A three filament mechanistic model of musculotendon force and impedance
Abstract
The force developed by actively lengthened muscle depends on different structures across different scales of lengthening. For small perturbations, the active response of muscle is well captured by a lineartimeinvariant (LTI) system: a stiff spring in parallel with a light damper. The force response of muscle to longer stretches is better represented by a compliant spring that can fix its end when activated. Experimental work has shown that the stiffness and damping (impedance) of muscle in response to small perturbations is of fundamental importance to motor learning and mechanical stability, while the huge forces developed during long active stretches are critical for simulating and predicting injury. Outside of motor learning and injury, muscle is actively lengthened as a part of nearly all terrestrial locomotion. Despite the functional importance of impedance and active lengthening, no single muscle model has all these mechanical properties. In this work, we present the viscoelasticcrossbridge activetitin (VEXAT) model that can replicate the response of muscle to length changes great and small. To evaluate the VEXAT model, we compare its response to biological muscle by simulating experiments that measure the impedance of muscle, and the forces developed during long active stretches. In addition, we have also compared the responses of the VEXAT model to a popular Hilltype muscle model. The VEXAT model more accurately captures the impedance of biological muscle and its responses to long active stretches than a Hilltype model and can still reproduce the forcevelocity and forcelength relations of muscle. While the comparison between the VEXAT model and biological muscle is favorable, there are some phenomena that can be improved: the low frequency phase response of the model, and a mechanism to support passive force enhancement.
eLife assessment
This is a valuable study that develops a new model of the way muscle responds to perturbations, synthesizing models of how it responds to small and large perturbations, both of which are used to predict how muscles function for stability but also how they can be injured, and which tend to be predicted poorly by classic Hilltype models. The evidence presented to support the model is solid, since it outperforms Hilltype models in a variety of conditions. Although the combination of phenomenological and mechanistic aspects of the model may sometimes make it challenging to interpret the output, the work will be of interest to those developing realistic models of the stability and control of movement in humans or other animals.
https://doi.org/10.7554/eLife.88344.4.sa0Introduction
The stiffness and damping of muscle are properties of fundamental importance for motor control, and the accurate simulation of muscle force. The CNS exploits the activationdependent stiffness and damping (impedance) of muscle when learning new movements (Franklin et al., 2003), and when moving in unstable (Trumbower et al., 2009) or noisy environments (Selen et al., 2009). Reaching experiments using haptic manipulanda show that the CNS uses cocontraction to increase the stiffness of the arm when perturbed by an unstable force field (Burdet et al., 2001). With time and repetition, the force field becomes learned and cocontraction is reduced (Franklin et al., 2003).
The force response of muscle is not uniform, but varies with both the length and time of perturbation. Under constant activation and at a consistent nominal length, Kirsch et al., 1994 were able to show that muscle behaves like a lineartimeinvariant (LTI) system in response to small perturbations (see Appendix 9, Note 1): a springdamper of best fit captured over 90% of the observed variation in muscle force for small perturbations (1–3.8% optimal length) over a wide range of bandwidths (4–90 Hz). When active muscle is stretched appreciably, titin can develop enormous forces (Herzog and Leonard, 2002; Leonard et al., 2010), which may prevent further lengthening and injury. The stiffness that best captures the response of muscle to the small perturbations of Kirsch et al., 1994 is far greater than the stiffness that best captures the response of muscle to large perturbations (Herzog and Leonard, 2002; Leonard et al., 2010). Since everyday movements are often accompanied by both large and small kinematic perturbations, it is important to accurately capture these two processes.
However, there is likely no single muscle model that can replicate the force response of muscle to small (Kirsch et al., 1994) and large perturbations (Herzog and Leonard, 2002; Leonard et al., 2010) while also retaining the capability to reproduce the experiments of Hill, 1938 and Gordon et al., 1966. Unfortunately, this means that simulation studies that depend on an accurate replication of the perturbation response may reach conclusions well justified in simulation but not in reality. In this work, we focus on formulating a mechanistic muscle model (see Appendix 9, Note 2), that can replicate the force response of active muscle to length perturbations both great and small.
There are predominantly three classes of models that are used to simulate musculoskeletal responses: phenomenological models constructed using the famous forcevelocity relationship of Hill, 1938, mechanistic Huxley (Huxley, 1957; Huxley and Simmons, 1971; Kosta et al., 2022) models in which individual elastic crossbridges are incorporated, and linearized muscle models (Hogan, 1985; MussaIvaldi et al., 1985) which are accurate for small changes in muscle length. Kirsch et al., 1994 demonstrated that, for small perturbations, the force response of muscle is well represented by an activationdependent spring and damper that are connected in parallel. Neither Hill nor Huxley models are likely to replicate the experiments of Kirsch et al., 1994 because a Hill muscle model (Zajac, 1989; Millard et al., 2013) does not contain any active spring elements; while a Huxley model lacks an active damping element. Although linearized muscle models can replicate the experiment of Kirsch et al., 1994, these models are only accurate for small changes in length and cannot replicate the nonlinear forcevelocity relation of Hill, 1938, nor the nonlinear forcelength relation of Gordon et al., 1966. However, there have been significant improvements to the canonical forms of phenomenological, mechanistic, and linearized muscle models that warrant closer inspection.
Several novel muscle models have been proposed to improve upon the accuracy of Hilltype muscle models during large active stretches. Forcinito et al., 1998 modeled the velocity dependence of muscle using a rheological element (see Appendix 9, Note 3) and an elastic rack rather than embedding the forcevelocity relationship in equations directly, as is done in a typical Hill model (Zajac, 1989; Millard et al., 2013). This modification allows the model of Forcinito et al., 1998 to more faithfully replicate the force development of active muscle, as compared to a Hilltype model, during ramp length changes of ≈10% (see Appendix 9, Note 4) of the optimal CE length, and across velocities of 4–11% of the maximum contraction velocity (see Appendix 9, Note 5). Tamura and Saito, 2002 extended the work of Forcinito et al., 1998 by formulating a rheological muscle model with two Maxwell elements (a springdamper in series) where one develops force quickly (high stiffness) and the other develops force slowly (low stiffness). By carefully selecting the dynamics that drive the two elements, the model of Tamura and Saito, 2002 replicated the forcelengthvelocity relations (Gordon et al., 1966; Hill, 1938) as well as qualitatively reproducing the force and stiffness profiles (Tamura et al., 2005) of forceenhancement and forcedepression as measured by Sugi and Tsuchiya, 1988. Haeufle et al., 2014 made use of a serialparallel network of springdampers to allow their model to reproduce the forcevelocity relationship of Hill, 1938 mechanistically rather than embedding the experimental curve directly in their model. Günther et al., 2018 evaluated how accurately a variety of springdamper models were able to reproduce the microscopic increases in crossbridge force in response to small length changes. While each of these models improves upon the force response of the Hill model to ramp length changes, none are likely to reproduce the experiment of Kirsch et al., 1994 because the linearized versions of these models lead to a serial, rather than a parallel, connection of a spring and a damper: Kirsch et al., 1994 specifically showed (see Figure 3 of Kirsch et al., 1994) that a serial connection of a springdamper fails to reproduce the phase shift between force and length present in their experimental data.
Titin (Maruyama, 1976; Wang et al., 1979) has been more recently investigated to explain how lengthened muscle can develop active force when lengthened both within, and beyond, actinmyosin overlap (Leonard et al., 2010). Titin is a gigantic multisegmented protein that spans a halfsarcomere, attaching to the Zline at one end and the middle of the thick filament at the other end (Maruyama et al., 1985). In skeletal muscle, the two sections nearest to the Zline, the proximal immunoglobulin (IgP) segment and the PEVK segment — rich in the amino acids proline (P), glutamate (E), valine (V) and lysine (K) — are the most compliant (Trombitás et al., 1998b) since the distal immunoglobulin (IgD) segments bind strongly to the thick filament (Houmeida et al., 1995). Titin has proven to be a complex filament, varying in composition and geometry between different muscle types (Tomalka et al., 2019; Boldt et al., 2020), widely between species (Lindstedt and Nishikawa, 2017), and can apply activation dependent forces to actin (Kellermayer and Granzier, 1996). It has proven challenging to determine which interactions dominate between the various segments of titin and the other filaments in a sarcomere. Experimental observations have reported titinactin interactions at myosinactin binding sites (Astier et al., 1998; Niederländer et al., 2004), between titin’s PEVK region and actin (Bianco et al., 2007; Nagy et al., 2004), between titin’s N2A region and actin (Dutta et al., 2018), and between the PEVKIgD regions of titin and myosin (DuVall et al., 2017). This large variety of experimental observations has led to a correspondingly large number of proposed hypotheses and models, most of which involve titin interacting with actin (Rode et al., 2009; Nishikawa et al., 2012; SchappacherTilp et al., 2015; Tahir et al., 2018; Heidlauf et al., 2016; Heidlauf et al., 2017), and more recently with myosin (DuVall et al., 2016).
The addition of a titin element to a model will result in more accurate force production during large active length changes, but does not affect the stiffness and damping of muscle at modest sarcomere lengths because of titin’s relatively low stiffness. At sarcomere lengths of $1.62{\mathcal{l}}_{\mathrm{o}}^{\mathrm{M}}$ or less, the stiffness of the actinmyosin load path with a single attached crossbridge (0.22–1.15pN/nm) equals or exceeds the stiffness of 6 passive titin filaments (0.0348–0.173pN/nm), and our estimated stiffness of 6 active titin filaments (0.0696–0.346pN/nm, see Appendix 1 for further details). When fully activated, the stiffness of the actinmyosin load path (4.05–18.4pN/nm) far exceeds that of both the passive titin (0.0348–0.173pN/nm), and our estimated active titin ($0.06960.346\text{pN/nm}$) load paths. Since titinfocused models have not made any changes to the modeled myosinactin interaction beyond a Hill (Zajac, 1989; Millard et al., 2013) or Huxley (Huxley, 1957; Huxley and Simmons, 1971) model, it is unlikely that these models would be able to replicate the experiments of Kirsch et al., 1994.
