Rheological basis of skeletal muscle work loops

Skeletal muscle is subjected to simultaneous time-varying neural stimuli and length changes in vivo. Work loops are experimental representations of these in vivo conditions and exhibit force versus length responses that are not explainable using either soft matter rheology or the classical isometric and isotonic characterizations of muscle. These gaps in our understanding have often prompted the search for new muscle phenomena. However, we presently lack a framework to explain the mechanical origins of work loops that integrates multiple facets of current understanding of muscle, as a rheological material and also a stimulus-responsive actuator. Here we present a new hypothesis that work loops emerge by splicing together force versus length loops corresponding to different constant stimuli. Using published muscle datasets and a detailed sarcomere model, we find that the hypothesis accurately predicts work loops and helps understand them in terms of rheological behaviors measured at fixed-stimuli. Importantly, this framework identifies conditions under which a rheological understanding of muscle fails to explain the emergent work loops, and new muscle phenomena may be necessary to explain its in vivo function. Introduction Rheology, or how materials respond to being deformed, is a central consideration for living materials (Gardel et al., 2008; Kollmannsberger and Fabry, 2011; Kasza et al., 2007). Many biological tissues may be considered tunable because their functionality arises from a modulation of their rheological properties by an external control parameter or a stimulus (Banerjee et al., 2020). One tunable material of considerable relevance to animals (Lindstedt and Hoppeler, 2016; Nishikawa et al., 2018), and the object of much engineering mimicry (de Gennes, 1997; Madden et al., 2004; Hines et al., 2017), is skeletal muscle. A skeletal muscle’s rheological behavior is actively regulated by the nervous system and is crucial for how animals control their body movements (Dickinson et al., 2000; Nguyen et al., 2018). There are several broad groups of experimental characterizations of skeletal muscle rheology that each provide a different cross-sectional view of muscle phenomena. Isometric and isotonic measurements capture the steady forces developed bymuscle as a function of its length and shortening velocity (James et al., 1996; Zajac, 1989). Transient forces in response to step length, step velocity, and twitch perturbations capture the initial fast response and history-dependent relaxation to steady-state (Sandercock and Heckman, 1997; Rassier and Herzog, 2004; Lakie and Campbell, 2019;Huxley and Simmons, 1971). Oscillatory rheological experiments, or sinusoidal analysis, measure the frequency-dependent storage and loss moduli under fixed neural or electrical stimulation and length oscillations (Machin and Pringle, 1960; Kawai and Brandt, 1980). And lastly, work loop analysis capture the mechanical actuation of muscles when they are simultaneously subjected to time-varying stimulation and length oscillations (Josephson, 1985; Ahn, 2012). Underlying all these 1 of 24 ar X iv :2 00 5. 07 23 8v 3 [ co nd -m at . s of t] 3 0 D ec 2 02 1 Manuscript submitted to eLife broad groups are biophysical models of actomyosin crossbridges that form the internal motor machinery of muscle (Huxley, 1957; Huxley and Simmons, 1971; Nguyen and Venkadesan, 2021). In this paper, we hypothesize and test a connection between two of the broad groups, namely oscillatory rheological experiments and work loops. Such connections between different rheological characterizations can help illuminate new understandings of muscle phenomena and, in this case, provide a rheological basis for the shape of skeletal muscle work loops. Work loop analysis is prevalently applied to study a skeletalmuscle’s forces in response to an externally imposed oscillatory length perturbation while it is actively regulated by neural or electrical stimuli (Josephson, 1985; Ahn, 2012). The trajectory obtained by plotting the imposed oscillatory perturbation with the recorded force response forms the work loop, which is a graphical signature of the dynamics and work-producing capabilities of skeletal muscles (Nishikawa et al., 2018; James et al., 1996). The shape of work loops is an important determinant of the biomechanical functions that the muscle provides (Dickinson et al., 2000). It depends on the precise timing of the stimulus, the frequency of oscillation, the muscle’s physiological properties, and other factors that are still vigorously debated (Josephson, 1985; James et al., 1996; Dickinson et al., 2000; Sawicki et al., 2015). As a result, we currently lack a cohesive framework to understand and predict the emergence of complex work loop shapes. For example, two cockroach leg extensor muscles that appear nearly identical under isometric or isotonic force characterizations generate loops of markedly different shapes, implying different functional consequences to the cockroach (Ahn and Full, 2002; Ahn et al., 2006). The difference between oscillatory rheological experiments and work loop analysis is that one involves fixed neural or electrical stimulation whereas the other involves time-varying stimuli. A question that arises is if a detailed accounting of the time-varying stimuli permits a characterization of work loops as a superposition of rheological properties measured at fixed-stimuli. The tunable nature and nonlinear rheological properties of skeletal muscles, however, are several hurdles. For example, under fixed-stimuli, a muscle follows force-length trajectories that shares similarities with other soft and passive materials (Tschoegl, 1989; de Gennes, 1997; Madden et al., 2004, figure 1a) and can be non-elliptic in shape for large-amplitude which suggest a nonlinear oscillatory rheology. And when the rheology is changing in time due to variable stimuli, the trajectory is far more complex and exhibits loop features like self-intersections and directional changes between clock-wise and counter-clockwise loops that passive materials do not show (figure 1b). Here, we build upon current fixed-stimuli rheological characterizations and admit tunability under a specific hypothesis and examine to what extent measurements from oscillatory rheological experiments can explain the emergent shape of skeletal muscle work loops. In this paper, we develop and test the hypothesis that skeletal muscle work loops measured under time-varying neural or electrical stimulation emerge by transitioning between underlying fixed-stimulus force-length loops. We call this transition between different loops as splicing, and term the framework developed here as the splicing hypothesis. The paper starts with pedagogical preliminaries to introduce and define ideas of oscillatory rheology and linear viscoelasticity. We then present themathematical formulation of the splicing hypothesis and derive aminimal parameterization for the space of all possible loop shapes obtained using a singular version of the splicing hypothesis where the transition between force-length loops is assumed to be instantaneous. The space of loop shapes lends insights into differentmodes ofmechanical actuation that can arise as a result of a skeletal muscle’s tunable rheology and forms the premise for understanding the shapes that emerge under more gradual transitions. We then show two parallel validations of our mathematical framework using published work loops of short-horn sculpin (Myoxocephalus scorpius) abdominal muscles and using direct numerical simulations of a detailed biophysical sarcomere model. We then discuss the implications of our framework for muscle modeling and experiments.