1, 225-236, (2002)
[83] KoshlandJr., D. E.; Némethy, G.; Filmer, D., Comparison of experimental binding data and theoretical models in proteins containing subunits, Biochemistry, 5, 1, 365-385, (1966)
[84] Laird, A. K., Dynamics of tumor growth, British J. Cancer, 18, 3, 490-502, (1964)
[85] Langmuir, I., The adsorption of gases on plane surfaces of Glass, mica and platinum, J. Am. Chem. Soc., 40, 9, 1361-1403, (1918)
[86] Levenberg, K., A method for the solution of certain problems in least squares, Q. Appl. Math., 2, 2, 164-168, (1944) · Zbl 0063.03501
[87] Lieber, R. L., Skeletal muscle structure, function, and plasticity, 3, (2009), Lippincott Williams and Wilkins, New York
[88] Lillo, R. S.; Himm, J. F.; Weathersby, P. K.; Temple, D. J.; Gault, K. A.; Dromsky, D. M., Using animal data to improve prediction of human decompression risk following air-saturation dives, J. Appl. Physiol., 93, 1, 216-262, (2002)
[89] 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)
[90] Lombardi, V.; Piazzesi, G.; Linari, M., Rapid regeneration of the actin-myosin power stroke in contracting muscle, Nature, 355, 6361, 638-641, (1992)
[91] Ma, S.; Zahalak, G. I., Activation dynamics for a distribution-moment model of skeletal model, Math. Comput. Modell., 11, 1, 778-782, (1988)
[92] Marquart, D., An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11, 2, 431-441, (1963) · Zbl 0112.10505
[93] Marsden, C. D.; Meadows, J. C.; Merton, P. A., Isolated single motor units in human muscle and their rate of discharge during maximal voluntary effort, J. Physiol., 217, Suppl., 12P-13P, (1971)
[94] Martin, J. H.; Jessell, T. M., Modality coding in the somatic sensory system, (Kandel, E. R.; Schwartz, J. H.; Jessell, T. M., Principles of Neural Science, (1991), North-Holland Elsevier Science Publishers B.V. Amsterdam), 341-352
[95] McDonald, K. S., Ca\({}^{2 +}\) dependence of loaded shortening in rat skinned cardiac myocytes and skeletal muscle fibres, J. Physiol., 525, 1, 169-181, (2000)
[96] McMahon, T. A., Muscles, reflexes, and locomotion, (1984), Princeton University Press Princeton, NJ
[97] McPhedran, A. M.; Wuerker, R. B.; Henneman, E., Properties of motor units in a homogeneous red muscle (soleus) of the cat, J. Neurophysiol., 28, 1, 71-84, (1965)
[98] Metcalfe, B., Metcalfe’s law after 40 years of Ethernet, IEEE Comput. Soc., 46, 12, 26-31, (2013)
[99] Michaelis, L.; Menten, M. L., The kinetics of invertase action, Biochemische Zeitschrift, 49, 1, 333-369, (1913)
[100] Milner-Brown, H. S.; Stein, R. B.; Yemm, R., Changes in firing rate of human motor units during linearly changing voluntary contractions, J. Physiol., 230, 2, 371-390, (1973)
[101] Monod, J.; Wyman, J.; Changeux, J.-P., On the nature of allosteric transitions: a plausible model, J. Mol. Biol., 12, 1, 88-118, (1965)
[102] 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)
[103] Moss, R. L., Sarcomere length-tension relations of frog skinned muscle fibres during calcium activation at short lengths, J. Physiol., 292, 1, 177-192, (1979)
[104] Motulsky, H.; Christopoulos, A., Fitting models to biological data using linear and nonlinear regression: A practical guide to curve Fitting, (2003), GraphPad Software Inc., San Diego CA · Zbl 1081.62100
[105] Natori, R., Effects of na and ca ions on the excitability of isolated myofibrils, (Ebashi, S., Molecular biology of muscular contraction, (1965), Elsevier, Amsterdam)
[106] Needham, D. M., Machina carnis. the biochemistry of muscular contraction in its historical development, (1971), Cambridge University Press Cambridge, MA
[107] Peachey, L. D., The sarcoplasmic reticulum and transverse tubules of the frog’s sartorius, J. Cell Biol., 25, 3, 209-231, (1965)
