Rheological basis of 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 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 tunability belies and damping to steady dynamical conditions [19, the rheological origins of work loops. 20]. To do so, the material’s force (or stress) re- We develop and test the hypothesis that sponse to sinusoidal length (or strain) oscillations 3 of different frequencies and amplitudes are charac- by combining all three elements. The Lissajous fig- terized by a Lissajous figure [24] of force versus ure upon summing the elastic and viscous elements length that provides a graphical signature of the is a sheared ellipse (figure 2b, Supplement §S1), material’s rheology [19]. The Lissajous figures are and the ideal force element shifts the entire figure approximately elliptic loops for small amplitude os- up or down. Under a time-varying stimulus that cillations, but are typically non-elliptic for larger modulates all three elements, a new Lissajous figure amplitudes (figure 1a). For small-amplitude oscilla- emerges by switching between two sheared ellipses tions, the complex modulus E(ω) = E0(ω)+i E00(ω) (figure 2c). captures the material’s dynamic response and gen- Sign convention: Following the muscle literature erally depends on the oscillatory frequency ω [19]. [11], increasing length is positive but positive forces The storage modulus E0(ω) and the loss modulus imply the opposite sense, namely contraction. So a E00(ω) are, respectively, the in-phase and out-of- positive or counter-clockwise loop is when the mate- phase components of the measured force divided by rial performs work on the environment, and a nega- the imposed length amplitude. Non-elliptic shapes tive or clockwise loop is when the material absorbs for large-amplitude oscillations are understood as work (‘+’ and ‘ ’ regions in figures 1 and 2). length-dependence of the moduli or as higher or- Stimulus and activation:− The experimenter applies der terms of a Fourier expansion whose leading-order an external “stimulus” but “activation” is an inter- terms are E0 and E00 [20, 21]. nal variable that more directly affects rheology. For The storage and loss moduli have been widely ap- example, the stimulus is external temperature and plied to muscle [9, 10, 25, 26] and other natural activation is the material’s internal temperature, or and engineered tunable materials [27–29], so long as the stimulus is a neural spike and activation is the the external stimulus is held constant. Although in engagement of muscle’s motor machinery. We later vivo muscle strain is often greater than the small- show how to experimentally infer activation in mus- amplitudes used in oscillatory tests, the loss and cle. storage moduli have helped develop predictive mod- a c els for muscle’s dynamic response [25, 30, 31] and Length - guided the interpretation of in vivo data [26, 32]. + Tunable oscillatory rheology Linear damper Ideal force generator Activation

Current rheological methods under a fixed stim- - Force ulus do not directly accommodate the complex Lis- + φA φB time sajous figures that arise under a time-varying stimu- + Force lus (figure 1b for example). We extend current meth- Length Linear spring − ods and incorporate a time-varying element using b 1 a new splicing hypothesis. According to this hy- E + pothesis, Lissajous figures measured under a con- A-ellipse E - 0

stant stimulus are spliced at different junctions to 2 L + construct and predict the more complex Lissajous = figures. -- B-ellipse − Force We first illustrate the splicing hypothesis using Force idealized materials, a Hookean spring with tunable Length Length stiffness and neutral length, a Newtonian damper FIG. 2. Splicing hypothesis. a, Example work loops with tunable viscosity, and an ideal force genera- for idealized tunable elements. b, The force-length loop tor with tunable force output. The force genera- for a linear material is a sheared ellipse, which is the tor accommodates the work-producing actuator ca- sum of a sloped line (elastic component) and a clockwise pabilities of tunable materials. The Lissajous fig- horizontal ellipse (viscous component). c, Splicing the ures for these idealized elements under a constant state A ellipse and state B ellipse results in a new work stimulus are a sloped line for the Hookean spring, loop. a horizontal clockwise ellipse for the Newtonian damper, and a flat line for the ideal force generator. Results and discussion Phasically modulating the properties of the ideal- Splicing hypothesis ized elements—stiffness, neutral length, damping, or We model the effect of tuning a material using force level—results in more complicated loops that an external stimulus as transitioning across differ- exhibit reversals and self-intersections (figure 2a). ent rheological states. In general, activation could We apply this idea to more general tunable materials vary continuously but we first illustrate the splicing 4

a 1.0Hz b hypothesis using two distinct levels or states A and 90 60 B. Consider a tunable material that has a greater 120 30 150 storage modulus, loss modulus, and contractile force 2.0Hz at state A compared to B, implying a more inclined, 5% strain 5 180 wider, and vertically offset ellipse for A than B (fig- 345

3.0Hz ure 2c). Periodically changing the activation states 210 from B to A and back to B would result in switch- 320

