New Results

Active hydraulics and odd elasticity of muscle fibers

Suraj Shankar, L. Mahadevan
doi: https://doi.org/10.1101/2022.02.20.481216

Abstract

Muscle is a complex hierarchically organized soft contractile engine. To understand the limits on the rate of contraction and muscle energetics, we construct a coarse-grained multiscale model that integrates over molecular details and describes muscle as an active sponge. Our analysis of existing experiments highlights the importance of spatially heterogeneous strains and local volumetric deformations in muscular contractions across species and muscle type. The minimal theoretical model shows how contractions generically induce intracellular fluid flow and power active hydraulic oscillations, which determine the limits of ultrafast muscular contractions. We further demonstrate that the viscoelastic response of muscle is naturally nonreciprocal – or odd – owing to its active and anisotropic nature. This points to an alternate mode of muscular power generation from periodic cycles in spatial strain alone, contrasting with previous descriptions based on temporal cycles. Our work suggests the need for a revised view of muscle dynamics that emphasizes the multiscale spatio-temporal origins of soft hydraulic power, with potential implications for physiology, biomechanics and locomotion.

1. Introduction

Muscle is the primary driver of nearly all motion across the animal kingdom. Since the pioneering work by H. E. Huxley, A. F. Huxley and others reviewed in [1, 2], much work has focused on elucidating the biochemical details of the molecular rate-limiting steps involved in the contractile machinery, i.e., the binding kinetics of actomyosin crossbridges [3, 4, 5] and calcium signaling that controls motor kinetics [6, 7]. But muscle fibers are more than molecular motors, being spatially and heirarchically organized across multiple scales [8] (Fig. 1A) with complex structural and mechanical properties. They are just as responsible for the slow movements of the sloth bear as the rapid contractions associated with high frequency sound production by rattlesnakes, fish swim-bladders and songbirds, and the flapping wings of insects, where operating frequencies can range between 102 −103 Hz [9, 10]. Such extremely fast motions naturally raise the question of the maximal rate at which muscle contracts, and the limits on its energetics. To understand these questions requires us to integrate processes across scales. On mesoscopic scales, striated muscle fibers are soft, wet and active materials composed of a dense, anisotropic and actively contracting polymeric lattice (sarcomeres forming the myofibril), bathed in cytosol. The dominant component of muscle fibers, by far, is water (0.7 − 0.9 volume fraction [11, 12]) and assumed to simply play a permissive role, subservient to cellular signalling and biochemical processes. But in recent years, intracellular fluid flows have increasingly been recognized for their central role in dictating cellular morphology, motility and physiology [13, 14, 15], e.g., in rapid nonmuscular movements in plants [16, 17].

Figure 1.
Figure 1. Muscle fibers are soft, wet, active and multiscale.

(A) Striated muscle consists of long, multinucleated fibers that span several spatial scales, ranging from the contractile machinery arranged in periodically repeating sarcomeres (∼ 1 μm) to individual macroscopic fibers that can span 1 − 10 cm (figure prepared using Servier Medical Art images from Bioicons licensed under CC-BY 3.0 Unported). (B) A multiscale mathematical model for muscle fibers (Eqs. 1-5). On the molecular scale, motor heads along the myosin thick filament stochastically bind to the actin thin filament and generate a contractile force (Fm) upon hydrolyzing ATP, with both radial and axial components for a finite crossbridge binding angle (θ). The kinetic rates (ωon/off ) decrease with increasing inter-filament spacing (h) and motor force (Fm), with κ, β > 0 measuring the linearized feedback response at resting, stall conditions (see SI Sec. I for details). On the mesoscale, the myofilament lattice is an anisotropic elastic network with a characteristic shear modulus E and pore size 𝓁p, that is permeated by a fluid of viscosity η, and contracts under the action of a molecularly generated active stress (σa). (C) Actomyosin threads contract by expelling water, changing both shape and size when placed in boiled muscle juice (a source of ATP) (adapted with permission from Ref. [18]).(D) In-vivo contraction of an electrically stimulated skeletal muscle (murine gastrocnemius) displaying spatial strain heterogeneities (right: heatmap of the 2D areal strain) measured using the stained nuclei as fiducial markers (green: undeformed, red: deformed; adapted with permission from Ref. [19]).