Although most motor control simulations (Kearney et al., 1997; Perreault et al., 2004; Schouten et al., 2008; Trumbower et al., 2009; Mitrovic et al., 2010) make use of the canonical linearized muscle model, phenomenological muscle models have also been used and modified to include stiffness. Sartori et al., 2015 modeled muscle stiffness by evaluating the partial derivative of the force developed by a Hilltype muscle model with respect to the contractile element (CE) length. Although this approach is mathematically correct, the resulting stiffness is heavily inﬂuenced by the shape of the forcelength curve and can lead to inaccurate results: at the optimal CE length this approach would predict an active muscle stiffness of zero since the slope of the forcelength curve is zero; on the descending limb this approach would predict a negative active muscle stiffness since the slope of the forcelength curve is negative. In contrast, CE stiffness is large and positive near the optimal length (Kirsch et al., 1994), and there is no evidence for negative stiffness on the descending limb of the forcelength curve (Herzog and Leonard, 2002). Although the stiffness of the CE can be kept positive by shifting the passive forcelength curve, which is at times used in finiteelementmodels of muscle (Heidlauf et al., 2017), this introduces a new problem: the resulting passive CE stiffness cannot be lowered to match a more ﬂexible muscle. In contrast, De Groote et al., 2017 and De Groote et al., 2018 modeled shortrangestiffness using a stiff spring in parallel with the active force element of a Hilltype muscle model. While the approach developed in De Groote et al., 2017 and De Groote et al., 2018 likely does improve the response of a Hilltype muscle model for small perturbations, there are several drawbacks: the shortrangestiffness of the muscle sharply goes to zero outside of the specified range whereas in reality the stiffness is only reduced (Kirsch et al., 1994, see Figure 9A); the damping of the canonical Hillmodel has been left unchanged and likely differs substantially from biological muscle (Kirsch et al., 1994).
In this work, we propose a model that can capture the force development of muscle to perturbations that vary in size and timescale, and yet is described using only a few states making it well suited for largescale simulations. When active, the response of the model to perturbations within actinmyosin overlap is dominated by a viscoelastic crossbridge element that has different dynamics across timescales: over brief timescales the viscoelasticity of the lumped crossbridge dominates the response of the muscle (Kirsch et al., 1994), while over longer timescales the forcevelocity (Hill, 1938) and forcelength (Gordon et al., 1966) properties of muscle dominate. To capture the active forces developed by muscle beyond actinmyosin overlap we added an active titin element which, similar to the models of Rode et al., 2009 and (SchappacherTilp et al., 2015), features an activationdependent (see Appendix 9, Note 6) interaction between titin and actin. To ensure that the various parts of the model are bounded by reality, we have estimated the physical properties of the viscoelastic crossbridge element as well as the active titin element using data from the literature.
While our main focus is to develop a more accurate muscle model, we would like the model to be well suited to simulating systems that contain tens to hundreds of muscles. Although Huxley models have been used to simulate wholebody movements such as jumping (van Soest et al., 2019), the memory and processing requirements associated with simulating a single muscle with thousands of states is high. Instead of modeling the force development of individual crossbridges, we lump all of the crossbridges in a muscle together so that we have a small number of states to simulate per muscle.
To evaluate the proposed model, we compare simulations of experiments to original data. We examine the response of active muscle to small perturbations over a wide bandwidth by simulating the stochastic perturbation experiments of Kirsch et al., 1994. The activelengthening experiments of Herzog and Leonard, 2002 are used to evaluate the response of the model when it is actively lengthened within actinmyosin overlap. Next, we use the activelengthening experiments of Leonard et al., 2010 to see how the model compares to reality when it is actively lengthened beyond actinmyosin overlap. In addition, we examine how well the model can reproduce the forcevelocity experiments of Hill, 1938 and forcelength experiments of Gordon et al., 1966. Since Hilltype models are so commonly used, we also replicate all of the simulated experiments using the Hilltype muscle model of Millard et al., 2013 to make the differences between these two types of models clear.
Model
We begin by treating whole muscle as a scaled halfsarcomere that is pennated at an angle $\alpha$ with respect to a tendon (Figure 1A). The assumption that mechanical properties scale with size is commonly used when modeling muscle (Zajac, 1989) and makes it possible to model vastly different musculotendon units (MTUs) by simply changing the architectural and contraction properties: the maximum isometric force $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, the optimal CE length $\mathcal{l}}_{\text{o}}^{\text{M}$ (at which the CE develops $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$), the pennation angle $\alpha}_{\text{o}$ of the CE (at a length of $\mathcal{l}}_{\text{o}}^{\text{M}$) with respect to the tendon, the maximum shortening velocity $v}_{max}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ of the CE, and the slack length of the tendon $\mathcal{l}}_{\text{s}}^{\text{T}$. Many properties of sarcomeres scale with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and $\mathcal{l}}_{\text{o}}^{\text{M}$: $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ scales with physiological crosssectional area (Maganaris et al., 2001), the forcelength property scales with $\mathcal{l}}_{\text{o}}^{\text{M}$ (Winters et al., 2011), the maximum normalized shortening velocity of different CE types scales with $\mathcal{l}}_{\text{o}}^{\text{M}$ across animals great and small (Rome et al., 1990), and titin’s passiveforcelength properties scale from single molecules to myofibrils (Herzog et al., 2012; Prado et al., 2005).
The proposed model has several additional properties that we assume scale with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and inversely with $\mathcal{l}}_{\text{o}}^{\text{M}$: the maximum active isometric stiffness $k}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ and damping $\beta}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$, the passive forces due to the extracellular matrix (ECM), and passive forces due to titin. As crossbridge stiffness is well studied (Kaya and Higuchi, 2013), we assume that muscle stiffness due to crossbridges scales such that
where $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ is the maximum normalized stiffness. This scaling is just what would be expected when many crossbridges (Kaya and Higuchi, 2013) act in parallel across the crosssectional area of the muscle, and act in series along the length of the muscle. Although the intrinsic damping properties of crossbridges are not well studied, we assume that the linear increase in damping with activation observed by Kirsch et al., 1994 is due to the intrinsic damping properties of individual crossbridges which will also scale linearly with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and inversely with $\mathcal{l}}_{\text{o}}^{\text{M}$
where $\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ is the maximum normalized damping. For the remainder of the paper, we refer to the proposed model as the VEXAT model due to the viscoelastic (VE) crossbridge (X) and activetitin (AT) elements of the model.
To reduce the number of states needed to simulate the VEXAT model, we lump all of the attached crossbridges into a single lumped crossbridge element (XE) that attaches at $\mathcal{l}}^{\text{S}$ (Figure 1A) and has intrinsic stiffness and damping properties that vary with the activation and forcelength properties of muscle. The active force developed by the XE at the attachment point to actin is transmitted to the main myosin filament, the Mline, and ultimately to the tendon (Figure 1B). In addition, since the stiffness of actin (Higuchi et al., 1995) and myosin filaments (Tajima et al., 1994) greatly exceeds that of crossbridges (Veigel et al., 1998), we treat actin and myosin filaments as rigid to reduce the number of states needed to simulate this model. Similarly, we have lumped the six titin filaments per halfsarcomere (Figure 1A) together to further reduce the number of states needed to simulate this model.
The addition of a titin filament to the model introduces an additional active loadpath (Figure 1C) and an additional passive loadpath (Figure 1D). As is typical (Zajac, 1989; Millard et al., 2013), we assume that the passive elasticity of these structures scale linearly with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and inversely with $\mathcal{l}}_{\text{o}}^{\text{M}$. Since the VEXAT model has two passive load paths (Figure 1D), we further assume that the proportion of the passive force due to the extracellularmatrix (ECM) and titin does not follow a scaledependent pattern, but varies from muscletomuscle as observed by Prado et al., 2005.
As previously mentioned, there are several theories to explain how titin interacts with the other filaments in activated muscle. While there is evidence for titinactin interaction near titin’s N2A region (Dutta et al., 2018), there is also support for a titinactin interaction occurring near titin’s PEVK region (Bianco et al., 2007; Nagy et al., 2004), and for a titinmyosin interaction near the PEVKIgD region (DuVall et al., 2017). For the purposes of our model, we will assume a titinactin interaction because current evidence weighs more heavily towards a titinactin interaction than a titinmyosin interaction. Next, we assume that the titinactin interaction takes place somewhere in the PEVK segment for two reasons: first, there is evidence for a titinactin interaction (Bianco et al., 2007; Nagy et al., 2004) in the PEVK segment; and second, there is evidence supporting an interaction at the proximal end of the PEVK segment (Dutta et al., 2018, N2Aactin interaction). We have left the point within the PEVK segment that attaches to actin as a free variable since there is some uncertainty about what part of the PEVK segment interacts with actin.
The nature of the mechanical interaction between titin and the other filaments in an active sarcomere remains uncertain. Here, we assume that this interaction is not a rigid attachment, but instead is an activation dependent damping to be consistent with the observations of Kellermayer and Granzier, 1996 and Dutta et al., 2018: adding titin filaments and calcium slowed, but did not stop, the progression of actin filaments across a plate covered in active crossbridges (an in vitro motility assay). When activated, we assume that the amount of damping between titin and actin scales linearly with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and inversely with $\mathcal{l}}_{\text{o}}^{\text{M}$.
After lumping all of the crossbridges and titin filaments together, we are left with a rigidtendon MTU model that has two generalized positions
and an elastictendon MTU model that has three generalized positions
Given these generalized positions, the path length $\mathcal{l}}^{\text{P}$, and a pennation model, all other lengths in the model can be calculated. Here, we use a constant thickness
pennation model to evaluate the pennation angle
of a CE with a rigidtendon, and
to evaluate the pennation angle of a CE with an elastictendon. We have added a small compressive element KE (Figure 1A) to prevent the model from reaching the numerical singularity that exists as $\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ approaches $\stackrel{~}{\mathcal{l}}}_{min}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, the length at which $\alpha \to {90}^{\circ}$ in Equation 6 and Equation 7. The tendon length
of an elastictendon model is the difference between the path length and the CE length along the tendon. The length of the XE
is the difference between the halfsarcomere length and the sum of the average point of attachment $\mathcal{l}}^{\text{S}$ and the length of the myosin filament $\text{L}}^{\text{M}$. The length of $\mathcal{l}}^{\text{2}$, the lumped PEVKIgD segment, is
the difference between the halfsarcomere length and the sum of the length from the Zline to the actin binding site on titin ($\mathcal{l}}^{\text{1}}+{\text{L}}^{\text{T12}$) and the length of the IgD segment that is bound to myosin ($\text{L}}^{\text{IgD}$). Finally, the length of the extracellularmatrix $\mathcal{l}}^{\text{ECM}$ is simply
half the length of the CE since we are modeling a halfsarcomere.