20 mN 245 280 ing between the two ellipses. For this illustration, 10% strain Rat papillary muscle work loop assume that the time taken to switch and settle into Passive rat papillary muscle the new state is negligible compared to the period c Pearson’s correlation coefficient 0.96 of the oscillation. Generically, there are two tar- Predicted work loop 0.95 get points to jump to upon switching, but only one 0.96 Estimated basis loops of them will traverse the loop in a direction consis- FA(t) tent with its loss modulus, thus fully defining a new 0.91 spliced loop built up from basis ellipses. This spliced 0.99 FB(t) 0.97 loop exhibits self-intersections and net positive work 0.99 although both the A and B ellipses are individually 0.90 dissipative. When phasically stimulated during φ ωt 0.99 0.94 A ≤ ≤ φB, the force response F (t) is found by splicing the 1.0 Twitch data Smooth fit forces FB(t) and FA(t) corresponding to the two ac- 0.99 Superimposed 0.95 tivation states, respectively, and expressed as, twitches activation F (t) for φ ωt φ , A A B 0.0 200 F (t) = ≤ ≤ (1) electrical stimuli Time since first spike (ms) (FB(t) otherwise. FIG. 3. Work loops, basis loops, and gradual acti- The force-length loops of F (t) and F (t) introduce A B vation. a, Passive force-length loops and works loops of the notion of basis loops that are produced under rat papillary muscle show that low-stimulus segments of constant activation and underlie the spliced loops. the measured work loops trace closely along the passive So the basis loops are independent of the particular loop [adapted from 33]. Yellow dots indicate electrical choice of phasic stimulus that is used to produce the spikes that activates the muscle’s motor machinery. b, work loop. In addition to the activation level, these Work loops from a short-horned sculpin abdominal mus- basis loops depend on the oscillation frequency and cle measured at 5Hz and 5% strain amplitude and for different stimulation protocols [adapted from 35]. Three amplitude. They are elliptic for linear materials but electrical spikes were applied in each cycle (yellow dots). could generally be non-elliptic. We refer to the con- Work loop locations on the sine wave indicate the phase struction of work loops from basis loops as splicing difference between the first stimulus and the length os- (figure 2c, equation 1). cillation. c, A leave-one-out analysis to estimate the ac- Basis loops and splicing in muscle tivated basis loop (grey shaded ellipses), used to predict Work loops of rat papillary muscles show evidence the work loop (blue), and compare with measurement for the existence of one of the basis loops, namely (black). Gradually varying activation is modeled using the twitch response to a single electrical spike found from the analog to the B-ellipse in figure 2c that has low separate measurements of the short-horned sculpin ab- activation. Baxi et al. [33] overlaid work loops and dominal muscle (bottom-left inset) [adapted from 36]. passive force-length loops of rat papillary muscles The Pearson’s correlation coefficient is shown above each on the same plot. The work loop closely traces the predicted loop (supplement §S2.1). passive loop whenever the stimulus is low (figure 3a). So the passive loop is a basis loop, as predicted by the splicing hypothesis. But Baxi et al. [33] did not basis loops are not available, we show a means to in- measure a basis loop when the papillary muscle was fer the activated basis loop by pooling data from stimulated. This could be because stretching highly multiple work loop measurements. By using the in- activated muscle often induces tissue damage [34], ferred basis loop we show how to generate a pre- thus making it experimentally infeasible to directly dicted work loop and compare the prediction with measure basis loops at high activations and large measured work loops that were not part of the data strains. used to infer the basis loop. For this analysis, we use Although direct measurements of highly activated a previously published dataset consisting of twelve 5 work loops from a short-horn sculpin (fish) muscle date the inferred continuously varying activation by [35, Methods]. In that study, three successive elec- using a family of basis loops that lie between the trical spikes spaced 20 ms apart were applied per cy- two extremes of FA(t) when the normalized acti- cle and the onset of the first stimulus relative to vation a(t) = 1 and FB(t) when a(t) = 0. The the length oscillation was systematically varied to geometric picture for this modification is that the produce twelve work loops that vary considerably measured force reflects an intermediate basis loop in their shape (figure 3b). We perform a leave-one- that the work loop is tracking at an instant in time. out analysis to test the splicing hypothesis by using The activation a(t) specifies this intermediate ba- eleven work loops to infer the active basis loop, pre- sis loop by interpolating between the extremes such dict the twelfth work loop, and compare that with that when a(t) gradually changes from zero to one, measurement (figure 3c, see Methods). the basis loop gradually varies from FB(t) to FA(t). A complication arises that unlike the simplified Using standard practice to linearly relate force and illustration of the splicing approach, the activation activation [39], and for an onset time t0 when the of muscle is not instantaneous nor is it constant for stimulus is first applied, we obtain the entire activated duration. Instead, work loop F (t) = a(t t )F (t) + (1 a(t t ))F (t). (2) measurements typically apply impulsive neural or − 0 A − − 0 B electrical spikes that cause a gradual rise and fall In the particular dataset of the fish muscle, the in intracellular calcium concentration, which in turn passive force F (t) is negligible compared to the ac- leads to a gradual engagement and disengagement of B tive force F (t). So we set F (t) = 0. In general, the muscle’s motor machinery. The engagement of A B the measured passive loop would define F (t) much the motor machinery is the internal activation that B like in the Baxi et al. study [33]. Furthermore, we affects force production, but it is experimentally in- parameterize the activated basis loop F (t) using an accessible. We overcome this inaccessibility and es- A ideal force term F , storage modulus E , and loss timate the internal activation in response to a spike A0 A0 modulus E . These three parameters will be used by using independent experimental measurements A00 in the leave-one-out analysis to estimate the active of the force twitch response to a single spike un- basis loop (Methods). Thus, for an applied length der isometric (constant length) conditions [36]. Be- oscillation of amplitude ∆L, frequency ω and a stim- cause the length is constant, the measured force re- ulus onset time t , the spliced response is given by, sponse reflects the internal activation state and not 0 the rheological response of the tissue. In the isomet- F (t) = a(t t0)(FA0 + ∆LEA0 sin ωt + ∆LEA00 cos ωt). ric twitch study [36], sculpin muscle was held at a − (3) fixed length and subjected to a single impulsive elec- trical spike while measuring the time-course of the The splicing approach accurately predicts the ex- force development and decay that reflects the acti- perimentally measured work loops and shows several vation of the internal motor machinery. The pulse complexities inherent in the data including loop re- duration in the isometric twitch study was 1 ms, dif- versals and self-intersections (figure 3c). The Pear- ferent from 2 ms that was used in the sculpin dataset. son’s correlation coefficients for the predictions, vis- But both are much smaller than the 20 ms inter-spike `a-visthe data, range from 0.90–0.99. The estimated interval, the 100 ms force rise and relaxation time, ∼ active basis loop is almost unchanging across all or the 200 ms time period of the oscillation. So we twelve leave-one-out studies, further supporting the treat the stimuli in both studies as impulses. Fur- splicing hypothesis and a simple rheological expla- thermore, the work loop protocol used three succes- nation for the complex shape of work loops (top- sive electrical spikes. We find the total response to right inset of figure 3c, Supplement table §S1). Fu- three spikes using a superposition of three twitch re- ture investigation is needed, but we speculate that sponses and rescale the response to lie between zero the slightly worse predictions when the stimulation and one to yield the normalized activation variable phase is around 0◦ might be due to stretch-induced a(t) (bottom left inset of figure 3c, and supplement doublet-potentiation in muscle where the activa- §S2). This procedure uses the impulse response of tion dynamics are modified when highly activated the activation dynamics to construct the response muscle is externally stretched [13]. Thus, in addi- to more generalized inputs, three spikes in this case. tion to revealing the rheological origins of muscle Our approach is similar to past applications of im- work loops, the splicing hypothesis may help identify pulse responses in neural systems involving sensory circumstances when tissue-specific phenomena such [37] and motor activation dynamics [38]. as doublet-potentiation become functionally conse- We modify the splicing approach and accommo- quential. 6