In animals, this begs a natural question: how important are spatial hydraulic effects in the dynamics of contracting muscle fibers? A coarse-grained view of muscle (as shown in Fig. 1B) suggests that muscle fibers behave as an active fluid-filled sponge. When a muscle fiber contracts, there must be relative movement of the actomyosin filament lattice relative to the ambient fluid. More specifically, intact muscle fibers cannot contract everywhere homogeneously (due to global incompressibility in the presence of an intact sarcolemma), but they can do so locally by slowly squeezing fluid through the pores of the myofilament lattice, a process that is potentially rate-limiting. The dynamical consequences of this process are typically neglected as most in-vitro studies focus on glycerinated (permeabilized) muscle fibers that allow free drainage of fluid and assume locally incompressible deformations [11]. That water movement may occur during muscle contraction (for various reasons including osmolyte imbalances) has been noted in old studies [20, 21], and in early experiments by Szent-Györgi [22] on extracted actomyosin threads that undergo syneresis by expelling water as they “violently contract” (Fig. 1C). More pertinently, active crossbridges are also known to produce transverse (radial) forces in addition to longitudinal ones, that can change the local volume of the sarcomere [23] and necessitate fluid redistribution. And finally, spatially nonuniform strains (Fig. 1D) have been reported during tetanic contractions of intact muscle fibres, both ex-vivo [24, 25] and in-vivo [26, 19], suggesting that local strain gradients and the accompanying pressure gradients and fluid hydraulics may all be relevant dynamically.

Recent experiments probing the rapid dynamics of the myofilament lattice using optical microscopy and small-angle X-ray scattering techniques [27, 28, 29, 30, 31, 32, 33] now allow for a quantification of some of these observations. We reanalyzed data on active oscillations of muscle fibers from various experiments performed across different muscle types and species to obtain time traces of local transverse (∈⊥) and longitudinal (∈zz) strains in the sarcomere (Fig. 2, see SI Sec. VA for details). Spontaneous contractile oscillations with frequency ω ∼ 1 Hz can emerge in glycerinated skeletal muscle fibers (rabbit psoas, Fig. 2A-B) [27, 28]. The resulting periodic strains are not volume preserving, i.e., they are non-isochoric (Fig. 2, black line), despite the fibers being permeabilized. This is likely because the fibers are slender enough to lack radial gradients in deformation (see SI Sec. IIIB).

Figure 2.
Figure 2. Volume changes and strain cycles are generic in self-oscillating muscle.

(A-B) Skinned rabbit psoas muscle fibers exhibit spontaneous sarcomeric oscillations in-vitro under different ATP concentrations and sarcomere resting lengths [27, 28]. Deformations are generically not isochoric (∈zz + ∈⊥ = 0, black line), as a result, fluid flow is inevitable. At low ATP concentrations (A), the axial (∈zz) and transverse strains (∈⊥) oscillate nearly antiphase with frequency ω ≈ 0.6 Hz [28], while at higher ATP concentrations (B), faster oscillations (ω ≈ 1 Hz) develop a strain cycle [27] that encloses a significant (signed) area, see SI Sec. VA. The direction of circling the loop is shown by the red arrow. (C) In-vivo measurements of strain in asynchronous flight muscle (DLM: dorsal longitudinal) of a fruit fly in tethered flight (ω ≈ 156 Hz) [30, 29, 31]. Axial strains plotted are obtained from high-speed optical measurements of muscle length [29] and transverse strains (computed from time-resolved X-ray diffractometry measurements with subnanometer precision, see SI Sec. VA for details) remain negligible at all times [30]. (D) Ex-vivo measurements of strain in intact synchronous flight muscle (DLM) from Manduca sexta (hawkmoth) [32], displaying nonisochoric strain cycles. The isolated, whole muscle is subject to oscillatory stretch (4% axial strain, ω = 25 Hz) and electrical stimulation (activation phase of 0.5), matching physiological in-vivo conditions. The transverse strain is estimated from time-resolved X-ray diffraction measurements of the lattice spacing. The data are presented as mean with shaded regions and error bars in A-C denoting one standard deviation from averaging over multiple periods, and a 95% confidence interval in D (sample size N = 1 in all). See SI Sec. VA for further details of the analysis. All data are digitally extracted and reanalyzed from Refs. [28, 27, 29, 32]. Animal silhouettes are adapted from the online open database PhyloPic and individual image credits are in the SI Sec. VD.

In-vivo measurements of the sarcomere geometry in intact asynchronous flight muscle of Drosophila [29, 30, 31] shows that, remarkably, the lattice contracts with a constant lattice spacing [30], hence ∈⊥ ≈ 0 under natural flight conditions (wing-beat frequency ω ≈ 156 Hz; Fig. 2C). Additionally, intact synchronous flight muscle of Manduca sexta displays periodic lattice dilations and contractions under physiological conditions (ω ≈ 25 Hz; Fig. 2D), both in-vivo [33] and ex-vivo [32]. In all of the above examples (with and without a membrane), deformations of the myofilament lattice fail to preserve its local volume (∈zz + ∈⊥ ≠ 0), thereby necessitating fluid movement through the sarcomere and thus lead to spatiotemporal heterogeneities in strain.