We have some freedom to choose the state vector of the model and the differential equations that define how the muscle responds to length and activation changes. The experiments we hope to replicate depend on phenomena that take place at different timescales: the stochastic perturbations of Kirsch et al., 1994 evolve over brief timescales, while all of the other experiments take place at much longer timescales. Here, we mathematically decouple phenomena that affect brief and long timescales by making a secondorder model that has states of the average point of crossbridge attachment $\mathcal{l}}^{\text{S}$, and velocity $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$. When the activation state $a$ and the titinactin interaction model are included, the resulting rigidtendon model has a total of four states
and the elastictendon model has
five states. For the purpose of comparison, a Hilltype muscle model with a rigidtendon has a single state ($a$), while an elastictendon model has two states ($a$ and $\mathcal{l}}^{\text{M}$) (Millard et al., 2013).
Before proceeding, a small note on notation: throughout this work we will use an underbar to indicate a vector, bold font to indicate a curve, a tilde for a normalized quantity, and a capital letter to indicate a constant. Unless indicated otherwise, curves are constructed using $C}^{2$ continuous (see Appendix 9, Note 7) Bézier splines so that the model is compatible with gradientbased optimization. Normalized quantities within the CE follow a specific convention: lengths and velocities are normalized by the optimal CE length $\mathcal{l}}_{\text{o}}^{\text{M}$, forces by the maximum active isometric tension $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, stiffness and damping by $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$. Velocities used as input to the forcevelocity relation $\mathbf{f}}^{\text{V}$ are further normalized by $v}_{\text{max}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and annotated using a hat: $\hat{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}={v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{v}_{\text{max}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$. Tendon lengths and velocities are normalized by $\mathcal{l}}_{\text{s}}^{\text{T}$ tendon slack length, while forces are normalized by $f}_{\mathrm{o}}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{M}$.
To evaluate the state derivative of the model, we require equations for $\dot{a}$, $\dot{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$, $v}^{\phantom{\rule{thinmathspace}{0ex}}1$, and $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ if the tendon is elastic. For $\dot{a}$ we use of the first order activation dynamics model described in Millard et al., 2013 (see Appendix 9, Note 8) which uses a lumped firstorder ordinarydifferentialequation (ODE) to describe how a fused tetanus electrical excitation leads to force development in an isometric muscle. We formulated the equation for $\dot{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$ with the intention of having the model behave like a springdamper over small timescales, but to converge to the tension developed by a Hilltype model
over longer timescales, where ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}(\cdot )$ is the activeforcelength curve (Figure 2A), ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}(\cdot )$ is the passiveforcelength curve (Figure 2A), and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{V}}(\cdot )$ is the forcevelocity curve (Figure 2B).
The normalized tension developed by the VEXAT model
differs from that of a Hill model, Equation 14, because it has no explicit dependency on $\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, includes four passive terms, and a lumped viscoelastic crossbridge element. The four passive terms come from the ECM element ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$ (Figure 3A), the PEVKIgD element ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}2}({\stackrel{~}{\mathcal{l}}}^{2})$ (Figure 3A and B), the compressive term ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{KE}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$ (prevents ${\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}\mathrm{cos}\alpha$ from reaching a length of 0), and a numerical damping term $\stackrel{~}{\beta}}^{\phantom{\rule{thinmathspace}{0ex}}\u03f5}{\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ (where $\stackrel{~}{\beta}}^{\phantom{\rule{thinmathspace}{0ex}}\u03f5$ is small). The active force developed by the XE’s lumped crossbridge $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}{\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}+{\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}{\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ is scaled by the fraction of the XE that is active and attached, $a{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}({\stackrel{~}{\mathcal{l}}}^{\text{S}}+{\stackrel{~}{\text{L}}}^{\text{M}})$, where ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}(\cdot )$ is the activeforcelength relation (Figure 2A). We evaluate $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}$ using $\stackrel{~}{\mathcal{l}}}^{\text{S}}+{\stackrel{~}{\text{L}}}^{\text{M}$ in Equation 15, rather than $\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ as in Equation 14, since actinmyosin overlap is independent of crossbridge strain. With $\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ derived, we can proceed to model the acceleration of the CE, $\dot{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$, so that it is driven over time by the force imbalance between the XE’s active tension and that of a Hill model.
We set the first term of $\dot{\stackrel{~}{v}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$ so that, over time, the CE is driven to develop the same active tension as the Hilltype model of Millard et al., 2013 (terms highlighted in blue)
where $\tau}^{\text{S}$ is a time constant and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{V}}({\hat{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}})$ is the forcevelocity curve (Figure 2B). The rate of adaptation of the model’s tension, to the embedded Hill model, is set by the time constant $\tau}^{\text{S}$: as $\tau}^{\text{S}$ is decreased the VEXAT model converges more rapidly to a Hilltype model; as $\tau}^{\text{S}$ is increased the active force produced by the model will look more like a springdamper. Our preliminary simulations indicate that there is a tradeoff to choosing $\tau}^{\text{S}$: when $\tau}^{\text{S}$ is large the model will not shorten rapidly enough to replicate Hill’s experiments, while if $\tau}^{\text{S}$ is small the lowfrequency response of the model is compromised when the experiments of Kirsch et al., 1994 are simulated.
The remaining two terms, $\text{D}{\stackrel{~}{v}}^{\text{S}}$ and ${\text{e}}^{(a/{a}_{\text{L}}{)}^{2}}({\text{G}}_{\text{L}}{\stackrel{~}{\mathcal{l}}}^{\text{X}}+{\text{G}}_{\text{V}}{\stackrel{~}{v}}^{\text{X}})$, have been included for numerical reasons specific to this model formulation rather than muscle physiology. We include a term that damps the rate of actinmyosin translation, $\text{D}{\stackrel{~}{v}}^{\text{S}}$, to prevent this secondorder system from unrealistically oscillating (see Appendix 9, Note 9). The final term ${\text{e}}^{(a/{a}_{\text{L}}{)}^{2}}({\text{G}}_{\text{L}}{\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}+{\text{G}}_{\text{V}}{\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}})$, where $\text{G}}_{\text{L}$ and $\text{G}}_{\text{V}$ are scalar gains, and $a}_{\text{L}$ is a lowactivation threshold ($a}_{\text{L}$ is $0.05$ in this work). This final term has been included as a consequence of the generalized positions we have chosen. When the CE is nearly deactivated (as $a$ approaches $a}_{\text{L}$), this term forces $\stackrel{~}{\mathcal{l}}}^{\text{S}$ and $\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{S}$ to shadow the location and velocity of the XE attachment point. This ensures that if the XE is suddenly activated, that it attaches with little strain. We had to include this term because we made $\mathcal{l}}^{\text{S}$ a state variable, rather than $\mathcal{l}}^{\text{X}$. We chose $\mathcal{l}}^{\text{S}$ as a state variable, rather than $\mathcal{l}}^{\text{X}$, so that the states are more equally scaled for numerical integration.
The passive force developed by the CE in Equation 15 is the sum of the elastic forces (Figure 3A) developed by the forcelength curves of titin (${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$) and the ECM (${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$). We model titin’s elasticity as being due to two serially connected elastic segments: the first elastic segment ${\mathbf{f}}^{1}({\stackrel{~}{\mathcal{l}}}^{1})$ is formed by lumping together the IgP segment and a fraction $\text{Q}$ of the PEVK segment, while the second elastic segment ${\mathbf{f}}^{2}({\stackrel{~}{\mathcal{l}}}^{2})$ is formed by lumping together the remaining $(1\text{Q})$ of the PEVK segment with the free IgD section. Our preliminary simulations of the active lengthening experiment of Herzog and Leonard, 2002 indicate that a $\text{Q}$ value of 0.5, positioning the PEVKactin attachment point that is near the middle of the PEVK segment, allows the model to develop sufficient tension when actively lengthened. The large section of the IgD segment that is bound to myosin is treated as rigid.
The curves that form ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$, ${\mathbf{f}}^{1}({\stackrel{~}{\mathcal{l}}}^{1})$, and ${\mathbf{f}}^{2}({\stackrel{~}{\mathcal{l}}}^{2})$ have been carefully constructed to satisfy three experimental observations: that the total passive forcelength curve of titin and the ECM match the observed passive forcelength curve of the muscle (Figure 2A and Figure 3A) as in the experiments of Prado et al., 2005; that the proportion of the passive force developed by titin and the ECM (Figure 3A) is within experimental observations of Prado et al., 2005; and that the Ig domains and PEVK residues show the same relative elongation (Figure 3C) as observed by Trombitás et al., 1998a. Even though the measurements of Trombitás et al., 1998b come from human soleus titin, we can construct the forcelength curves of other titin isoforms if the number of proximal Ig domains, PEVK residues, and distal Ig domains are known (see Appendix 2.4). Since the passive–forcelength relation and the results of Trombitás et al., 1998b are at modest lengths, we consider two different extensions to the forcelength relation to simulate extreme lengths: first, a simple linear extrapolation; second, we extend the forcelength relation of each of titin’s segments to follow a wormlikechain (WLC) model similar to Trombitás et al., 1998b (see Appendix 2.4 for details on the WLC model). With titin’s passive forcelength relations defined, we can next consider what happens to titin in active muscle.
When active muscle is lengthened, it produces an enhanced force that persists long after the lengthening has ceased called residual force enhancement (RFE) (Herzog and Leonard, 2002). For the purposes of the VEXAT model, we assume that RFE is produced by titin. Experiments have demonstrated RFE on both the ascending limb (Peterson et al., 2004) and descending limb of the forcelength (Herzog and Leonard, 2002) relation. The amount of RFE depends both on the final length of the stretch (Hisey et al., 2009) and the magnitude of the stretch: on the ascending limb the amount of RFE varies with the final length but not with stretch magnitude, while on the descending limb RFE varies with stretch magnitude.
To develop RFE, we assume that some point of the PEVK segment bonds with actin through an activationdependent damper. The VEXAT model’s distal segment of titin, $\mathcal{l}}^{\text{2}$, can contribute to RFE when the titinactin bond is formed and CE is lengthened beyond $\stackrel{~}{\mathcal{l}}}_{s}^{\text{M}$, the shortest CE length at which the PEVKactin bond can form. In this work, we set $\stackrel{~}{\mathcal{l}}}_{s}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{M}$ to be equal to the slack length of the CE’s forcelength relation $\stackrel{~}{\mathcal{l}}}_{\text{s}}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\mathrm{E}$ (see Table 1E and H). To incorporate the asymmetric length dependence of RFE observed by Hisey et al., 2009, we introduce a smooth stepup function
that transitions from zero to one as $\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ extends beyond $\stackrel{~}{\mathcal{l}}}_{s}^{\text{M}$, where the sharpness of the transition is controlled by $\text{R}$. Similar to the experimental work of Hisey et al., 2009, active lengthening on the ascending limb will produce similar amounts of RFE since $\stackrel{~}{\mathcal{l}}}_{s}^{\text{M}}<{\mathcal{l}}_{\text{o}}^{\text{M}$ and the titinactin bond is prevented from forming below $\stackrel{~}{\mathcal{l}}}_{s}^{\text{M}$. In contrast, the amount of RFE on the descending limb increases with increasing stretch magnitudes since the titinactin bond is able to form across the entire descending limb.
