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Rheological Basis of Skeletal Muscle Work Loops

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Rheological Basis of Skeletal Muscle Work Loops Rheological basis of skeletal muscle work loops Khoi D. Nguyen and Madhusudhan Venkadesan∗ Department of Mechanical Engineering & Materials Science, Yale University, New Haven, CT, USA This paper proposes a hypothesis for the tunable rheology of skeletal muscle and tests it using published datasets and simulations. Skeletal muscle’s rheology, how it deforms under forces, is neu- rally modulated and crucial for animal movement. Muscle’s force-length response is well-studied under fixed and time-varying stimulation. Under fixed stimulation, oscillatory force measurements characterize muscle rheology using dynamic material moduli. Under time-varying stimulation, work loops characterize work production and absorption using force-length curves obtained under peri- odic length oscillations. But unlike fixed-stimulus measurements, work loops exhibit unusual but functionally critical features like work-producing loop reversals and self-intersections. We tested the hypothesis that work loops emerge by splicing the rheological responses obtained under fixed stimulation and found that it accurately predicts the loops in experimental data on sculpin skele- tal muscle and in numerical sarcomere simulations. Thus, classical rheological loops and splicing underlie the emergent shape of muscle work loops under steady dynamical conditions. Keywords: oscillatory rheology, Lissajous figures, work loops, muscle, tunable materials Introduction time-varying stimulus while the oscillating force re- Rheology, or how materials deform under forces, sponse is recorded. The length versus force loops, is a central consideration for living materials. Many called work loops, are simulacra of periodic motor biological tissues may be considered tunable because behaviors like locomotion, and help to characterize their functionality arises from the modulation of rhe- the dynamic, work-producing capabilities of muscle ological properties by an external stimulus. One [2, 8]. such material of considerable relevance to animals The shape of the work loop is a graphical signa- [1, 2], and the object of much engineering mimicry ture of the muscle’s biomechanical function [6]. It [3–5], is skeletal muscle. Skeletal muscle’s response depends upon the precise timing of the stimulus, to perturbations is actively tuned and regulated by the frequency of oscillation, the muscle’s physiolog- the nervous system and is crucial for how animals ical properties, and other factors that are still vig- control their body movement [6, 7]. Therefore, in orously debated [6, 8, 11, 16]. As a result, we cur- addition to isometric or isotonic characterization of rently lack a cohesive framework to understand and muscle’s force producing capabilities [8], dynamic predict the emergence of complex work loop shapes. measurements have been used to characterize its re- For example, two muscles that appear nearly iden- sponse to perturbations. These dynamic measure- tical under isometric or isotonic force response mea- ments may be categorized into three broad groups: surements may generate work loops of markedly dif- frequency-dependent rheology with a fixed stimu- ferent shapes, implying different functional conse- lus and sinusoidal length oscillations [9, 10], work quences to the animal [17, 18]. But commonalities loops under time-varying stimulation and sinusoidal in testing protocols suggest parallels between mus- length oscillations [11, 12], and transient non-steady cle work loops and oscillatory rheological tests at phenomena including history-dependence that are a fixed stimulus. So, we investigated muscle as a not captured by the former two steady dynamical tunable rheological material and developed a frame- characterizations [13–15]. In this paper, we show work that expands traditional oscillatory rheology the relationship between the two dynamic steady- to ask whether muscle’s fixed-stimulus rheology can arXiv:2005.07238v2 [cond-mat.soft] 6 Mar 2021 state characterizations, namely fixed-stimulus rheol- explain work loops. ogy and work loops under time-varying stimulation. Oscillatory testing, in shear or extension, is used Work loop analysis is prevalently used to study to characterize the rheology of a wide variety of pas- muscle’s perturbation response to externally im- sive materials, including, common elastic solids and posed length changes while it is actively regu- viscous fluids, and complex materials like gels and lated by an external neural or electrical stimulus non-Newtonian fluids [19, 20]. In these tests, the ma- [11, 12]. In this method, muscle is simultaneously terial is subjected to oscillatory extensional or shear subjected to oscillatory length perturbations and a strains and their force