Given the empirical evidence summarized above, how can we build a model that captures and explains the three-dimensional (3D) and volumetric deformations in muscle fibers? Here we do this and demonstrate two separate consequences of such spatial deformations in muscle fibers - (i) the dynamics of active contraction is constrained by flows within a fluid-filled fiber, and (ii) the 3D mechanical response of a muscle fiber is nonreciprocal [34, 35] allowing it to function as a soft engine using strain cycles.

Current approaches to explore multi-scale phenomena in muscle employ detailed, spatially explicit, computational models at both the cellular [36, 37, 38, 39, 40] and whole tissue level [41, 42, 43]. Other studies have recently begun addressing the role of fluid dynamics in this question by focusing on intramuscular pressure [44] and fluid flow within muscle microstructure [45, 46, 32], but largely ignore the multiscale, elastic, active and spatial aspects of the problem. Here we adopt a complementary perspective by developing a minimal continuum model that integrates these aspects, identifies the relevant coarse-grained variables and key dimensionless parameters, and highlights general biophysical principles about cellular constraints on muscular performance limits.

To do this, we turn to a framework to describe fluid-filled sponge-like materials, poroelasticity, originally developed to understand the mechanics of water-logged soils [47] and other geophysical problems [48], but has since been applied to describe cartilage [49], passive muscle [50], living cells [14, 15, 13], rapid nastic motions [16, 17], active gels [51, 52, 53, 54] etc. In Fig. 1B, we schematize a generalization of this framework to integrate the molecular actomyosin kinetics with the anisotropic elasticity, activity and flow in the cell to describe muscle fibers as an active self-squeezing sponge.

2 Biophysical model

At a minimal level, we model the muscle fiber as a cylinder (length L, radius R) of a biphasic mixture of an active porous solid (ϕ: solid fraction) immersed in fluid (1 − ϕ: fluid fraction). Assuming that Embedded Image is along the long-axis of the fiber, the crystalline arrangement of interdigitated filaments and flexible proteins endows the sarcomere with a uniaxially anisotropic elastic stress (σel) that is linearly related to the strain tensor ∈ = [∇u + (∇u)T ]/2 (displacement Embedded Image, assuming axisymmetry), where the drained elastic moduli are computed as a function of ϕ using standard homogenization techniques (see SI Sec. IA for details). In addition, the interaction between the thick (myosin) and thin (actin) filaments also leads to an active stress (σa). We assume that the passive elastic response of the porous solid is both anisotropic and compressible, approaching the incompressible limit only as ϕ → 1. The fluid stress, on the other hand, is dominated on large scales by an isotropic pressure p, whose gradients drive a flow velocity (v) with viscous dissipation being consequential only on the scale of the hydraulic pore size 𝓁p ∼ 20 − 55 nm [11, 8] (see SI Sec. IA for details). We emphasize that viscous forces are important on small scales, not because they balance individual motor forces (they don’t [1, 46]), but because they balance large-scale spatial gradients (∼ 1/L) of the active stress. Mass and momentum conservation then collectively dictate overall force balance, global incompressibility and force balance in the fluid (Darcy’s law) as follows (see SI Sec. IA for details) Embedded Image Embedded Image Embedded Image where η is the fluid viscosity and Embedded Image is a ϕ-dependent anisotropic permeability tensor (see SI Sec. IA for details). We note that the anisotropic structure of the sarcomere ensures that the lattice becomes radially impermeable to fluid flow (K⊥ → 0 as ϕ → ϕ∗ ≈ 0.91, see SI Sec. IA for details) before becoming incompressible (ϕ → 1). Hence volumetric deformations leading to intracellular flow are inevitable for all physiologically relevant ϕ ∼ 0.1 − 0.22 [11, 8].

To determine the active stress σa due to the molecular kinetics of the force-generating myosin motors, we use a simple two state model with nm(x, t) being the coarse-grained fraction of bound myosin motors and ⟨ y ⟩ (x, t) being the average extension of the motor head, at a given time t and position x in the cell. In the mean-field limit, the coupled dynamics of the actomyosin crossbridges is then given by (see SI Sec. IB for details) Embedded Image Embedded Image where y0 ∼ 8 − 10 nm [55] is the motor prestrain generated during the powerstroke and λ ‖, λ⊥ are factors associated with the geometry of the binding crossbridge [56, 57] (see SI Sec. IB for details). The kinetic rates minimally incorporate biophysical feedback (Fig. 1B; see SI Sec. IB for details) through a load-dependent unbinding rate (ωoff (⟨y ⟩)) [58, 59, 60] which allows for stretch-activation [61], and a strain or lattice-spacing dependent binding rate (ωon(∈⊥)) due to mechanisms including lattice geometry [62, 36, 37], titin [63] etc., that allow for length-based regulation of force underlying the well-known Frank-Starling law [2]. As ωon/off represent effective coarse-grained kinetic rates, we do not distinguish between different microscopic mechanisms of feedback. Assuming the myosin head behaves like a spring with stiffness km, size dm and a linear density N along the thick filament, the active contractile force density is Fm = −kmnm⟨y ⟩ which gives rise to an anisotropic active stress Embedded Image [64]. The active stress importantly includes both axial and radial components of the active force which are governed by the average crossbridge binding angle θ0 [56, 57, 36] (Fig. 1B; see SI Sec. IB for details). Eqs. 1-5 supplemented by appropriate boundary and initial conditions complete the specification of our multiscale continuum model.