At very long CE lengths, the modeled titinactin bond can literally slip off of the end of the actin filament (Figure 1A) when the distance between the Zline and the bond, $\stackrel{~}{\mathcal{l}}}^{\text{1}}+{\stackrel{~}{\text{L}}}^{\text{T12}$, exceeds the length of the actin filament, $\stackrel{~}{\text{L}}}^{\text{A}$. To break the titinactin bond at long CE lengths, we introduce a smooth stepdown function
The stepdown function $u}^{\text{L}$ transitions from one to zero when the titinactin bond ($\stackrel{~}{\mathcal{l}}}^{\text{1}}+{\stackrel{~}{\text{L}}}^{\text{T12}$) reaches $\stackrel{~}{\text{L}}}^{\text{A}$, the end of the actin filament.
The strength of the titinactin bond also appears to vary nonlinearly with activation. Fukutani and Herzog, 2018 observed that the absolute RFE magnitude produced by actively lengthened fibers is similar between normal and reduced contractile force states. Since the experiments of Fukutani and Herzog, 2018 were performed beyond the optimal CE length, titin could be contributing to the observed RFE as previously described. The consistent pattern of absolute RFE values observed by Fukutani and Herzog, 2018 could be produced if the titinactin bond saturated at its maximum strength even at a reduced contractile force state. To saturate the titinactin bond, we use a final smooth step function
where $\text{A}}_{\circ$ is the threshold activation level at which the bond saturates. While we model the strength of the titinactin bond as being a function of activation, which is proportional Ca^{2+} concentration (Millar and Homsher, 1990), this is a mathematical convenience. The work of Leonard et al., 2010 makes it clear that both Ca^{2+} and crossbridge cycling are needed to allow titin to develop enhanced forces during active lengthening: no enhanced forces are observed in the presence of Ca^{2+} when crossbridge cycling is chemically inhibited. Putting this all together, the active damping acting between the titin and actin filaments is given by the product of $u}^{\text{A}}{u}^{\text{S}}{u}^{\text{L}}{\beta}_{\text{A}}^{\text{PEVK}$, where $\beta}_{\text{A}}^{\text{PEVK}$ is the maximum damping coefficient.
With a model of the titinactin bond derived, we can focus on how the bond location moves in response to applied forces. Since we are ignoring the mass of the titin filament, the PEVKattachment point is balanced by the forces applied to it and the viscous forces developed between titin and actin
due to the active ($u}^{\text{A}}{u}^{\text{S}}{u}^{\text{L}}{\beta}_{\text{A}}^{\text{PEVK}$) and a small amount of passive damping ($\beta}_{\text{P}}^{\text{PEVK}$). Since Equation 20 is linear in $\stackrel{~}{v}}^{\text{1}$, we can solve directly for it
The assumption of whether the tendon is rigid or elastic affects how the state derivative is evaluated and how expensive it is to compute. While all of the position dependent quantities can be evaluated using Equations 6–11 and the generalized positions, evaluating the generalized velocities of a rigidtendon and elastictendon model differ substantially. The CE velocity $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and pennation angular velocity $\dot{\alpha}$ of a rigidtendon model can be evaluated directly given the path length, velocity, and the time derivatives of Equation 6 and Equation 8. After $v}^{\text{1}$ is evaluated using Equation 21, the velocities of the remaining segments can be evaluated using the time derivatives of Equations 9–11.
Evaluating the CE rate of lengthening, $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, for an elastictendon muscle model is more involved. As is typical of lumped parameter muscle models (Zajac, 1989; Thelen, 2003; Millard et al., 2013), here we assume that difference in tension, $\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\u03f5$, between the CE and the tendon
is negligible (see Appendix 9, Note 10). During our preliminary simulations, it became clear that treating the tendon as an idealized spring degraded the ability of the model to replicate the experiment of Kirsch et al., 1994 particularly at high frequencies. Kirsch et al., 1994 observed a linear increase in the gain and phase profile between the output force and the input perturbation applied to the muscle. This pattern in gain and phase shift can be accurately reproduced by a spring in parallel with a damper. Due to the way that impedance combines in series (see Appendix 9, Note 11), the models of both the CE and the tendon need to have parallel spring and damper elements so that the entire MTU, when linearized, appears to be a spring in parallel with a damping element. We model tendon force using a nonlinear spring and damper model
where the damping coefficient $U{\hat{\mathbf{k}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}})$, is a linear scaling of the normalized tendon stiffness $\hat{\mathit{k}}}^{\text{T}$ by $U$, a constant scaling coefficient. We have chosen this specific damping model because it fits the data of Netti et al., 1996 and captures the structural coupling between tendon stiffness and damping (see Appendix 2.2 and Figure 1 for further details).
Now that all the terms in Equation 22 have been explicitly defined, we can use Equation 22 to solve for $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$. Equation 22 becomes linear in $v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ after substituting the force models described in Equation 23 and Equation 15, and the kinematic model described in Equation 8, Equation 9 and Equation 11 (along with the time derivatives of Equations 8–11). After some simplification we arrive at
allowing us to evaluate the final state derivative in $\underset{\_}{\dot{x}}$. During simulation the denominator of $\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ will always be finite since ${\stackrel{~}{\beta}}^{\u03f5}>0$, and $\alpha <{90}^{\circ}$ due to the compressive element. The evaluation of $\underset{\_}{\dot{x}}$ in the VEXAT model is free of numerical singularities, giving it an advantage over a conventional Hilltype muscle models (Millard et al., 2013). In addition, the VEXAT’s $\underset{\_}{\dot{x}}$ does not require iteration to numerically solve a root, giving it an advantage over a singularityfree formulation of the Hill model (Millard et al., 2013). As with previous models, initializing the model’s state is not trivial and required the derivation of a modelspecific method (see Appendix 3 for details).
Biological benchmark simulations
In order to evaluate the model, we have selected three experiments that capture the responses of active muscle to small, medium, and large length changes. The small (1–3.8% $\mathcal{l}}_{\text{o}}^{\text{M}$) stochastic perturbation experiment of Kirsch et al., 1994 demonstrates that the impedance of muscle is well described by a stiff spring in parallel with a damper, and that the springdamper coefficients vary linearly with active force. The active impedance of muscle is such a fundamental part of motor learning that the amount of impedance, as indicated by cocontraction, is used to define how much learning has actually taken place (Franklin et al., 2003; Franklin et al., 2008): cocontraction is high during initial learning, but decreases over time as a task becomes familiar. The active lengthening experiment of Herzog and Leonard, 2002 shows that modestly stretched (7–21% $\mathcal{l}}_{\text{o}}^{\text{M}$) biological muscle has positive stiffness even on the descending limb of the active forcelength curve (${\mathcal{l}}_{\text{o}}^{\text{M}}>1$). In contrast, a conventional Hill model (Zajac, 1989; Millard et al., 2013) can have negative stiffness on the descending limb of the activeforcelength curve, a property that is both mechanically unstable and unrealistic. The final active lengthening experiment of Leonard et al., 2010 unequivocally demonstrates that the CE continues to develop active forces during extreme lengthening (329% $\mathcal{l}}_{\text{o}}^{\text{M}$) which exceeds actinmyosin overlap. Active force development beyond actinmyosin overlap is made possible by titin, and its activationdependent interaction with actin (Leonard et al., 2010). The biological benchmark simulations conclude with a replication of the forcevelocity experiments of Hill, 1938 and the forcelength experiments of Gordon et al., 1966.
The VEXAT model requires the architectural muscle parameters ($f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, $\mathcal{l}}_{\text{o}}^{\text{M}$, $\alpha}_{\text{o}$, $v}_{max}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, and $\mathcal{l}}_{\text{s}}^{\text{T}$) needed by a conventional Hilltype muscle model as well as additional parameters. The additional parameters are needed for these component models: the compressive element (Equation 15 and Equation 24), the lumped viscoelastic XE (Equation 1 and Equation 2), XEactin dynamics (Equation 16), the twosegment active titin model (Figure 3), titinactin dynamics (Equation 21), and the tendon damping model (Equation 23). Fortunately, there is enough experimental data in the literature that values can be found, fitted, or estimated directly for our simulations of experiments on cat soleus (Table 1F.I.), and rabbit psoas fibrils (see Appendix 2 and Appendix 8 for rabbit psoas fibril model parameters). The parameter values we have established for the cat soleus (Table 1F.I.) can serve as initial values when modeling other mammalian MTU’s because these parameters have been normalized (by $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, $\mathcal{l}}_{\text{o}}^{\text{M}$, and $\mathcal{l}}_{\text{s}}^{\text{T}$ where appropriate) and will scale appropriately given the architectural properties of a different MTU. By making use of these default values, the VEXAT model can be made to represent another MTU using exactly the same number of parameters as a Hilltype muscle model (Table 1A.E.).
Stochastic length perturbation experiments
In the insitu experiment of Kirsch et al., 1994, the force response of a cat’s soleus muscle under constant stimulation was measured as its length was changed by small amounts. Kirsch et al., 1994 applied stochastic length perturbations (Figure 4A, blue line) to elicit force responses from the muscle (in this case a springdamper Figure 4A, black line) across a broad range of frequencies (4–90 Hz) and across a range of small length perturbations (1–3.8% $\mathcal{l}}_{\text{o}}^{\text{M}$). The resulting timedomain signals can be quite complicated (Figure 4A) but contain rich measurements of how muscle transforms changes in length into changes in force.