The quality of the predictions point to the verac- fect of slow versus fast stimulus-activation dynamics ity of the splicing hypothesis and the rheological ba- relative to the oscillation time period (figure 4c). As sis for the emergence of work loops in muscle. It is the case with real muscle, there is no single vari- also underscores the robustness of three linearity as- able in the model that specifies its activation. For sumptions, namely, (i) linear rheology for the acti- example, although intracellular calcium concentra- vated basis loop (ellipse-shaped), (ii) the principle of tion is accessible in this biophysical model, it does superposition to construct the activation dynamics not dictate force because there are complex neighbor from the isometric twitch response, and (iii) the in- interactions between crossbridges that lead to more termediate basis loops as a linear interpolation of the active crossbridges than the calcium concentration minimal and maximal basis loops to accommodate alone would predict. To separately identify the con- the slowly varying internal activation. The robust- tribution of muscle’s rheology versus the activation ness of the linear rheological assumption indicates dynamics to the work loop, we generate two differ- that although muscle is a nonlinear material, an el- ent work loop predictions. The first is the idealized lipse captures the essence of the basis loop under model of instantaneous change in activation, which high activation. The robustness of a linear theory shows how the rheology of the basis loops affects for constructing the activation dynamics and using the shape. The second uses a time-varying activa- that to infer the dynamic stiffness of muscle at vary- tion that is estimated from isometric experiments ing activations is also to be expected based on proven and interpolates between the two basis loops in a antecedents in capturing neural stimulation dynam- slowly time-varying manner to show the contribu- ics in sensory systems [37] and studies on dynamic tion of the activation dynamics (equation 2). stiffness of ankle muscles [38]. Thus we conclude Previously, in the experimental data with sculpin that the basis loops are the building blocks for the muscle, we used the isometric force twitch upon ap- work loop in muscle, in agreement with the splicing plying a single spike as the impulse response and hypothesis. constructed the time-varying a(t) by superimposing Muscle biophysical model individual impulse responses. That same procedure To further test the splicing hypothesis and to could be used with the muscle model but we now guide experimental design of future studies to infer demonstrate a different means to find a(t) when it is the underlying basis loops, we use direct numerical possible to conduct fresh experiments. The work simulations of a detailed biophysical model of the loops use a rectangular pulse train stimulus that contractile machinery in muscle that was developed varies low-high-low instead of discrete spikes. Such by Walcott [40]. Walcott’s model incorporates real- stimulus profiles are common when applying tetanic istic aspects such as thin filament activation dynam- electrical stimuli [e.g. 8]. So we use the isometric ics and spatial coupling between actomyosin cross- response to a pulse train to capture the relationship bridges, in addition to the latest advances in mod- between the external stimulus and the slowly time- elling the Lymn-Taylor actomyosin crossbridge cy- varying force. We average the pulse train response cle that underlies force production in muscle [2, 41]. over multiple periods to find the slowly time-varying Muscle is comprised of bundles of fibrils and a fib- activation response to this stimulus, a(t). ril is organized into repeating contractile structures, The smoothed splicing accurately predicts the called sarcomeres, of approximately 2.5 µm length work loop at all oscillation frequencies and the ide- that contract when muscle is stimulated. Within alized loop is shown to underlie the overall shape a single sarcomere, the contraction is powered by of the smoothed loop. Not surprisingly, the sharp myosin motors that interact with actin filaments corners in the idealized splicing construction are by forming force-bearing crossbridges when myosin smoothed out by incorporating the pulse train which stochastically binds to actin, strokes and unbinds captures asymmetries in the timescales of force rise (figure 4a). The activation-dependent rheological versus decay. If direct measurement of the acti- response of muscle depends upon the dynamics of vated basis loop could damage the tissue, we find these crossbridges. that small amplitude measurements of the activated We assess the splicing hypothesis by predicting basis loop could be used in lieu of the true large- work loops using the basis loops and comparing with amplitude basis loop with little loss of accuracy, at those found using phasic stimulation (figure 4b). To least for this detailed biophysical model of muscle probe the model’s nonlinearities, we use peak-to- (Supplement §S2). Thus, work loops are comprised peak stretch amplitudes that are twice the power of two components: (i) an idealized spliced loop stroke length of myosin. We repeat this for three formed from basis loops that can be found from different oscillation frequencies to elucidate the ef- the rheological response at constant stimulus, and 7