3 Active hydraulic oscillations

To understand the dynamical consequences of the model, we consider two simple limits. In the passive, isotropic limit (σa = 0, K ‖= K⊥), we recover the classical result [47] that the hydrostatic pressure (p) equilibrates across a length L diffusively on a poroelastic timescale τp ∼ (η/E)(L/𝓁p)2 (see SI Sec. II for details), that combines the material (viscosity η, elastic modulus E) with the microstructural (pore size 𝓁p). Anisotropy generalizes this result by distinguishing axial from radial flow. In contrast, in the active case, upon neglecting spatial heterogeneities and fluid flow (∇p ∼ 0), our model matches previous kinetic theories of molecular motor assemblies [65, 66, 67, 68], where the kinetic timescale, τk = [ωon + ωoff (y0)]−1 (see SI Sec. II for details) controls the residence time of bound motors and the rate of buildup of active stress. In this situation, if the load-dependent feedback is strong enough Embedded Image, the molecular reaction develops an oscillatory instability with a characteristic frequency ω ∼ 1/τk (see SI Sec. III for details). Combining the two limits and noting that in an anisotropic fiber, radial flow dominates axial flow, a key dimensionless parameter emerges - the radial poroelastic Damköhler number Embedded Image that captures the relative importance of radial fluid permeation (a mesoscopic time) to actomyosin kinetics (a molecular time). A similar measure for axial flow can also be constructed, see SI Sec. II. Assuming typical values for E ∼ 0.1 − 10 MPa, η ∼ 10−3 Pa.s, 𝓁p ∼ 20 − 60 nm, τk ∼ 1 − 10 ms and R ∼ 5 − 100 μm [11, 6, 69], we obtain a wide range of Da⊥ ∼ 10−2 − 103 that is accessible to and seems to be exploited by the evolutionary range of muscle physiology (Fig. 3B).

Figure 3.
Figure 3. Active hydraulics limits muscular contraction rates.

(A) Active hydraulic oscillations involving a periodic buildup of spatial gradients in active stresses along with a sloshing of fluid occur generically with a frequency Embedded Image that combines the fast crossbridge cycling time (τk) with the slow fluid permeation time (τp). (B) Characteristic oscillation frequencies (ω) of contractions in fast sonic (blue), flight (red), cardiac (orange) and skeletal (green) muscles across species plotted against the estimated radial Damköhler number (Da⊥, Eq. 6). The data are compiled in SI Table 1 and presented as geometric means with the error bars representing the full range of estimated parameters. Active hydraulic oscillations dominate for Da⊥ ≥ 1 (shaded red region) with the theoretical scaling limit ωmax from Eq. 7 plotted (dashed line with constant q0 ∼ 3.83 computed for the longest wavelength radial mode, see SI Sec. VB and Fig. S2 for details). When Da⊥ « 1 (shaded blue region), flow is irrelevant and kinetics dominates, so the oscillation frequency saturates. Consistent with our prediction, the rates of spontaneous muscular contractions in the representative examples do not exceed the limit set by active hydraulics. Animal silhouettes are adapted from the online open database PhyloPic and individual image credits are in the SI Sec. VD.