As long as muscle can be considered to be linear (a sinusoidal change in length produces a sinusoidal change in force), then system identification methods (Oppenheim et al., 1996; Koopmans, 1995) can be applied to extract a relationship between length $x(t)$ and force $y(t)$. We will give a brief overview of system identification methods here to make methods and results clearer. First, the timedomain signals ($x(t)$ and $y(t)$) are transformed into an equivalent representation in the frequencydomain ($X(s)$ and $Y(s)$) as a sum of scaled and shifted sine curves (Figure 4B and C) using a Fourier transform (Oppenheim et al., 1996). In the frequency domain, we identify an LTI system of best fit $H(s)$ that describes how muscle transforms changes in length into changes in force such that $Y(s)=H(s)X(s)$. Next, we evaluate how $H(s)$ scales the magnitude (gain) and shifts the timing (phase) of a sinusoid in $X(s)$ into a sinusoid of the same frequency in $Y(s)$ (Figure 4D). This process is repeated across all frequencymatched pairs of input and output sinusoids to build a function of how muscle scales (Figure 4E) and shifts (Figure 4F) input length sinusoids into output force sinusoids. The resulting transformation turns two complicated timedomain signals (Figure 4A) into a clear relationship in the frequencydomain that describes how muscle transforms length changes into force changes: a very slow (≈0 Hz) length change will result in an output force that is scaled by 4.5 and is in phase (Figure 4E and F), a 35 Hz sinusoidal length change will produce an output force that is scaled by 4.9 and leads the input signal by 24° (Figure 4E and F), and frequencies between 0 Hz and 35 Hz will be between these two signals in terms of scaling and phase. These patterns of gain and phase can be used to identify a network of springdampers that is equivalent to the underlying linear system (the system in Figure 4A, E and F is a 4.46 N/mm spring in parallel with a 0.0089 Ns/mm damper). Since experimental measurements often contain noise and small nonlinearities, the mathematical procedure used to estimate $H(s)$ and the corresponding gain and phase profiles is more elaborate than we have described (see Appendix 4 for details).
Kirsch et al., 1994 used system identification methods to identify LTI mechanical systems that best describes how muscle transforms input length waveforms to output force waveforms. The resulting LTI system, however, is only accurate when the relationship between input and output is approximately linear. Kirsch et al., 1994 used the coherence squared, $({C}_{xy}{)}^{2}$, between the input and output to evaluate the degree of linearity: $Y(s)$ is a linear transformation of $X(s)$ at frequencies in which $({C}_{xy}{)}^{2}$ is near one, while $Y(s)$ cannot be described as a linear function of $X(s)$ at frequencies in which $({C}_{xy}{)}^{2}$ approaches zero. By calculating $({C}_{xy}{)}^{2}$ between the length perturbation and force waveforms, Kirsch et al., 1994 identified the bandwidth in which the muscle’s response is approximately linear. Kirsch et al., 1994 set the lower frequency of this band to 4 Hz, and Figure 3 of Kirsch et al., 1994 suggests that this corresponds to $({C}_{xy}{)}^{2}\ge 0.67$ though the threshold for $({C}_{xy}{)}^{2}$ is not reported. The upper frequency of this band was set to the cutoff frequency of the lowpass filter applied to the input (15, 35, or 90 Hz). Within this bandwidth, Kirsch et al., 1994 compared the response of the specimen to several candidate models and found that a parallel springdamper fit the muscle’s frequency response best. Next, Kirsch et al., 1994 evaluated the stiffness and damping coefficients that best fit the muscle’s frequency response. Finally, Kirsch et al., 1994 evaluated how much of the muscle’s timedomain response was captured by the springdamper of best fit by evaluating the varianceaccountedfor (VAF) between the two timedomain signals
Astonishingly, Kirsch et al., 1994 found that a springdamper of best fit has a VAF of between 78% and 99% (see Appendix 9, Note 12) when compared to the experimentally measured forces $f}^{\phantom{\rule{thinmathspace}{0ex}}\text{EXP}$. By repeating this experiment over a variety of stimulation levels (using both electrical stimulation and the crossedextension reﬂex) Kirsch et al., 1994 showed that these stiffness and damping coefficients vary linearly with the active force developed by the muscle. Furthermore, Kirsch et al., 1994 repeated the experiment using perturbations that had a variety of lengths (0.4 mm, 0.8 mm, and 1.6 mm) and bandwidths (15 Hz, 35 Hz, and 90 Hz) and observed a peculiar quality of muscle: the damping coefficient of best fit increases as the bandwidth of the perturbation decreases (see Figures 3 and 10 of Kirsch et al., 1994 for details). Here, we simulate the experiment of Kirsch et al., 1994 to determine, first, the VAF of the VEXAT model and the Hill model in comparison to a springdamper of best fit; second, to compare the gain and phase response of the models to biological muscle; and ﬁnally, to see if the springdamper coefficients of best ﬁt for both models increase with active force in a manner that is similar to the cat soleus that Kirsch et al., 1994 studied.
To simulate the experiments of Kirsch et al., 1994 we begin by creating the 9 stochastic perturbation waveforms used in the experiment that vary in perturbation amplitude (0.4 mm, 0.8mm, and 1.6 mm) and bandwidth (0–15 Hz, 0–35 Hz, and 0–90 Hz) (see Appendix 9, Note 13). The waveform is created using a vector that is composed of random numbers with a range of [1,1] that begins and ends with a series of zerovalued padding points. Next, a forward pass of a secondorder Butterworth ﬁlter is applied to the waveform and ﬁnally the signal is scaled to the appropriate amplitude (Figure 5). The muscle model is then activated until it develops a constant tension at a length of $\mathcal{l}}_{\text{o}}^{\text{M}$. The musculotendon unit is then simulated as the length is varied using the previously constructed waveforms while activation is held constant. To see how impedance varies with active force, we repeated these simulations at ten evenly spaced tensions from 2.5N to 11.5N. Ninety simulations are required to evaluate the nine different perturbation waveforms at each of the ten tension levels. The timedomain length perturbations and force responses of the modeled muscles are used to evaluate the coherence squared of the signal, gain response, and phase responses of the models in the frequencydomain. Since the response of the models might be more nonlinear than biological muscle, we select a bandwidth that meets $({C}_{xy}{)}^{2}>0.67$ but otherwise follows the bandwidths analyzed by Kirsch et al., 1994 (see Appendix 4 for details).
When coupled with an elastictendon, the 15 Hz perturbations show that neither model can match the VAF of the analysis of Kirsch et al., 1994 (compare Figure 6A to G), while at 90 Hz the VEXAT model reaches a VAF of 89% (Figure 6D) which is within the range of 78–99% reported by Kirsch et al., 1994. In contrast, the Hill model’s VAF at 90 Hz remains low at 58% (Figure 6J). While the VEXAT model has a gain proﬁle in the frequencydomain that closer to the data of Kirsch et al., 1994 than the Hill model (compare Figure 6B to H and E to K), both models have a greater phase shift than the data of Kirsch et al., 1994 at low frequencies (compare Figure 6C to I and F to L). The phase response of the VEXAT model to the 90 Hz perturbation (Figure 6F) shows the consequences of Equation 16: at low frequencies the phase response of the VEXAT model is similar to that of the Hill model, while at higher frequencies the model’s response becomes similar to a springdamper. This frequency dependent response is a consequence of the ﬁrst term in Equation 16: the value of $\tau}^{\text{S}$ causes the response of the model to be similar to a Hill model at lower frequencies and mimic a springdamper at higher frequencies. Both models show the same perturbationdependent phaseresponse, as the damping coefficient of best ﬁt increases as the perturbation bandwidth decreases: compare the damping coefficient of best ﬁt for the 15 Hz and 90 Hz proﬁles for the VEXAT model (listed on Figure 6A. and D.) and the Hill model (listed on Figure 6G. and J., respectively).
The closeness of each model’s response to the springdamper of best ﬁt changes when a rigidtendon is used instead of an elastictendon. While the VEXAT model’s response to the 15 Hz and 90 Hz perturbations improves slightly (compare Figure 6A–F to Figure 1A–F in Appendix 6), the response of the Hill model to the 15 Hz perturbation changes dramatically with the timedomain VAF rising from 55% to 85% (compare Figure 6G–L to Figure 1G–L in Appendix 6). Although the Hill model’s VAF in response to the 15 Hz perturbation improved, the frequency response contains mixed results: the rigidtendon Hill model’s gain response is better (Figure 1H in Appendix 6), while the phase response is worse in comparison to the elastictendon Hill model. While the rigidtendon Hill model produces a better timedomain response to the 15 Hz perturbation than the elastictendon Hill model, this improvement has been made with a larger phase shift between force and length than biological muscle (Kirsch et al., 1994).
The gain and phase proﬁles of both models deviate from the springdamper of best ﬁt due to the presence of nonlinearities, even for small perturbations. Some of the VEXAT model’s nonlinearities in this experiment come from the tendon model (compare Figure 6A–F to Figure 1A–F in Appendix 6), since the response of the VEXAT model with a rigidtendon stays closer to the springdamper of best ﬁt. The Hill model’s nonlinearities originate from the underlying expressions for stiffness and damping of the Hill model, which are particularly nonlinear with a rigidtendon model (Figure 1G–L in Appendix 6). The stiffness of a Hill model’s CE
is heavily inﬂuenced by the partial derivative of $\frac{\text{d}{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}}{\text{d}{\mathcal{l}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}}$ which has a region of negative stiffness. Although $\frac{\text{d}{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}}{\text{d}{\mathcal{l}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}}$ is well approximated as being linear for small length changes, $\frac{\text{d}{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}}{\text{d}{\mathcal{l}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}}$ changes sign across $\mathcal{l}}_{\text{o}}^{\text{M}$. The damping of a Hill model’s CE
also suffers from high degrees of nonlinearity for small perturbations about ${v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}=0$ since the slope of $\frac{\text{d}{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{V}}}{\text{d}{v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}}$ is positive and large when shortening, and positive and small when lengthening (Figure 2B). While Equation 26 and Equation 27 are mathematically correct, the negative stiffness and wide ranging damping values predicted by these equations do not match experimental data (Kirsch et al., 1994). In contrast, the stiffness
and damping
of the VEXAT’s CE do not change so drastically because these terms do not depend on the slope of the forcelength relation, or the forcevelocity relation (see Appendix 2.5 for derivation).