(ii) stimulation-activation dynamics that smooth or internal activation. So to broaden the applicability regularize the idealized loop and can be found using of the splicing hypothesis, we investigated further to isometric studies. derive a parameterized shape-space for all possible shapes that an idealized spliced loop could take on. a Ensemble of cycling crossbridges Kinetics Length g(x) For this derivation, we use linear theory where oscillations detachment rate Thin filament the material under constant activation is fully de-

x scribed by an ideal force term, and the storage and

Thick filament left nearest neighbor, i = 2 attachment rate f(x, i, j, ε) loss moduli. The analysis for nonlinear rheologies is right nearest neighbor, j = 1 b a direct extension of the the linear theory but adds Basis loops Stimulus-activation dynamics Work loop additional storage and loss moduli parameters that Pulse train Phase-averaged correspond to higher harmonics (Supplement §S3). A-loop 1.0 Stimulus The oscillatory force response F (t) of a tunable ma- Isometric tension terial that is phasically activated between A and B B-loop 0.0

time Stimulus, Activation φA φD time depends on the active force terms FA0 and FB0, the storage moduli EA0 and EB0 , the loss moduli EA00 and Splice Predict loop by incorporating Compare stimulus-activation dynamics EB00 , and length amplitude ∆L, according to,

FA0 + ∆L (EA0 sin ωt + EA00 cos ωt) , for φ ωt φ , F (t) =  A ≤ ≤ B c F + ∆L (E sin ωt + E cos ωt) , Basis loops Spliced loop Predicted loop Work loop  B0 B0 B00 ω = 0.01 ka ω = 0.05 ka ω = 0.10 ka  otherwise.   (4) Atypical loop shapes, which are not seen in non- tunable materials, emerge from phasic changes in rheology between the two activation states. There- fore, it is the difference between the rheologies of

Tension per crossbridge Tension the two states that introduces new loop features 2(powerstroke distance) (figure 5a). We subtract the force response FB(t)

Length from F (t), and derive nondimensional expressions using the length scale ∆L, force scale (F F ), A0 − B0 FIG. 4. Work loop prediction in muscle: a model- and timescale 1/ω. In terms of the nondimensional based illustration. a, The sarcomere is simulated as phase φ = ωt, the activated nondimensional force an ensemble of crossbridges that cycle between attached response fA(φ) = (FA(t) FB(t))/(FA0 FB0) and detached states depending on its internal displace- − − is expressed using the difference in moduli ∆e0 = ment x, distance to its left nearest neighbor i, distance ∆L(E0 E0 )/(F F ) and ∆e00 = ∆L(E00 to its right nearest neighbor j, and a coupling strength A − B A0 − B0 A − E00 )/(F F ) as, ε that also captures the variable stimulus in this sys- B A0 − B0 tem [40, Methods]. b, Workflow to predict work loops based on basis loops and isometric responses, and thus fA(φ) = 1 + ∆e0 sin φ + ∆e00 cos φ. (5) assess the splicing hypothesis. c, Loops at three differ- ent oscillatory frequencies using Monte Carlo simulation We recast the modulus parameters ∆e0 and ∆e00 of 500,000 crossbridges. Tension is in units of the pow- in terms of the nondimensional work w0 and w00 that erstroke force. Time is in units of 1/ka, where ka is a is performed by the material as a result of switch- myosin attachment rate parameter that is around 40Hz ing the storage and loss moduli respectively, and an in chicken pectoralis at 25◦C. additional work w0 as a result of switching the ideal force component between FB0 and FA0. Thus, the Shape-space of spliced loops nondimensional response f(φ) is, The idealized spliced loop is an important part of F (t) F (t) the work loop and encapsulates its rheological ori- f(φ) = − B F F gins. The activation dynamics then alter the ideal- A0 − B0 ized loop and give rise to the final work loop. The w0 2 w00 `B `A 1 + w ` +` sin φ + w −β cos φ, construction of the idealized loop does not rely on 0 A B 0 = for φA φ φB, any muscle-specific knowledge, unlike the activation  ≤ ≤ 0, otherwise. dynamics that specifically relies on the relationship  (6) between the stimulation delivered to muscle and its  8