When poroelastic, active and kinetic effects are all incorporated, we obtain a novel oscillatory instability that relies on spatial gradients in strain (Fig. 3A). A local buildup of active stress (on timescale τk) squeezes the sarcomere, forcing fluid to flow and distend neighbouring regions of the lattice (on a timescale τp), which in turn induces further buildup of myosin via stretch activation (in space). This mechanism of ‘active hydraulics’ intrinsically couples intracellular fluid flow, spatial deformation gradients and active stress generation through mechanical feedback. To see this, a minimal one-dimensional (1D) description with ∈⊥ = 0 suffices (appropriate for Drosophila flight muscle [30]). Axial force balance along the muscle fiber implies Embedded Image, while poroelastic flow dictates Embedded Image . Considering the slowest modes on scale of the system (L), these equations together (see SI Sec. IIIA for details) yield Embedded Image . In the limit when the active stress is slaved to the density of bound motors (see SI Sec. IIIA for details), we can write Embedded Image and linearize the kinetics about the steady state motor density Embedded Image to obtain Embedded Image that includes the feedback mechanism through Embedded Image (see SI Sec. IIIA for details). For strong enough activity the coupled dynamics undergoes a Hopf bifurcation resulting in the spontaneous emergence of active hydraulic oscillations with a characteristic frequency Embedded Image which predicts remarkably well the fruit fly wing-beat frequency ω ≃ 150 − 160 Hz upon using estimates of τk ∼ 0.3 ms [4] and τp ∼ 5 − 6 s (see SI Sec. IIIA for details).

For general 3D deformations, fluid is more easily shunted radially rather than axially due to a smaller hydraulic resistance across a slender fiber (R2/K⊥ « L2/K‖) and the reduced permeability of Z-discs, a feature that survives the inclusion of heterogeneity in pore size. Hence, the radial, i.e., fastest poroelastic time (not the slowest) controls pressure relaxation and is hydraulically rate-limiting. Upon extending the previous 1D instability calculation to allow radial flow, spontaneous oscillations now emerge with a scaled characteristic frequency (for large Da⊥) Embedded Image that involves both the kinetic and poroelastic time scales; a careful calculation shows that this phenomenon persists more generally (see SI Sec. III for details). Strikingly, spatiotemporal volumetric deformations trigger active hydraulic oscillations (inevitable for Da⊥ ≥ 1), even when the instability mechanism is kinetic (see SI Sec. III for details), hence offering a natural explanation for the experimental data in Fig. 2. Thus, we see that active hydraulics, rather than just kinetics, determines the fastest rate of spontaneous muscle contraction; in Fig. 3B, we plot the scaling relation in Eq. 7 and compare it to existing experimental data on muscular contractions across the animal kingdom. As our analysis focuses on rate limitations intrinsic to muscle fibers, we neglect constraints set by calcium and neuromuscular control, that can be surpassed in ultrafast contractions, for e.g., in asynchronous insect flight muscle [69]. Note that, Ca+2 cycling only introduces additional microscopic time scales that effectively replace actomyosin kinetic rates, if slower. Using representative estimates of the poroelastic, kinetic and contraction time scales in fast sonic (blue), flight (red), cardiac (orange) and skeletal (green) muscles (see SI Sec. VB and Table 1 for details), we find that while synchronous muscles are typically dominated by kinetics (Da⊥ < 1, shaded blue region), asynchronous muscles responsible for insect flight are often hydraulically dominated (Da⊥ ≥ 1, shaded red region), and the data is consistent with the maximal contraction rate being set by active hydraulics.

While features such as Ca+2 regulation and external inertial loads that are neglected here play a role in dictating muscle contraction rates, our analysis suggests that active hydraulics is also an important limiter of contraction rates in physiological settings. Direct measurements of spatial gradients of deformation and intracellular fluid flow in contracting muscle would provide a concrete test of these predictions.

4 Nonreciprocal mechanics of an odd elastic engine

We now show that spatial 3D deformations also have unusual mechanical and energetic implications. The mechanical response of muscle is quantified by the relation between stresses (forces) and strains (deformations) which can be time (or frequency) dependent in complex viscoelastic materials [70]. For small deformations, we can linearize the dynamical equations (Eqs. 1-5) about the resting state, Fourier transform in time (∈ (ω) =∫ dt e−iωt ∈ (t)) and obtain an effective (visco)elastic constitutive relation σij(ω) = 𝒜ijkl(ω)Ekl(ω) that linearly relates the total stress (σ = ϕ(σel + σa) − (1 − ϕ)p1) to the strain tensor, shown pictorially for the shear and isotropic components in Fig. 4A (see SI Sec. IV for details). The frequency dependent complex modulus 𝒜 includes both an (in-phase) elastic response Re[𝒜] and an (out-of-phase) viscous response Im[𝒜]. In a passive system, time-reversal symmetry enforces 𝒜ijkl(ω) = 𝒜klij(ω) [71], but in an active system such as muscle, the lack of energy conservation allows new nonsymmetric terms (recently christened ‘odd (visco)elasticity’ [34, 72] in chiral and active media) that violate a fundamental property of mechanics - Maxwell-Betti reciprocity [71, 73]. Thus, in addition to the bulk (Beff (ω)) and Young’s moduli (Yeff (ω)) that govern the standard compressive and shear response and acquire frequency dependent corrections from motor kinetics, muscle’s uniaxial anisotropy couples isotropic dilations to shear via a passive anisotropic modulus C(ω) along with an exclusively active and nonreciprocal ‘odd-modulus’ ζ(ω) (Fig. 4A). While complete expressions for the frequency dependent moduli in terms of microscopic parameters are complicated and provided in SI Sec. IV, one simplifying limit assumes a uniform horizontal motor binding geometry (θ0 = 0) which yields the following expression for the odd-modulus