By repeating the stochastic perturbation experiments across a range of isometric forces, Kirsch et al., 1994 were able to show that the stiffness and damping of a muscle varies linearly with the active tension it develops (see Figure 12 of Kirsch et al., 1994). We have repeated our simulations of the experiments of Kirsch et al., 1994 at ten nominal forces (spaced evenly between 2.5 N and 11.5 N) and compared how the VEXAT model and the Hill model’s stiffness and damping coefficients (Figure 7) compare to Figure 12 of Kirsch et al., 1994. The stiffness and damping proﬁle of the VEXAT model deviates a little from the data of Kirsch et al., 1994 because XE’s dynamics at 35 Hz are still inﬂuenced by the Hill model embedded in Equation 16 (see Appendix 2.5). Despite this, the VEXAT model develops similar stiffness and damping proﬁle with either a viscoelastictendon (Figure 7A and B) or a rigidtendon (Figure 7C and D). In contrast, when the Hill model is coupled with an elastictendon both its stiffness and damping are larger than what is reported by Kirsch et al., 1994 at the higher tensions (Figure 7A and B). This pattern changes when simulating a Hill model with a rigidtendon: the model’s stiffness is slightly negative (Figure 7C), while the model’s ﬁnal damping coefficient is nearly three times the value measured by Kirsch et al., 1994 (Figure 7D). Though a negative stiffness may seem surprising, Equation 26 shows a negative stiffness is possible at the nominal CE length of these simulations: just past $\mathcal{l}}_{\text{o}}^{\text{M}$ the slope of the active forcelength curve is negative and the slope of the passive forcelength curve is negligible. The tendon model also affects the VAF of the Hill model to a large degree: the elastictendon Hill model has a low VAF 30–51% (Figure 7A & B) while the rigidtendon Hill model has a much higher VAF of 86%. Although the VAF of the rigidtendon Hill model is acceptable, these forces are being generated in a completely different manner than those obtained from biological muscle, as the data of Kirsch et al., 1994 indicate (Figure 7C and D).
When the VAF of the VEXAT and Hill model is evaluated across a range of nominal tensions (ten values from 2.5 to 11.5 N), frequencies (15 Hz, 35 Hz, and 90 Hz), amplitudes (0.4 mm, 0.8 mm, and 1.6 mm), and tendon types (rigid and elastic) two things are clear: ﬁrst, that the VEXAT model’s 64–100% VAF is close to the 78–99% VAF reported by Kirsch et al., 1994 while the Hill model’s 28–95% VAF differs (Figure 8); and second, that there are systematic variations in VAF, stiffness, and damping across the different perturbation magnitudes and frequencies (see Appendix 5—table 1 and Appendix 5—table 2). Both models produce worse VAF values when coupled with an elastictendon (Figure 8A, B and C), although the Hill model is affected most: the mean VAF of the elastictendon Hill model is 67% lower than the mean VAF of the rigidtendon model for the 0.4 mm 15 Hz perturbations (Figure 8A). While the VEXAT model’s lowest VAF occurs in response to the low frequency perturbations (Figure 8A) with both rigid and elastictendons, the Hill model’s lowest VAF varies with both tendon type and frequency: the rigidtendon Hill model has its lowest VAF in response to the 1.6 mm 90 Hz perturbations (Figure 8C) while the elastictendon Hill model has its lowest VAF in response to the 0.4 mm 15 Hz perturbations (Figure 8A). It is unclear if biological muscle displays systematic shifts in VAF since Kirsch et al., 1994 did not report the VAF of each trial.
Active lengthening on the descending limb
We now turn our attention to the active lengthening insitu experiments of Herzog and Leonard, 2002. During these experiments, cat soleus muscles were actively lengthened by modest amounts (7–21% $\mathcal{l}}_{\text{o}}^{\text{M}$) starting on the descending limb of the activeforcelength curve (${\mathcal{l}}^{\text{M}}/{\mathcal{l}}_{\text{o}}^{\text{M}}>1$ in Figure 2A). This starting point was chosen speciﬁcally because the stiffness of a Hill model may actually change sign and become negative because of the inﬂuence of the activeforcelength curve on $k}^{\text{M}$ as shown in Equation 26 as $\mathcal{l}}^{\text{M}$ extends beyond $\mathcal{l}}_{\text{o}}^{\text{M}$. The experiment of Herzog and Leonard, 2002 is important for showing that biological muscle does not exhibit negative stiffness on the descending limb of the activeforcelength curve. In addition, this experiment also highlights the slow recovery of the muscle’s force after stretching has ceased, and the phenomena of passive force enhancement after stimulation is removed. Here we will examine the 9 mm/s ramp experiment in detail because the simulations of the 3 mm/s and 27 mm/s ramp experiments produces similar stereotypical patterns (see Appendix 7 for details).
When the active lengthening experiment of Herzog and Leonard, 2002 is simulated (Figure 9A), both models produce a force transient initially (Figure 9B), but for different reasons. The VEXAT model’s transient is created when the lumped crossbridge spring (the $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}{\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ term in Equation 15) is stretched. In contrast, the Hill model’s transient is produced, not by spring forces, but by damping produced by the forcevelocity curve as shown in Equation 26.
After the initial force transient, the response of the two models diverges (Figure 9B): the VEXAT model continues to develop increased tension as it is lengthened, while the Hill model’s tension drops before recovering. The VEXAT model’s continued increase in force is due to the titin model: when activated, a section of titin’s PEVK region remains approximately ﬁxed to the actin element (Figure 1C). As a result, the $\mathcal{l}}^{\text{2}$ element (composed of part of PEVK segment and the distal Ig segment) continues to stretch and generates higher forces than it would if the muscle were being passively stretched. While both the elastic and rigidtendon versions of the VEXAT model produce the same stereotypical ramplengthening response (Figure 9B), the rigidtendon model develops slightly more tension because the strain of the MTU is solely borne by the CE.
In contrast, the Hill model develops less force during lengthening when it enters a small region of negative stiffness (Figure 9B and C) because the passiveforcelength curve is too compliant to compensate for the negative slope of the active forcelength curve. Similarly, the damping coefficient of the Hill model drops substantially during lengthening (Figure 9D). Equation 27 and Figure 2B shows the reason that damping drops during lengthening: $d{\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{V}}/d{v}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$, the slope of the line in Fig. Figure 2B, is quite large when the muscle is isometric and becomes quite small as the rate of lengthening increases.
After the ramp stretch is completed (at time 3.4 s in Figure 9B), the tension developed by the cat soleus recovers slowly, following a proﬁle that looks strikingly like a ﬁrstorder decay. The large damping coefficient acting between the titinactin bond slows the force recovery of the VEXAT model. We have tuned the value of $\beta}_{\text{A}}^{\text{PEVK}$ to $71.9{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/({\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/s)$ for the elastictendon model, and $77.7{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/({\mathcal{l}}_{\text{o}}^{\text{M}}/s)$ for the rigidtendon model, to match the rate of force decay of the cat soleus in the data of Herzog and Leonard, 2002. The Hill model, in contrast, recovers to its isometric value quite rapidly. Since the Hill model’s force enhancement during lengthening is a function of the rate of lengthening, when the lengthening ceases, so too does the force enhancement.
Once activation is allowed to return to zero, the data of Herzog and Leonard, 2002 shows that the cat soleus continues to develop a tension that is $\mathrm{\Delta}{f}_{B}$ above passive levels (Figure 9B for $t>8.5s$). The force $\mathrm{\Delta}{f}_{B}$ is known as passive force enhancement, and is suspected to be caused by titin binding to actin (Herzog, 2019). Since we model titinactin forces using an activationdependent damper, when activation goes to zero our titin model becomes unbound from actin. As such, both our model and a Hill model remain $\mathrm{\Delta}{f}_{B}$ below the experimental data of Herzog and Leonard (Figure 9B) after lengthening and activation have ceased.
Active lengthening beyond actinmyosin overlap
One of the great challenges that remains is to decompose how much tension is developed by titin (Figure 1C) separately from myosin (Figure 1B) in an active sarcomere. The activelengthening experiment of Leonard et al., 2010 provides some insight into this force distribution problem because they recorded active forces both within and far beyond actinmyosin overlap. The data of Leonard et al., 2010 shows that active force continues to develop linearly during lengthening, beyond actinmyosin overlap, until mechanical failure. When activated and lengthened, the myoﬁbrils failed at a length of $3.38{\mathcal{l}}_{\text{o}}^{\text{M}}$ and force of $5.14{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$, on average. In contrast, during passive lengthening myoﬁbrils failed at a much shorter length of $2.86{\mathcal{l}}_{\text{o}}^{\text{M}}$ with a dramatically lower tension of of $1.31{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$. To show that the extraordinary forces beyond actinmyosin overlap can be ascribed to titin, Leonard et al., 2010 repeated the experiment but deleted titin using trypsin: the titindeleted myoﬁbrils failed at short lengths and insigniﬁcant stresses. Using the titin model of Equation 20 (Figure 1A) as an interpretive lens, the huge forces developed during active lengthening would be created when titin is bound to actin leaving the distal segment of titin to take up all of the strain. Conversely, our titin model would produce lower forces during passive lengthening because the proximal Ig, PEVK, and distal Ig regions would all be lengthening together (Figure 3A).
Since the experiment of Leonard et al., 2010 was performed on skinned rabbit myoﬁbrils and not on whole muscle, both the VEXAT and Hill models had to be adjusted prior to simulation (see Appendix 8 for parameter values). To simulate a rabbit myoﬁbril we created a forcelength curve (Rassier et al., 1999) consistent with the ﬁlament lengths of rabbit skeletal muscle (Higuchi et al., 1995; 1.12 µm actin, 1.63 µm myosin, and 0.07 µm zline width) and ﬁt the forcelength relations of the two titin segments to be consistent with the structure measured by Prado et al., 2005 of rabbit psoas titin consisting of a 70–30% mix of a 3300kD and a 3400kD titin isoform (see Appendix 2.4 for ﬁtting details and Appendix 8 for parameter values). Since this is a simulation of a ﬁbril, we used a rigidtendon of zero length (equivalent to ignoring the tendon), and set the pennation angle to zero.
As mentioned in the ‘Model’ section, because this experiment includes extreme lengths, we consider two different forcelength relations for each segment of titin (Figure 10A): a linear extrapolation, and an extension that follows the WLC model. While both versions of the titin model are identical up to $\stackrel{~}{\mathcal{l}}}_{\text{toe}}^{\text{PE}$, beyond $\stackrel{~}{\mathcal{l}}}_{\text{toe}}^{\text{PE}$ the WLC model continues to develop increasingly large forces until all of the Ig domains and PEVK residues have been unfolded and the segments of titin reach a physical singularity: at this point the Ig domains and PEVK residues cannot be elongated any further without breaking molecular bonds (see Appendix 2.4 for details). Our preliminary simulations indicated that the linear titin model’s titinactin bond was not strong enough to support large tensions, and so we increased the value of $\beta}_{\text{A}}^{\text{PEVK}$ from 71.9 to 975 (compare Table 1H and Appendix 8—table 1G).
The Hill model was similarly modiﬁed, with the pennation angle set to zero and coupled with a rigidtendon of zero length. Since the Hill model lacks an ECM element the passiveforcelength curve was instead ﬁtted to match the passive forces produced in the data of Leonard et al., 2010. No adjustments were made to the active elements of the Hill model.