The shape factor β = φB cos2 φ dφ represents the a b φA dimensional 1.0 w partial area of the loss-ellipse traversed in switching n = R 3w the rheology, and `A = `(φA) and `B = `(φB) are 0 0 t ) the dimensionless lengths at activation and deactiva- ( /w F 0 tion, respectively. The nondimensional work w0 for L(t) w w n = switching the elastic modulus, w00 for switching the nondimensional −w 0 loss modulus, and w0 for switching the ideal force w n = w -1.0

term are, ) 0 φ ( `B f -1.0 0 1.0 1 2 2 (φ) w /w0 w0 = ∆e0 `d` = ∆e0(`A `B), (7) c − `A 2 − Z increasing frequency φB 2 Ideal force w00 = ∆e00 cos φdφ = ∆e00β (8) generator − − ZφA `B w = d` = ` ` , (9) 0 − A − B Z`A and the nondimensional and dimensional net work Loss modulus per loop are respectively,

wn = fd` = w0 + w0 + w00, and (10) Storage modulus I 2 Wn = wn∆L(FA0 FB0) πE00 ∆L . (11) − − B FIG. 5. Shape-space of spliced loops. a, Rep- Expressing the force response, and thus the loop resentative loops to illustrate nondimensionalization of shape, using work ratios aids in the interpretation of loop shape. b, Two work ratios define the shape-space the response under varying activation in terms of the of spliced loops, one measures the net mechanical work by variable elastic modulus and the another the work material’s functionality as a force actuator, elastic by variable loss modulus. The loops are scaled to have body, or viscous damper (figure 5b). It delineates the same width and height and shown here for one ac- the activation-dependent material properties from tivation protocol (φA = π/6 and φD = 5π/4). Contour the effect of the stimulus protocol that are captured (gray) lines of net work wn are also shown. c, Example of by the parameters (`B +`A)/2, and β/(`B `A). Fur- traversing the shape space for an ideal force generator in thermore, the idealized loop has an interpretation− as parallel with a tunable Maxwell body, a spring and dash- an invariant structure that defines the work loop as pot in series, which exhibits vastly different loop shapes simply by changing the oscillatory frequency (Methods). an emergent periodic orbit of a piecewise smooth dy- namical system (Supplement §S4). The material’s dynamics and rheological response at constant ac- tivation are the constitutive pieces of the piecewise of loop shapes a material can exhibit and conse- system. Thus, the shape-space identifies the proper- quently the exchange of mechanical work with its ties that govern the dynamics of the piecewise sys- environment. tem when the activation is varied. Conclusion Although future experiments are necessary to test We adapted current oscillatory rheological meth- the veracity of the shape-space, the splicing ap- ods to admit tunable properties by splicing proach provides a single framework in which to view piecewise-rheologies and showed its predictive ability vastly different loop shapes. For example, different using published data on sculpin skeletal muscle and loop shapes may arise from the same material sim- direct numerical simulations of a detailed sarcom- ply by varying the oscillatory frequency and hold- ere model. We found that most but not all of the ing all else constant (figure 5c). This is because sculpin muscle’s work loop is accounted for by an in- the material’s rheological properties are frequency- terpolation between rheological characterizations at dependent. The splicing approach enables the ap- fixed stimuli. Our method incorporates experimen- plication of rheological modeling to generate pre- tal rheological characterization of muscle into pre- dictions for tunable materials, including muscles, dictions under time-varying stimulus conditions, so that are subjected to varying stimuli. It may also that future investigations into new emergent muscle prove to be a tool for designing actuators and pro- phenomena can account for and filter the effects of grammable mechanical interfaces from tunable ma- steady-state rheology. In this manner, the splicing terials because it is now possible to predict the range hypothesis is a new tool to study muscle mechan- 9 ics and complements ongoing investigations into the skeletal muscles of rabbit, and crayfish. Jour- molecular mechanisms that underlie the emergent nal of Muscle Research & Cell Motility 1, 279–303 rheological properties. Finally, the shape-space of (1980). work loops provides a unified view of the vastly dif- [11] Josephson, R. K. Mechanical power output from striated muscle during cyclic contraction. Journal ferent loop shapes a muscle can exhibit to variably of Experimental Biology 114, 493–512 (1985). perform as actuators, springs, dampers, and combi- [12] Ahn, A. N. How muscles function – the work loop nations thereof based on its tunable rheology. technique. Journal of Experimental Biology 215, Acknowledgments: Funding support from the 1051–1052 (2012). Raymond and Beverly Sackler Institute for Bio- [13] Sandercock, T. G. & Heckman, C. Doublet poten- tiation during eccentric and concentric contractions logical, Physical and Engineering Sciences at Yale, of cat soleus muscle. Journal of Applied Physiology a National Institutes of Health training grant 82, 1219–1228 (1997). T32EB019941, and the Robert E. Apfel Fellowship [14] Rassier, D. E. & Herzog, W. Considera- awarded by Yale. 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[26] Palmer, B. M. et al. Enhancing diastolic function triphosphate hydrolysis by actomyosin. Biochem- by strain-dependent detachment of cardiac myosin istry 10, 4617–4624 (1971). crossbridges. Journal of General Physiology 152 [42] Virtanen, P. et al. SciPy 1.0: Fundamental Algo- (2020). rithms for Scientific Computing in Python. Nature [27] Gardel, M. L., Kasza, K. E., Brangwynne, C. P., Methods 17, 261–272 (2020). Liu, J. & Weitz, D. A. Mechanical response of cy- [43] Kv˚alseth,T. O. Cautionary note about R2. The toskeletal networks. In Biophysical Tools for Biolo- American Statistician 39, 279–285 (1985). gists, Volume Two: In Vivo Techniques, vol. 89 of Methods in Cell Biology, chap. 19, 487 – 519 (Aca- demic Press, 2008). Methods [28] Kollmannsberger, P. & Fabry, B. Linear and nonlin- ear rheology of living cells. Annual Review of Ma- Sculpin Work Loops terials Research 41, 75–97 (2011). [29] De Vicente, J., Klingenberg, D. 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Power out- lowing procedure:( 1) Superimpose the three twitch put of fish muscle fibres performing oscillatory responses separated by 20ms intervals.( 2) Scale the work: effects of acute and seasonal temperature superimposed response by its maximum so that ac- change. Journal of Experimental Biology 157, 409– 423 (1991). tivation ranges from 0 and 1. We refer to this in- [36] Altringham, J. & Johnston, I. The mechanical prop- termediate step as a∗(t).( 3) Impose periodicity by erties of polyneuronally innervated, myotomal mus- finding a baseline shift δ such that a∗(δ) = a∗(T +δ) cle fibres isolated from a teleost fish (myoxocephalus where T = 200ms is the period of the sculpin work scorpius). Pfl¨ugersArchiv 412, 524–529 (1988). loops and define the activation a(t) to be a∗(t + δ). [37] Gupta, P., Albeanu, D. F. & Bhalla, U. S. Olfactory The third step makes a minor correction to ensure bulb coding of odors, mixtures and sniffs is a linear that a(t) is periodic and is necessary because the sum of odor time profiles. Nature neuroscience 18, smooth fit to the data ctk exp( t/τ ) is not auto- 272 (2015). − a [38] Weiss, P., Hunter, I. & Kearney, R. Human ankle matically periodic. This is because, under steady joint stiffness over the full range of muscle activation periodic conditions, the initial level of the activa- levels. Journal of 21, 539–544 (1988). tion is not zero. So we use the initial time offset [39] Zajac, F. E. Muscle and tendon: properties, models, δ = 3.21ms, which captures a non-zero initial condi- scaling, and application to biomechanics and motor tion of a(0) = a(T ) = 0.00356. The final result a(t) control. Critical Reviews in Biomedical Engineering is plotted in as an inset in figure 3c. 17, 359–411 (1989). [40] Walcott, S. Muscle activation described with a dif- Given a(t), measured force response yi(t), and ferential equation model for large ensembles of lo- th cally coupled molecular motors. Physical Review E stimulus timing θi of the i sculpin work loop, a 90, 042717 (2014). set of values (FA0,EA0 ,EA00 ) were generated for the th [41] Lymn, R. & Taylor, E. W. Mechanism of adenosine i loop by excluding it and minimizing the error for the remaining loops. So the fitted values for the ith 11 loop are obtained by minimizing where C = 11. The detachment rate g of a cross- bridge with internal displacement x is