Embedded Image where the stall force Embedded Image is the average active force generated by unloaded motors and Embedded Image is the steady-state bound motor fraction (zero load duty-ratio). It is worth emphasizing that spatial anisotropy and activity are sufficient for odd (visco)elasticty to emerge in muscle, even without chiral effects that are often invoked [34]. Interestingly, the possible presence of such an odd-modulus in muscle was noted in passing in old work [74], though its implications were unrecognized.

Figure 4.
Figure 4. Odd elasticity and strain cycle engine in muscle.

(A) A pictorial representation of a stress-strain relation involving shear and volume changes, for a 3D continuum material. As muscle fibers are active, the matrix of effective moduli can be nonsymmetric, with the asymmetry quantified by an odd or nonreciprocal modulus ζ. While the bulk (Beff ) and Young’s (Yeff ) moduli are present in isotropic solids, the passive coupling C is only present in anisotropic solids and ζ requires both anisotropy and activity to be present. (B) The presence of ζ ≠ 0 allows work to be actively produced by cycling strains in different directions, say axial and radial (right). The work done by the odd-modulus (Wodd, Eq. 9) is proportional to odd-elasticity (Re[ζ]) and the signed area enclosed by the closed loop in strain space. (C) The average quasistatic elastic response of a crossbridge is nonreciprocal. For the simple binding geometry with θ0 = 0, axial forces (Fz, magenta) include both active and passive (spring constant kz) contributions, while radial forces (F⊥, green) are purely passive (spring constant k⊥). As crossbridge binding kinetics depends on filament spacing (h), the average axial active force decreases with increasing radial stretch (δh), but the radial force is unchanged by an axial stretch (δ𝓁), hence the response is nonreciprocal. Linearized force response about rest state (with constant stall force Fstall) shown for small radial and axial stretch, with the nonreciprocal odd spring constant ko shown. (D) A strain cycle in axial and radial deformations generates work proportional to the area of the cycle and the odd spring constant ko, i.e., W = koΔhΔ𝓁. The corresponding axial (Fz, magenta) and radial (F⊥, green) forces are plotted, showing conventional work loops.

A unique consequence of nonreciprocal mechanics is the ability to generate work from cycles of strain (Fig. 4B). Spontaneous strain cycles in muscle fibers (see Fig. 2B,D) have been previously interpreted as a time-varying Poisson ratio [32], but in an active medium, these cycles perform work. Upon writing Embedded Image in terms of ‘even’ and ‘odd’ tensors (under the exchange of ij and kl indices), we can compute the cumulative mechanical work in a cycle as W = −∮σijdEij (W > 0 when work is produced, W < 0 when dissipated), which includes two qualitatively different terms W = Weven + Wodd (see SI Sec. IV for details). The even viscous term Embedded Image depends on the traversal frequency ω and strain rates (⋵) whereas the odd elastic term Embedded Image depends on strain (see SI Sec. IV for details). While Weven is simply an anisotropic generalization of standard viscous dissipation (Im[𝒜]/ω is like a viscosity), an intuitive explanation for Wodd is that, in the absence of energy conservation, cyclic deformations in different directions do not bring the system back to its initial energy, hence work can be either produced or absorbed. In the axisymmetric limit, relevant for muscle, we can simplify Wodd as (see SI Sec. IV for details) Embedded Image i.e., work from odd elasticity depends on the odd-modulus ζ and the area enclosed by a loop in the space of axial and transverse strains (Fig. 4B). Crucially this mechanism of power generation relies on 3D spatial deformations (axial and radial), rather than temporal variations.

What is the microscopic origin of muscular odd elasticity? We show that a crossbridge, upon averaging over stochastic binding events, behaves as a nonreciprocal mechanical element. To illustrate this point, we consider the simplifying limit of rate-independent (quasistatic) deformations and a horizontal binding geometry (θ0 = 0), so all active forces are purely axial and molecular kinetics are equilibriated (Fig. 4C). As the binding rate depends on the filament spacing (h), a radial stretch (δh) modifies the active axial force (Fz) exerted, but an axial stretch (δ𝓁) generates no radial force (F⊥), which is purely passive when θ0 = 0. This asymmetry in response generalizes to arbitrary binding geometries and highlights the mechanical nonreciprocity of a crossbridge. Notably, this previously seemingly unrecognized general property is present in most microscopic crossbridge models that account for lattice deformations [75, 36, 37, 27].