When the slow active stretch (0.1 µm/sarcomere/s) of the experiment of Leonard et al., 2010 is simulated, only the VEXAT model with the linear titin element can match the experimental data of Leonard et al., 2010 (Figure 10B). The Hill model cannot produce active force for lengths greater than $1.62{\mathcal{l}}_{\text{o}}^{\text{M}}$ since the active forcelength curve goes to zero (Figure 2A and Figure 10B) and the model lacks any element capable of producing force beyond this length. In contrast, the linear titin model continues to develop active force until a length of 3.38 $\mathcal{l}}_{\text{o}}^{\text{M}$ is reached, at which point the titinactin bond is pulled off the end of the actin ﬁlament and the active force is reduced to its passive value.
The WLC titin model is not able to reach the extreme lengths observed by Leonard et al., 2010. The distal segment of the WLC titin model approaches its contour length early in the simulation and ensures that the the titinactin bond is dragged off the end of the actin ﬁlament at 1.99 $\mathcal{l}}_{\text{o}}^{\text{M}$ (Figure 10B). After 1.99 $\mathcal{l}}_{\text{o}}^{\text{M}$ (Figure 10B), the tension of the WLC titin model drops to its passive value but continues to increase until the contour lengths of all of the segments of titin are reached at 2.32 $\mathcal{l}}_{\text{o}}^{\text{M}$. Comparing the response of the linear model to the WLC titin model two things are clear: the linear titin model more faithfully follows the data of Leonard et al., 2010, but does so with titin segment lengths that exceed the maximum contour length expected for the isoform of titin in a rabbit myoﬁbril.
This simulation has also uncovered a surprising fact: the myoﬁbrils in the experiments of Leonard et al., 2010 do not fail at 2.32 $\mathcal{l}}_{\text{o}}^{\text{M}$, as would be expected by the WLC model of titin, but instead reach much greater lengths (Figure 10B). Physically, it may be possible for a rabbit myoﬁbril to reach these lengths (without exceeding the contour lengths of the proximal Ig, PEVK, and distal Ig segments) if the bond between the distal segment of titin and myosin breaks down. This would allow the large Ig segment, that is normally bound to myosin, to uncoil and continue to develop the forces observed by Leonard et al., 2010. Unfortunately the mechanism which allowed the samples in Leonard et al.’s experiments to develop tension beyond titin’s contour length remains unknown.
Forcelength and forcevelocity
Although the active portion of the Hill model is embedded in Equation 16, it is not clear if the VEXAT model can still replicate forcevelocity experiments of Hill, 1938 and the forcelength experiments of Gordon et al., 1966. Here, we simulate both of these experiments using the cat soleus model that we have used for the simulations described in the ‘Model’ section (‘Stochastic length perturbation experiments’) and compare the results to the forcelength and forcevelocity curves that are used in the Hill model and in Equation 16 of the VEXAT model.
The forcevelocity experiment of Hill, 1938 is simulated by activating the model, and then by changing its length to follow a shortening ramp and a lengthening ramp. During shortening experiments, the CE shortens from $1.1{\mathcal{l}}_{\text{o}}^{\text{M}}$ to $0.9{\mathcal{l}}_{\text{o}}^{\text{M}}$ with the measurement of active muscle force is made at $\mathcal{l}}_{\text{o}}^{\text{M}$. Lengthening experiments are similarly made by measuring muscle force midway through a ramp stretch that begins at $0.9{\mathcal{l}}_{\text{o}}^{\text{M}}$ and ends at $1.1{\mathcal{l}}_{\text{o}}^{\text{M}}$. When an elastictendon model is used, we carefully evaluate initial and terminal path lengths to accommodate for the stretch of the tendon so that the CE still shortens from $1.1{\mathcal{l}}_{\text{o}}^{\text{M}}$ to $0.9{\mathcal{l}}_{\text{o}}^{\text{M}}$ and lengthens from $0.9{\mathcal{l}}_{\text{o}}^{\text{M}}$ to $1.1{\mathcal{l}}_{\text{o}}^{\text{M}}$.
The VEXAT model produces forces that differ slightly from the $\mathbf{f}}^{\text{V}$ that is embedded in Equation 16 while the Hill model reproduces the curve (Figure 11). The maximum shortening velocity of the VEXAT model is slightly weaker than the embedded curve due to the series viscoelasticity of the XE element. Although the model can be made to converge to the $\mathbf{f}}^{\text{V}$ curve more rapidly by decreasing $\tau}^{\text{S}$ this has the undesirable consequence of degrading the lowfrequency response of the model when simulating the experiments of Kirsch et al., 1994 (particularly Figure 6C., and F.). These small differences can be effectively removed by scaling $v}_{max}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ by $\text{s}}^{\text{V}$ (Figure 11A has ${\text{s}}^{\text{V}}=0.95$) to accommodate for the small decrease in force caused by the viscoelastic XE element.
The forcelength experiments of Gordon et al., 1966 were simulated by ﬁrst passively lengthening the CE, and next by measuring the active force developed by the CE at a series of ﬁxed lengths. Prior to activation, the passive CE was simulated for a brief period of time in a passive state to reduce any history effects due to the active titin element. To be consistent with the experiment of Gordon et al., 1966, we subtracted off the passive force from the active force before producing the activeforcelength proﬁle.
The simulation of Gordon et al., 1966 shows that the VEXAT model (Figure 12A, blue dots) produces a forcelength proﬁle that is shifted to the right of the Hill model (Figure 12A, red line) due to the series elasticity introduced by the XE. We can solve for the size of this rightwards shift by noting that Equation 16 will drive the $\stackrel{~}{\mathcal{l}}}^{\text{S}$ to a length such that the isometric force developed by the XE is equal to that of the embedded Hill model
allowing us to solve for
the isometric strain of the XE. Since there are two viscoelastic XE elements per CE, the VEXAT model has an active forcelength characteristic that shifted to the right of the embedded $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}$ curve by a constant $\frac{2}{{\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}}$. This shift, $\mathrm{\Delta}}^{\text{L}$, can be calibrated out of the model (Figure 12A, magenta squares) by adjusting the ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{L}}(\cdot )$ curve so that it is $\frac{2}{{\stackrel{~}{k}}_{\text{o}}^{\text{X}}}$ to the left of its normal position. Note that all simulations described in the previous sections made use of the VEXAT model with the calibrated forcelength relation and the calibrated forcevelocity relation.
Discussion
A muscle model is deﬁned by the experiments it can replicate and the mechanisms it embodies. We have developed the VEXAT muscle model to replicate the force response of muscle to a wide variety of perturbations (Kirsch et al., 1994; Herzog and Leonard, 2002; Leonard et al., 2010) while also retaining the ability to reproduce the forcevelocity experiment of Hill, 1938 and the forcelength experiment of Gordon et al., 1966. The model we have developed uses two mechanisms to capture the force response of muscle over a large variety of time and length scales: ﬁrst, a viscoelastic crossbridge element that over brief timescales appears as a springdamper, and at longer timescales mimics a Hillmodel; second, a titin element that is capable of developing active force during large stretches.
The viscoelastic crossbridge and titin elements we have developed introduce a number of assumptions into the model. While there is evidence that the activationdependent stiffness of muscle originates primarily from the stiffness of the attached crossbridges (Veigel et al., 1998), the origins of the activationdependent damping observed by Kirsch et al., 1994 have not yet been established. We assumed that, since the damping observed by Kirsch et al., 1994 varies linearly with activation, the damping originates from the attached crossbridges. Whether this damping is intrinsic or is due to some other factor remains to be established. Next, we have also assumed that the force developed by the XE converges to a Hill model (Millard et al., 2013) given enough time (Equation 16). A recent experiment (see Figure 7 of Tomalka et al., 2021) suggests the force developed by the XE might decrease during lengthening rather than increasing as is typical of a Hill model (Millard et al., 2013). If the observations of Tomalka et al., 2021 can be replicated, the VEXAT model will need to be adjusted so that the XE element develops less force during active lengthening while the activetitin element develops more force. Finally, we have assumed that actinmyosin sliding acceleration (due to crossbridge cycling) occurs when there is a force imbalance between the external force applied to the XE and the internal force developed by the XE as shown in Equation 16. This assumption is a departure from previous models: Hilltype models (Zajac, 1989; Millard et al., 2013) assume that the tension applied to the muscle instantaneously affects the actinmyosin sliding velocity; Huxley models (Huxley, 1957) assume that the actinmyosin sliding velocity directly affects the rate of attachment and detachment of crossbridges.
The active titin model that we have developed makes assumptions similar to Rode et al., 2009 and SchappacherTilp et al., 2015: some parts of the PEVK segment bond to actin, and this bond cannot do any positive work on titin. The assumption that the bond between titin and actin cannot do positive work means that titin cannot be actively preloaded: it can only develop force when it is stretched. In contrast, two mechanisms have been proposed that make it possible for titin to be preloaded by crossbridge cycling: the winding ﬁlament theory of Nishikawa et al., 2012 and the titin entanglement hypothesis of DuVall et al., 2016. If titin were signiﬁcantly preloaded by crossbridge cycling, the titin load path would support higher forces and the myosinactin load path would bear less force. Accordingly, the overall stiffness of the CE would be reduced, affecting our simulations of Kirsch et al., 1994: lower myosinactin loads mean fewer attached crossbridges, since crossbridges are stiff in comparison to titin, the stiffness of the CE would decrease (see Appendix 1). Hopefully experimental work will clarify if titin can be actively preloaded by crossbridges in the future.
Both the viscoelastic crossbridge and active titin elements include simple myosinactin and titinactin bond models that improve accuracy but have limitations. First, the viscoelastic crossbridge element has been made to represent a population of crossbridges in which the contribution of any single crossbridge is negligible. Although it may be possible for the XE model to accurately simulate a maximally activated single sarcomere (which has roughly 20 attached crossbridges per half sarcomere Huxley and Simmons, 1971; Howard, 1997), the accuracy of the model will degrade as the number of attached crossbridges decreases. When only a single crossbridge remains, the XE model’s output will be inaccurate because it can only generate force continuously while a real crossbridge generates force discretely each time it attaches to, and detaches from, actin. Next, we have used two equations, Equation 16 and Equation 21, that assume myosinactin and titinactin interactions are temperatureinvariant and scale linearly with size ($\mathcal{l}}_{\text{o}}^{\text{M}$ and $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$). In contrast, myosinactin interactions and some titinactin interactions are temperaturesensitive (Minajeva et al., 2002; Roots et al., 2012) and may not scale linearly with size. In the ‘Active lengthening beyond actinmyosin overlap’ section we had to adjust the active titin damping parameter, $\beta}_{\text{A}}^{\text{PEVK}$, to simulate the myoﬁbril experiments of Leonard et al., 2010, perhaps because the assumptions of temperatureinvariance and sizelinearity were not met: the initial value for $\beta}_{\text{A}}^{\text{PEVK}$ came from ﬁtting to insitu experimental data (Herzog and Leonard, 2002) from whole muscle that was warmer (35 − 36.5°C vs 20 − 21°C) and larger ($\mathcal{l}}_{\text{o}}^{\text{M}$ of 42.9mm vs. 10 − 15μm) than the myoﬁbrils (Leonard et al., 2010). While the cat soleus XE and titin model parameters (Table 1G, H1) can be used as rough default values, these parameters should be reﬁt to accurately simulate muscle that differs in scale or temperature from cat soleus. Finally, the VEXAT model in its current form ignores phenomena related to submaximal contractions: the shift in the peak of the forcelength relation (Stephenson and Wendt, 1984), and the scaling of the maximum shortening velocity (Chow and Darling, 1999). We hope to include these phenomena in a later version of the VEXAT model to more accurately simulate submaximal contractions.