response is recorded. Mus- cle is also characterized using oscillatory rheological tests when the stimulus is held fixed [9, 10]. These ∗ Email for correspondence: [email protected] muscle tests differ from work loops in that work 2 a b Non-tunable Tunable work loops under time-varying stimulation emerge material material Load cell Load cell by splicing or transitioning between underlying Length Length oscillations Stimulus oscillations constant-stimulus rheological responses. We test Length Length this hypothesis by comparing predicted work loops with published experimental data in skeletal muscle Stimulus and using direct numerical simulations of a detailed Force biophysical model of a sarcomere. We then derive a Force minimal parameterization for the space of all possi- time time ble loop shapes obtained by splicing, to gain insight Katydid into different modes of functionality that can arise - Rabbit latissimus wing muscle small amplitude large amplitude dorsi as a result of changes in the underlying rheology. + + Finally, we relax the simplifying linearity assump- Cockroach leg tions in the formulation to accommodate nonlinear extensors 178 and 179 stress-strain relationships. + In examining the evidence presented here, the Pedal mucus of a terrestrial slug - - Force Force Length Length reader is alerted to some cautionary points. Mus- cle is not monolithic and considerable physiological FIG. 1. Comparison of force-length loops in differences arise between different types of muscle; non-tunable and tunable materials. a, Oscillatory skeletal, cardiac, smooth, fast or slow, and many rheology of a passive non-tunable material (pedal mu- other varieties. This paper tests the hypothesis us- cus of a terrestrial slug, Limax maximus) for small and ing one specific muscle (sculpin) for which the type large amplitudes [adapted from 21]. b, Work loops un- der time-varying stimuli of the wing muscle of a katydid of data needed are presently available, but future (Neoconocephalus triops), rabbit latissimus dorsi muscle, studies may expand that set. Furthermore, rheol- and cockroach leg extensor muscles 178 and 179 [adapted ogy is fundamentally a bulk property and only lends from 8, 11, 17, respectively]. Yellow dots and thick yel- partial insight into the molecular mechanisms under- low lines indicate discrete and continuous stimulation, lying the emergent rheological properties. So rheo- respectively. The loops have been rescaled for visual logical studies are complementary to ongoing stud- comparison. Positive and negative mechanical work out- ies and debates that are centered around the molec- put are shaded green and red, respectively. ular mechanisms behind intriguing emergent prop- erties such as history-dependence [13–15], length- loops emerge when the stimulus is also be varied dependent transitions [22], and other transient non- at the same time as the length oscillations. As a re- steady phenomena [2, 23]. But, insofar as sta- sult, it is unknown to what extent muscle work loops ble work-loops can be measured and are applica- may be explained by the well-established toolkit of ble to the motor function of animals, testing the muscle oscillatory rheology. But translating rheo- splicing hypothesis will lend insight into the appli- logical tools to muscle work loops is impeded by the cability of fixed-stimulus rheology to the function- tunable nature of muscle. For example, under con- ally more realistic case of a time-varying stimulus. stant stimulation, muscle’s oscillatory rheological re- In this manner, the work presented here takes a sponse shares similarities with other soft polymeric bottom-up and data-driven approach to assess how materials [3, 4, 19]. But when muscle’s stimulus is well fixed-stimulus rheology explains the data under also varied and its properties tuned, like in work time-varying stimulation. Furthermore, the splicing loop measurements, the responses are far more com- hypothesis presents a means to incorporate steady- plex than those of passive materials and exhibit fea- state dynamical rheology into predictions under non- tures like self-intersections and directional changes steady conditions so that future investigations can between clockwise and counter-clockwise loops (fig- unambiguously account for the role of steady-state ure 1). Rheological tools are founded upon the as- rheology before attributing measured responses to sumption that the material properties being stud- new phenomena. ied are nearly constant during measurement, which Mathematical preliminaries is the reason for applying them to muscle under a Oscillatory rheology fixed stimulus. So we build upon current rheological Oscillatory rheology characterizes materials with methods to admit tunability under a specific hypoth- invariant properties by generalizing static stiffness esis and examine whether muscle’s
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