To see this more explicitly, we linearize the averaged forces about stall conditions for small deformations, keeping θ0 = 0, to obtain Fz = Fstall − kzδ𝓁 − koδh and F⊥ = −k⊥δh, where kz (k⊥) are axial (radial) spring constants that account for passive elasticity (Fig. 4C). Non-reciprocity is then quantified by an “odd” spring constant Embedded Image which originates from the strain dependent binding kinetics (Embedded Image ) and is necessarily active (Fstall ≠ 0). Comparing with the macroscopic odd modulus in Eq. 8, we directly see ζ ∝ −ko.

Although ko is a rate-independent, elastic constant (i.e., coefficient relating force to deformation), it does not conserve energy. As a result mechanical work done by the crossbridge is history dependent. Performing a quasistatic deformation cycle generates work given by W = − ∮ (Fzd𝓁 + F⊥dh) which is equal to ko times the area enclosed by the strain cycle (Fig. 4D). A more conventional work loop analysis (i.e., area enclosed by the force-displacement curve) [76] recapitulates the same result, as long as both axial and radial deformations and forces are correctly accounted for (Fig. 4D).

Where might we see these effects in muscle? While a fully 3D characterization of muscle’s viscoelastic response is currently unavailable, we use our model to analyze existing muscle rheology experiments [77, 78, 79] in a simpler 1D geometry that measures the uniaxial response (σzz) in skinned muscle fibers subject to small amplitude oscillatory axial strains (∈zz). Hydraulic effects are assumed to be irrelevant in these experiments as the fibers are permeabilized. By compiling data across muscle types and species (Drosophila insect flight [77], mouse cardiac [78] and rabbit skeletal [79]), we fit our biophysical model to the measured effective (complex) Young’s modulus Yeff (ω) = σzz(ω)/ ∈zz(ω) (Fig. 5, see SI Sec. VC for analysis details).

Figure 5.
Figure 5. Active and nonreciprocal viscoelasticity across muscle types and species.

Model fits of linear response (lines) to experimental data (circles are the mean, shaded region is one standard deviation) of sinusoidal uniaxial rheology digitally extracted and compiled from Ref. [77] (Drosophila flight muscle, left), Ref. [78] (mouse cardiac muscle, center) and Ref. [79] (rabbit skeletal muscle, right). As the muscle fibers are small and permeabilized, we assume affine deformations and neglect pressure gradients (see SI Sec. VC for fit details). The real and imaginary parts of the response correspond to the elastic (Re[Yeff ], green) and the viscous (Im[Yeff ], magenta) moduli. Dashed black line marks zero. The low frequency response in the experiments sometimes includes a weak power law or logarithmic dependence on frequency, possibly arising from internal passive processes that we neglect. Model estimates using Eq. 8 of the nonreciprocal (odd) elastic modulus Re[ζ] are shown in blue, with shaded region denoting one standard deviation (see SI Sec. VC for details). Animal silhouettes are adapted from the online open database PhyloPic and individual image credits are in SI Sec. VD.

In all three cases, a common characteristic response is seen - the elastic modulus (Re[Yeff ], Fig. 5 green) transitions from a low frequency (largely passive) stiffness to a high-frequency rigor-like response from transiently crosslinked actomyosin, passing through an intermediary softening on time scales associated with crossbridge cycling, i.e., when the kinetic cycling of motors enhances compliance via filament sliding. The viscous modulus (Im[Yeff ], Fig. 5 magenta) on the other hand is negative at low frequency (hence active and work producing), switching to positive (dissipative) values at higher frequencies (skeletal muscle displays additional features at low frequency arising from passive dissipation in sarcomeric polymers that we neglect, Fig. 5 right). In the 1D setting, the negative viscous modulus offers the only route to produce positive work through temporal changes in strain and odd-elastic effects are absent. Surprisingly, Im[Yeff ] vanishes precisely at the frequency of pure kinetic oscillations (see SI Sec. VC for details), indicating that either hydraulics (which modifies the oscillation frequency) or 3D deformations (allowing odd elasticity) are essential for sustained power generation from cyclic contractions.