[108] Pearl, R.; Reed, L. J., On the rate of growth of the population of the united states Since 1790 and its mathematical representation, Proc. National Acad. Sci. United States of America, 6, 6, 275-288, (1920)
[109] 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)
[110] Pieples, K.; Arteaga, G.; Solaro, R. J.; Grupp, I.; Lorenz, J. N.; Biovin, G. P.; Jagatheesan, G.; Labitzke, E.; Detombe, P. P.; Konhilas, J. P.; Irving, T. C.; Wieczorek, D. F., Tropomyosin 3 expression leads to hypercontractility and attenuates myofilament length-dependent ca\({}^{2 +}\) activation, Am. J. Heart Circulatory Physiol., 283, 4, H1344-H1353, (2002)
[111] Rack, P. M.H.; Westbury, D. R., The effects of length and stimulus rate on tension in the isometric cat soleus muscle, J. Physiol., 204, 2, 443-460, (1969)
[112] Rang, H. P., The receptor concept: pharmacology’s big idea, British J. Pharmacol., 147, Suppl. 1, S9-S16, (2006)
[113] Richards, F. J., A flexible growth function for empirical use, J. Exp. Botany, 10, 2, 290-301, (1959)
[114] Robertson, S. P.; Kerrick, W. G., The effects of ph on ca\({}^{2 +}\)-activated force in frog skeletal muscle fibers, Pflügers Archiv: Euro. J. Physiol., 380, 1, 41-45, (1979)
[115] Rockenfeller, R., On the Application of Mathematical Methods in Hill-Type Muscle Modeling: Stability, Sensitivity and Optimal Control, (2016), Universität Koblenz-Landau, Ph.D. thesis
[116] Rockenfeller, R., How to model a muscle’s active force-length relation: A comparative study, Comput. Methods Appl. Mech. Eng., 313, 1, 321-336, (2017)
[117] Rockenfeller, R.; Götz, T., Optimal control of isometric muscle dynamics, J. Math. Fundam. Sci., 47, 1, 12-30, (2015)
[118] Rockenfeller, R.; Günther, M., Extracting low-velocity concentric and eccentric dynamic muscle properties from isometric contraction experiments, Math. Bio., 278, 1, 77-93, (2016) · Zbl 1346.92016
[119] Rockenfeller, R.; Günther, M.; Schmitt, S.; Götz, T., Comparative sensitivity analysis of muscle activation dynamics, Comput. Math. Methods Med., 2015, Article ID 585409, doi:10.1155/2015/585409, (2015) · Zbl 1335.92013
[120] 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
[121] Roszek, B.; Baan, G. C.; Huijing, P. A., Decreasing stimulation frequency-dependent length-force characteristics of rat muscle, J. Appl. Physiol., 77, 5, 2115-2124, (1994)
[122] Shiner, J. S.; Solaro, R. J., The Hill coefficient for the ca\({}^{2 +}\)-activation of striated muscle contraction, Biophys. J., 46, 4, 541-543, (1984)
[123] Shirangi, M. G.; Emerick, A. A., An improved TSVD-based Levenberg-Marquardt algorithm for history matching and comparison with Gauss-Newton, J. Pet. Sci. Eng., 143, 1, 258-271, (2016)
[124] Siebert, T.; Rode, C.; Herzog, W.; Till, O.; Blickhan, R., Nonlinearities make a difference: comparison of two common Hill-type models with real muscle, Biol. Cybern., 98, 2, 133-143, (2008) · Zbl 1149.92302
[125] Siebert, W. M., Stochastic limitations on sensory performance, (J. D. Cowan, Some Mathematical Questions in Biology - IV, Lectures on Mathematics in the Life Sciences, 5, (1973), American Mathematical Society (AMS) Providence, RI), 48-74
[126] Sieck, G. C.; Ferreira, L. F.; Reid, M. V.; Mantilla, C. B., Mechanical properties of respiratory muscles, Compr. Physiol., 3, 4, 1553-1567, (2013)
[127] Stensen (Stenonis), N., Elementorum myologiæ specimen, seu musculi descriptio geometrica, 2, (1667), Stellae Florence
[128] Stephenson, D. G.; Forrest, Q. G., Different isometric force-\([\text{Ca}^{2 +}]\) relationships in slow- and fast-twitch skinned muscle fibres of the rat, Biochim. Biophys. Acta, 589, 2, 358-362, (1980)