[a(t θ )(F + ∆LE0 sin(ωt ) 1 k − m A0 A k g(x) = exp( E(x + 1)) (16) m=i k K − X6 X 2 + ∆LE00 cos(ωt )) y (t )] (12) A k − m k where E = 1 and K = 0.2. A Monte Carlo simulation computed the behavior where k indexes the time steps in [0,T ]. of 500,000 crossbridges as the system is sheared by an oscillatory motion L(t) = A sin(ωt) with A = 1 Goodness of fit measures and a coupling strength ε that periodically alter- Pearson’s correlation coefficient for the ith loop is nates between 0.15 and exp( 5), which respectively calculated from the fitted values as corresponds to pCa values of− 6.17 and 7.04 [40, via Walcott’s equation 23]. At each timestep, cross- (y (t ) y¯ )(F (t ) F¯) bridge attachment and detachment are decided by r = k i k − i k − (13) ¯ a random number generator. The tension Tm trans- k(yi(tk) y¯i) k(F (tk) F ) P − − mitted across the system at the mth timestep is com- P P puted as where bars denote mean values and F (tk) is pre- dicted force response in equation (3) given the fitted Tm = (xk + 1)s (17) values (F ,E ,E ). We use Pearson’s correlation A0 A0 A00 k coefficient because of the well-known problems in as- X sessing goodness of fit using the coefficient of deter- for index k over the 500,000 crossbridges and where mination R2 that is employed for linear statistical s = 1 if a crossbridge is attached and zero if de- models [43]. tached. Monte Carlo simulations of crossbridge dy- All equations are nondimensional. The chosen namics timescale is 1/ka where ka is the attachment rate of a crossbridge with many attached neighbors. The We use a published crossbridge model to assess length scale is a powerstroke distance d. The stiff- the applicability of splicing [40]. The attachment ness scale is k, the stiffness of each crossbridge’s elas- rate f of a crossbridge to an internal displacement tic spring. Consequently, the tension scale is kd. in the interval [x, x + dx] is given by The Monte Carlo simulations were performed using Matlab version 9.8.0.1323502 (R2020a, Natick, MA). D D Traversing the shape space f(x, i, j, ε) = β(i, j, ε) exp x2 (14) 2π − 2 We document here the process of generating di- r   mensionless spliced loops illustrated in figure 5c for a tunable Maxwell body in parallel with an ideal where D = 100. The function β(i, j, ε) depends on force generator. The force response to an oscilla- coupling strength ε and the distance to the nearest tory motion of unit amplitude is given by equa- attached crossbridge on the left i and right j as tions (6-9) with φA = π/6 and φB = 7π/6. The ∆e0 and ∆e00 needed to calculate w0 and w00 are de- 1 if i + j C rived for a Maxwell body, a spring with unit stiff- ≤ ε(i+j C)/C if i, j < C&i + j > C ness in series with a dashpot with unit damping  − 2 2  i/C coefficient. Specifically, ∆e0 = ω /(ω + 1) and β(i, j, ε) = ε if i < C&j C 2  ≥ ∆e00 = ω/(ω + 1). The oscillatory frequencies ω are εj/C if i C&j < C ≥ chosen such the spliced loops are uniformly spread ε if i C&j C on the semicircle on the Nyquist plot. Lastly, we  ≥ ≥  (15) hold the ideal motor work constant at w0 = 1.5.  Supplementary Notes Rheology of tunable materials

