Abstract
Initiated by neural impulses and subsequent calcium release, skeletal muscle fibers contract (actively generate force) as a result of repetitive power strokes of acto-myosin cross-bridges. The energy required for performing these cross-bridge cycles is provided by the hydrolysis of adenosine triphosphate (ATP). The reaction products, adenosine diphosphate (ADP) and inorganic phosphate (Pi), are then used—among other reactants, such as creatine phosphate—to refuel the ATP energy storage. However, similar to yeasts that perish at the hands of their own waste, the hydrolysis reaction products diminish the chemical potential of ATP and thus inhibit the muscle's force generation as their concentration rises. We suggest to use the term “exhaustion” for force reduction (fatigue) that is caused by combined Pi and ADP accumulation along with a possible reduction in ATP concentration. On the basis of bio-chemical kinetics, we present a model of muscle fiber exhaustion based on hydrolytic ATP-ADP-Pi dynamics, which are assumed to be length- and calcium activity-dependent. Written in terms of differential-algebraic equations, the new sub-model allows to enhance existing Hill-type excitation-contraction models in a straightforward way. Measured time courses of force decay during isometric contractions of rabbit M. gastrocnemius and M. plantaris were employed for model verification, with the finding that our suggested model enhancement proved eminently promising. We discuss implications of our model approach for enhancing muscle models in general, as well as a few aspects regarding the significance of phosphate kinetics as one contributor to muscle fatigue.
Highlights
Muscles performing mechanical work become exhausted, that is, they fail to maintain high force levels for a longer time period
We introduce a model for hydrolytic adenosine triphosphate (ATP)-adenosine diphosphate (ADP)-Pi dynamics (Allen and Orchard, 1987) and the relative chemical potential μ ATP of ATP as the corresponding state variable that is diminished by both ATP depletion as well as Pi accumulation
We developed a model able to explain the time course of force decay, which occurs as a consequence of ongoing neural stimulation
Summary
Muscles performing mechanical work become exhausted, that is, they fail to maintain high force levels for a longer time period. Several state-of-the-art muscle models do not comprise the physiologically well-observed force decay over time, especially in fast-twitch fibers under high neural stimulation (Brown and Loeb, 2000; Rode et al, 2009; Blümel et al, 2012; Millard et al, 2013; Haeufle et al, 2014b; Schappacher-Tilp et al, 2015; Mörl et al, 2016). Biomechanical models, including force decay on the time scale of seconds, focus on rather phenomenological descriptions of typical force decay patterns (Liu et al, 2002; El ahrache et al, 2006; Xia and Frey-Law, 2008; Ma et al, 2009, 2012; Frey-Law et al, 2012; Rashedi and Nussbaum, 2015b), or try to cover the entirety of physiological processes (Shorten et al, 2007) which is computationally expensive
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