[129] Stephenson, D. G.; Wendt, I. R., Length dependence of changes in sarcoplasmic calcium concentration and myofibrillar calcium sensitivity in striated muscle fibres, J. Muscle Res. Cell Motil., 5, 3, 243-272, (1984)
[130] Stephenson, D. G.; Williams, D. A., Calcium-activated force responses in fast- and slow-twitch skinned muscle fibres of the rat at different temperatures, J.Physiol., 317, 1, 281-302, (1981)
[131] Stephenson, D. G.; Williams, D. A., Effects of sarcomere length on the force-pca relation in fast- and slow-twitch skinned muscle fibres from the rat, J. Physiol., 333, 1, 637-653, (1982)
[132] Stephenson, D. G.; Williams, D. A., Slow Amphibian muscle fibres become less sensitive to ca\({}^{2 +}\) with increasing sarcomere length, Eur. J. Physiol., 397, 3, 248-250, (1983)
[133] Stephenson, R. P., A modification of receptor theory, British J. Pharmacol., 11, 4, 379-393, (1956)
[134] Stienen, G. J.M.; Blangé, T.; Treijtel, B. W., Tension development and calcium sensitivity in skinned muscle fibres of the frog, Eur. J. Physiol., 405, 1, 19-23, (1985)
[135] Swammerdam, J., Versuche die besondere bewegung der fleischstränge am frosche betreffend, die überhaupt auf alle bewegung der fleischstränge an mensch und thier kan gedeutet werden (German text), Opuscula selecta Neerlandicorum de arte medica, 1, 1, 83-135, (1738)
[136] Thelen, D. G., Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults, ASME J. Biomech. Eng., 125, 1, 70-77, (2003)
[137] van Soest, A. J.; Bobbert, M. F., The contribution of muscle properties in the control of explosive movements, Biol. Cybern., 69, 3, 195-204, (1993)
[138] Verhulst, P. F., Notice sur la loi que la population poursuit dans son accroissement (French text), Correspondance mathématique et physique, 10, 1, 113-121, (1838)
[139] Walker, J. S.; Li, X.; Buttrick, P. M., Analysing force-pca curves, J. Muscle Res. Cell Moti., 31, 1, 59-69, (2010)
[140] Walmsley, B.; Hodgson, J. A.; Burke, R. E., Forces produced by medial gastrocnemius and soleus muscles during locomotion in freely moving cats, J. Neurophysiol., 41, 5, 1203-1216, (1978)
[141] Walsh, G. E., Physiology of the nervous system, (1957), Longmans, London
[142] Weiss, J. N., The Hill equation revisited: uses and misuses, Fed. Am. Soci. Exp. Biol. (FASEB) J., 11, 11, 835-841, (1997)
[143] Winters, J. M.; Stark, L., Analysis of fundamental human movement patterns through the use of in-depth antagonistic muscle models, IEEE Trans. Biomed. Eng., BME-32, 10, 826-839, (1985)
[144] Wuerker, R. B.; McPhedran, A. M.; Henneman, E., Properties of motor units in a heterogeneous pale muscle (m. gastrocnemius) of the cat, J. Neurophysiol., 28, 1, 85-99, (1965)
[145] Xing-Zhou, Z.; Jing-Jie, L.; Zhi-Wei, X., Tencent and facebook data validate metcalfe’s law, J. Comput. Sci. Technol., 30, 2, 246-251, (2015)
[146] Yifrach, O., Hill coefficient for estimating the magnitude of cooperativity in gating transitions of voltage-dependent ion channels, Biophys. J., 87, 2, 822-830, (2004)
[147] Zahalak, G. I.; Ma, S., Muscle activation and contraction: constitutive relations based directly on cross-bridge kinetics, J. Biomech. Eng., 112, 1, 52-62, (1990)
[148] Zajac, F. E., Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control, Critical Rev. Biomed. Eng., 17, 4, 359-411, (1989)
[149] Zuurbier, C. J.; Lee-de Groot, M. B.E.; Van der Laarse, W. J.; Huijing, P. A., Effects of in vivo-like activation frequency on the length-dependent force generation of skeletal muscle fibre bundles, Eur. J. Appl. Physiol., 77, 6, 503-510, (1998)
[150] Zwietering, M. H.; Jongenburger, I.; Rombout, F. M.; van’t Riet, K., Modeling of bacteria growth curve, Appl.Environ. Microbiol., 56, 6, 1875-1881, (1990)
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.