Khoi D. Nguyen1 and Madhusudhan Venkadesan∗1

1Department of Mechanical Engineering & Materials Science, Yale University, New Haven, CT, USA

Contents

S1 Linear viscoelastic response as a sheared ellipse 2

S2 Large amplitude loop prediction using ellipses 2

S3 Expansion with higher harmonics: 3

S4 Geometric viewpoint of splicing 3 S4.1 Nonlinear muscle model ...... 4

References 6 List of Figures S1 Small-amplitude basis loops to generate large-amplitude work loop predictions . . . 2 S2 Geometric viewpoint of splicing ...... 4 List of Tables S1 Fitted ellipses to work loop data ...... 5 arXiv:2005.07238v2 [cond-mat.soft] 6 Mar 2021

∗[email protected]

1 S1 Linear viscoelastic response as a sheared ellipse The objective is to show that the linear viscoelastic response of a material to a sinusoidal length oscillation is geometrically a sheared ellipse. The force F (t) is

F (t) = ∆L(E0 sin ωt + E00 cos ωt), (S1.1) where L(t) = ∆L sin ωt is the length input, ω is angular frequency, and t is time. The horizontal ellipse due to the loss modulus E00 is the plot of L0E00 cos(ωt) against L(t). The total response F (t) due to the addition of the storage modulus is the same as a shear transformation of this ellipse as given by, f(t) 1 E L E cos(ωt) = 0 0 00 . (S1.2) L(t) 0 1 L(t)       The 2x2 matrix is the shear transformation with E0 as a shear factor. The linear viscoelastic response is therefore a sheared ellipse. S2 Large amplitude loop prediction using ellipses

Basis loops Predicted loop Work loop Loop prediction using large amplitude basis loops

ω = 0.01 ka ω = 0.05 ka ω = 0.10 ka

2(powerstroke distance)

Loop prediction by scaling ellipses to match the large amplitude oscillation

ω = 0.01 ka ω = 0.05 ka ω = 0.10 ka Tension per crossbridge Tension

2(powerstroke distance)

Length

Fig. S1: Small-amplitude basis loops to generate large-amplitude work loop predictions. When large amplitude basis loops are inaccessible, scaling up linear ellipses fitted from small amplitude basis loops results in minimal loss of accuracy between predicted loops and work loops. The work loops (solid black lines) are identical in both panels and serve as a reference for comparison of predicted loops. All loops are generated from Monte Carlo simulations of 500,000 crossbridges following the flow chart of figure 4b in the main text.

2 S3 Expansion with higher harmonics: The generalization of splicing the constant-activation basis loops to include higher harmonics is a straightforward extension of splicing ellipses by using a Fourier series for the force response. The force response to sinusoidal length perturbations as given by equation (4) in the main text generalizes to

F + ∆L (E sin(kωt) + E cos(kωt)), for ωt [φ , φ ] A0 A,k0 A,k00 ∈ A B F (t) = k (S3.1)  P FB0 + ∆L (EB,k0 sin(kωt) + EB,k00 cos(kωt)), otherwise  k P for index k over the set of positive integers and where each higher harmonic introduces four ad- ditional moduli. Subtracting the state B force response and normalizing by length scale ∆L and force scale F F result in two difference of moduli of the kth harmonic: ∆e = ∆L(E A0 − B0 k0 A,k0 − E )/(F F ) and ∆e = ∆L(E E )/(F F ). Expanding equations (5)–(9) from B,k0 A0 − B0 k00 A,k00 − B,k00 A0 − B0 the main text to include the higher harmonics results in

w0 l l w00 l l 1 + k B − A sin kφ + k B − A cos kφ , for φ [φ , φ ] w0 αk w0 βk A B f(φ) = k ∈ (S3.2)  P   0, otherwise. φB  w0 = ∆e0 sin kφ cos φ dφ = ∆e0 α (S3.3) k − k − k k φZA