Using our model fit and known structural parameters, we estimate the frequency dependent odd-elastic modulus (Re[ζ]) for various muscle types (Fig. 5, blue lines; see SI Sec. VC for details). Notably, we see that the odd-modulus is predominantly negative and it remains nonvanishing at low frequencies, where its magnitude is controlled by the active stress and the strain based feedback on myosin kinetics (see Eq. 8). As Fig. 2B,D show, strain cycles (enclosing negative area) can be seen during spontaneous muscle contractions, which when combined with our model prediction of Re[ζ] < 0 and Eq. 9 are consistent with strain cycles actively producing work through odd elasticity. Using Re[ζ] ∼ −(0.1 − 10) MPa (ω < 10 Hz,

Fig. 5) and typical strain cycles in Fig. 2, we estimate the odd elastic work that can be produced to be Wodd ∼ 0.2 − 20 kPa. Assuming an oscillation frequency ω = 10 Hz and muscle mass density ρm = 103 kg/m3, odd elasticity can generate a mass specific power Podd ∼ ωWodd/ρm ∼ 2 − 200 W/kg which can be potentially significant in physiological conditions [76]. A direct test of our predictions of odd elasticity would be afforded by considering the 3D structural and force dynamics of muscle fibers using a combination of X-ray based methods [29, 23, 33] and force spectroscopy techniques [80, 81].

5 Discussion

Recognizing the importance of spatial strain gradients and the dynamics of fluid movement leads naturally to an emergent maximal rate of muscle contraction ωmax = ωmax(τk, 𝓁p, η, E, R, L, · · ·) based on active hydraulic oscillations that combines a molecular kinetic time scale (τk) with a mesoscopic poroelastic time scale (τp), incorporating micro-structural (𝓁p) and macro-geometric (R, L) scales as well as material properties (η, E). Our analysis of published 3D spatial deformation data complements previous temporal studies of muscle rheology and highlights how muscle functions as an active elastic engine, whereby work can be produced from strain cycles via an emergent nonreciprocal response naturally present in anisotropic active solids. Additionally, we may estimate the maximal power density Pmax based on a (size independent) maximal strain (∈max) and stress (σmax) of muscle [82], Pmax ∼ σmax∈maxωmax. When molecular kinetics dominates (Da⊥ < 1, ωmax ∼ 1/τk), muscular power is not constrained by size (Pmax ∝ L0). But in the active hydraulic regime (Da⊥ > 1,Embedded Image ), the maximal power density decreases with size (Pmax ∝ 1/L), so larger organisms may need additional spring based mechanisms to amplify power output [83].

Beyond muscle, the principles of active hydraulics may also apply to other soft contractile systems. For example, most current soft muscle mimetic actuators [84, 85, 86, 87] have contraction rates that are largely limited by the slow propagation of the actuation signal which is typically diffusive, with ωmax ∼ 1/L2 [88]. Instead, our work suggests that if soft actuators were triggered locally rather than externally, then active hydraulics enables an alternate mechanism for faster contractions with ωmax ∼ 1/L.

Our results highlight the importance of a spatiotemporally integrated and multiscale view of muscle physiology for understanding this ubiquitous effector of soft power underlying animal movement. While we have demonstrated general biophysical consequences of an active sponge-like description of muscle, the specific physiological regimes and organisms where active hydraulics and odd elasticity are functional remains to be explored. A natural system to study this might be insect flight muscle, given its evolutionary age, phylogenetic breadth, and relatively simple physiology [89]. Further work incorporating tissue-scale response, neural control, Ca+2 signaling, inertial loading response etc., is required to understand the generality of the phenomena studied here, and connect with more macroscopic approaches to comparative biomechanics [90]. Ultimately, we will need an integrative view of muscle that spans the range from molecular motors to whole tissues to determine its ultimate performance limits and its failure modes, and thence eventually uncover its evolutionary trajectories and physiological functions.

Author Contributions Statement

LM conceived the research topic/approach; SS and LM formulated the theoretical model; SS performed the analytical calculations; SS compiled and analyzed the data; SS and LM wrote the paper.

Competing Interests Statement

The authors declare no competing interests.

Methods

Supplementary Information provides detailed descriptions of the theoretical model, analytical calculations, data acquisition and reanalysis.

Data Availability

All original data supporting the findings of this work were obtained from published literature as indicated in the Supplementary Information (Sec. V). All reanalyzed versions of the data used in this work are available within the paper and Supplementary Information (Sec. V).

Acknowledgments

SS acknowledges support from the Harvard Society of Fellows, and LM acknowledges partial support from the NSF-Simons Center for Mathematical and Statistical Analysis of Biology 1764269, the Simons Foundation and the Henri Seydoux Fund. We thank Shriya Srinivasan for useful discussions.

Footnotes

  • Title, abstract, introduction and discussion revised to clarify scope and significance; Section on odd elasticity expanded; new Figure 4 added; author affiliations updated.

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Active hydraulics and odd elasticity of muscle fibers
Suraj Shankar, L. Mahadevan
bioRxiv 2022.02.20.481216; doi: https://doi.org/10.1101/2022.02.20.481216
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