φB

w00 = ∆e00 cos kφ cos φ dφ = ∆e00β (S3.4) k − k − k k φZA

`D w = d` = ` ` , (S3.5) 0 − A − B Z`A

φB φB where α = sin kφ cos φ dφ and β = cos kφ cos φ dφ are shape parameters. The terms w0 k φA k φA k th and wk00 are theR work by storage forces andR loss forces of the k harmonic, respectively. The net work now includes contributes from all higher harmonics as

wn = w0 + (wk0 + wk00). (S3.6) k P The relation between nondimensional net work wn and the dimensional net work remains unchanged according to equation (11) from the main text because only the first harmonic of the state B force response contributes to net work. S4 Geometric viewpoint of splicing The idealized spliced loop is based on the assumption that the time for switching between the two rheologies and settling into the new periodic response is negligible compared to the time period of the oscillation. Thus, it may be understood as a singular periodic orbit of a piecewise smooth dynamical system with two switching planes (figure S2a,b). The switching planes and PB are defined by the phases φB and φA when the activation is changed from A to B or vice PA versa. At constant activation B or A, the dynamics of the material when subjected to periodic length oscillations are governed by the composite functions (R1 R2 ) and (R2 R1 ), respectively B ◦ B A ◦ A (figure S2b), which map initial conditions on the plane to the plane and back onto PA PB PA

3 again. The oscillatory responses at a constant activation, B or A, correspond to stable periodic orbits of the two dynamical systems. The splicing of those two periodic orbits by instantaneously switching between them at the planes and yields a new periodic orbit for the spliced PB PA dynamical system (J R1 J R1 ). The instantaneous jumps J and J (figure S2b) between D ◦ A ◦ A ◦ B A B slowly varying trajectories resembles approaches from geometric singular perturbation theory for constructing relaxation oscillations in multiple-timescale systems.[1, 2] The work loop that emerges in real tunable materials may be viewed as a regularization or smoothing of our singular construction of the spliced periodic orbit.

a Basis loops Spliced loop b 1 RA

PA PA PB

2 RA

JA JB

Force 2 RB

imposed PB motion PA PB Velocity 1 Length RB

Fig. S2: Geometric viewpoint of splicing as a piecewise smooth dynamic system with jumps in rheology at two phases. The stable periodic orbits of the basis loops and the spliced loops are illustrated in a length-velocity-force state space. A nonlinear muscle model (§S4.1)[3] in which force depends on activation, length, and velocity was used to illustrate the basis loops.

S4.1 Nonlinear muscle model We use a previously published nonlinear muscle model [3] to illustrate the splicing between nonlinear rheologies shown in figure S2a. The force output F (a, l, v) is a function of activation a, length l and velocity v according to,

F (a, l, v) = B(a, l)FL(l)FV (l, v) + FP 1(l) + BFP 2(l), (S4.1) where the functions are defined as,

a Nf (l) B(a, l) = 1 exp , (S4.2) − −0.56N (l)  f  ! 1 N (l) = 2.11 + 4.16(l− 1), (S4.3) f − l1.93 1 1.87 F (l) = exp − , (S4.4) L − 1.03 ! 5.72 v − − , v 0 5.72+(1 .38+2.09l) v ≤ FV (l, v) = 0−.62 ( 3.12+4.21l 2.67l2)v (S4.5) ( − − 0.62+v − , v > 0 l 1.42 F (l) = 5.42 ln exp − + 1 , (S4.6) P 1 0.052     F (l) = 0.02 exp( 18.7(l 0.79) 1). (S4.7) P 2 − − − −

4 To generate the nonlinear force responses, a = 1 is used as the first activation state and a = 0.5 as the second. The dimensional length input is l(t) = 1 + 0.15 sin((3π/2)t), activation phase is 1 1 φA = sin− (0.5), and deactivation phase is φD = (sin− (0.4) + π). All simulations were performed using Matlab version 9.8.0.1323502 (R2020a, Natick, MA).

Table S1: Fitted ellipses to work loop data of main text figure 3b. The scale Fmax is defined as the difference in maximal and minimal force observed in the 5◦ work loop

Stimulus timing FA,0/Fmax ∆LEA0 /FA,0 ∆LEA00 /FA,0 5◦ 0.659 0.186 0.333 30◦ 0.678 0.231 0.300 60◦ 0.695 0.235 0.245 90◦ 0.701 0.217 0.226 120◦ 0.702 0.202 0.230 150◦ 0.692 0.221 0.248 180◦ 0.690 0.223 0.250 210◦ 0.685 0.224 0.245 245◦ 0.694 0.202 0.266 280◦ 0.677 0.209 0.236 320◦ 0.708 0.262 0.245 345◦ 0.708 0.254 0.238

5 References [1] Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. Piecewise-smooth dynamical systems: theory and applications, vol. 163 (Springer Science & Business Media, 2008).

[2] Grasman, J. Relaxation Oscillations, 1475–1488 (Springer New York, New York, NY, 2011).

[3] Todorov, E. On the role of primary motor cortex in arm movement control. Progress in Motor Control III 6, 125–166 (2003).

6