The model we have proposed can replicate phenomena that occur at a wide variety of time and length scales: the experiments of Kirsch et al., 1994 which occur over small time and length scales; and the active lengthening experiments of Herzog and Leonard, 2002 and Leonard et al., 2010 which occur over physiological and supraphysiological length scales. In contrast, we have shown in the ‘Biological benchmark simulations’ section that a Hilltype model compares poorly to biological muscle when the same set of experiments are simulated. We expect that a Huxley model (Huxley, 1957) is also likely to have difficulty reproducing the experiment of Kirsch et al., 1994 because the model lacks an active damping element. Since titin was discovered (Maruyama, 1976) long after Huxley’s model was proposed (Huxley, 1957), a Huxley model will be unable to replicate any experiment that is strongly inﬂuenced by titin such as the experiment of Leonard et al., 2010.
Although there have been several more recent muscle model formulations proposed, none have the properties to simultaneously reproduce the experiments of Kirsch et al., 1994, Herzog and Leonard, 2002, Leonard et al., 2010, Hill, 1938, and Gordon et al., 1966. Linearized impedance models (Hogan, 1985; MussaIvaldi et al., 1985) can reproduce the experiments of Kirsch et al., 1994, but these models lack the nonlinear components needed to reproduce the forcelength experiments of Gordon et al., 1966, and the forcevelocity experiments of Hill, 1938. The models of Forcinito et al., 1998, and Tahir et al., 2018 have a structure that places a contractile element in series with an elastictendon. While this is a commonly used structure, at high frequencies the lack of damping in the tendon will drive the phase shift between length and force to approach zero. The measurements and model of Kirsch et al., 1994, in contrast, indicate that the phase shift between length and force approaches ninety degrees with increasing frequencies. Although the Hilltype models of Haeufle et al., 2014 and Günther et al., 2018 have viscoelastic tendons, these models have no representation of the viscoelasticity of the CE’s attached crossbridges. Similar to the Hilltype muscle model evaluated in this work (Millard et al., 2013), it is likely that models of Haeufle et al., 2014 and Günther et al., 2018 will not be able to match the frequency response of biological muscle. While the model of Tamura and Saito, 2002 is one of the few models that can develop forceenhancement and forcedepression (Tamura et al., 2005), it is unlikely that this model will be able to reproduce the frequency response of biological muscle because it uses springdamping elements in series: Kirsch et al., 1994 showed that the frequency response of springdamping elements in series poorly fits the response of biological muscle. The models of De Groote et al., 2017 and De Groote et al., 2018 introduced a shortrangestiffness element in parallel to a Hill model to capture the stiffness of biological muscle. While the formulations presented in De Groote et al., 2017 and De Groote et al., 2018 improves upon a Hill model it is unlikely to reproduce the experiement of Kirsch et al., 1994 because we have shown in the ‘Active lengthening beyond actinmyosin overlap’ section that a Hill model has a frequency response that differs from biological muscle. The muscle model of Rode et al., 2009 uses a Hill model for the CE and so we expect that this model will have the same difficulties reproducing the experiment of Kirsch et al., 1994. The model of SchappacherTilp et al., 2015 extends a Huxley model (Huxley, 1957) by adding a detailed titin element. Similar to a Huxley model, the model of SchappacherTilp et al., 2015 will likely have difficulty reproducing the experiment of Kirsch et al., 1994 because it is missing an active damping element.
While developing this model, we have come across open questions that we hope can be addressed in the future. How does muscle stiffness and damping change across the forcelength curve? Does stiffness and damping change with velocity? What are the physical origins of the active damping observed by Kirsch et al., 1994? What are the conditions that affect passiveforce enhancement, and its release? In addition to pursuing these questions, we hope that other researchers continue to contribute experiments that are amenable to simulation, and to develop musculotendon models that overcome the limitations of our model. To help others build upon our work, we have made the source code of the model and all simulations presented in this paper available online (see the elife 2023 branch of https://github.com/mjhmilla/Millard2021ImpedanceMuscle; copy archived at Millard, 2024).
Appendix 1
The stiﬀness of the actinmyosin and titin load paths
A single halfmyosin can connect to the surrounding six actin ﬁlaments through 97.9 crossbridges. A 0.800 μm halfmyosin has a pair crossbridges over 0.700 μm of its length every 14.3nm which amounts to 97.9 per halfmyosin (Huxley, 1969). Although 97.9 crossbridges does not make physical sense, here we will evaluate the stiffness of the CE assuming that fractional crossbridges exist and that attached crossbridges can be perfectly distributed among the six available actin ﬁlaments: the alternative calculation is more complicated and produces stiffness values that differ only in the 3^{rd} signiﬁcant digit. Assuming a duty cycle of 20% (Howard, 1997) (values between 5–90% have been reported by Finer et al., 1994), at full actinmyosin overlap there will be 19.6 crossbridges attached to the six surrounding actin ﬁlaments. Assuming that these 19.6 crossbridges are evenly distributed between the six actin ﬁlaments, each single actin will be attached to 3.26 attached crossbridges.
At full overlap, the Zline is 1 actin ﬁlament length $\text{L}}^{\text{A}$ (1.12 μm in rabbits according to Higuchi et al., 1995) from the Mline. The average point of crossbridge attachment is in the middle of the halfmyosin at a distance of 0.45 μm from the Mline (0.1 μm is bare and 0.35 μm is half of the remaining length), which is $\mathrm{L}}^{\text{A}$ − 0.45 μm from the Zline (Figure 1, Appendix 1—figure 1). A single actin ﬁlament has a stiffness of 48 − 68pN/nm (Higuchi et al., 1995) while a single crossbridge has a stiffness of 0.69 ± 0.47pN/nm (Veigel et al., 1998). Since stiffness scales inversely with length, actin’s stiﬀness between the Zline and the average attachment point is 81.8 − 121pN/nm. Finally, the stiffness of each actin ﬁlament and its 3.26 attached crossbridges is 0.712 − 3.67pN/nm and all six together have a stiffness of 4.27 − 22.0pN/nm.
Myosin has a similar stiffness as a single actin ﬁlament (Tajima et al., 1994), with the section between the average attachment point and the Mline having a stiffness of 76.9 − 113pN/nm. The ﬁnal active stiffness of halfsarcomere is 4.05 − 18.4pN/nm which comes from comes from the series connection of the group of six actin ﬁlaments, with 19.6 crossbridges, and ﬁnally the single myosin ﬁlament. When this procedure is repeated assuming that only a single crossbridge is attached the stiffness drops to 0.22 − 1.15pN/nm, which is slightly less than the stiffness of a single crossbridge (see main_ActinMyosinAndTitinStiffness.m in the elife2023 branch of accompanying code repository for details).
The forcelength proﬁle of a single rabbit titin has been measured by Kellermayer et al., 1997 using laser tweezers to apply cyclical stretches. By digitizing Figure 4B (blue line) of Kellermayer et al., 1997 we arrive at a stiffness for titin of 0.0058 − 0.0288pN/nm at 2μm (for a total sarcomere length of 4μm or $1.62{\mathcal{l}}_{\text{o}}^{\text{M}}$), and 0.0505 − 0.0928pN/nm at 4μm (8μm or $3.25{\mathcal{l}}_{\text{o}}^{\text{M}}$). Since there are 6 titin ﬁlaments acting in parallel for each halfsarcomere, we end up with the total stiffness for titin ranging between 0.0348 − 0.173pN/nm at 2μm and 0.303 − 0.557pN/nm at 4μm. When activated, the stiffness of our rabbit psoas lineartitin model (described in the ‘Active lengthening beyond actinmyosin overlap’ section and ﬁtted in Appendix 2.4, and with the parameters shown in Appendix 8) doubles, which would increase titin’s stiffness to 0.0696 − 0.346pN/nm at 2μm and 0.606 − 1.11pN/nm at 4μm.
Comparing the actinmyosin and titin stiffness ranges (Figure 2, Appendix 1—figure 2) makes it clear that the stiffness of actinmyosin with 1 attached crossbridge (AM:Low in Figure 2, Appendix 1—figure 2) is comparable to the highest stiffness values we have estimated for titin (AT:High in Figure 2, Appendix 1—figure 2). When all 20% of the available crossbridges are attached (AM:High in Figure 2, Appendix 1—figure 2), the average stiffness of the actinmyosin load path is roughly one order of magnitude stiffer than the highest stiffness values of titin (AT:High in Figure 2, Appendix 1—figure 2), and two to three orders of magnitude higher than the lowest stiffness titin load path (PT:Low in Figure 2). Similarly, the maximum XE stiffness and titin stiffness in this work are separated by roughly an order of magnitude: the cat soleus model has a XE stiffness of $49.1{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$ and maximum active titin stiffness of $8.42{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\text{M}}$ (Table 1); while the rabbit psoas ﬁbril model has a XE stiffness of $49.1{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$ and maximum active titin stiffness of $5.25{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$ (Appendix 8).
Appendix 2
2.1. Model fitting
Many of the experiments simulated in this work (Kirsch et al., 1994; Herzog and Leonard, 2002) have been performed using cat soleus muscle. While we have been able to extract some architectural parameters directly from the experiments we simulate ($f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and $\mathcal{l}}_{\text{o}}^{\text{M}$ from Herzog and Leonard, 2002), we have had to rely on the literature mentioned in Table 1 for the remaining parameters. The remaining properties of the model can be solved by ﬁrst building a viscoelastic damping model of the tendon; next, by solving for the intrinsic stiffness and damping properties of the CE; and ﬁnally, by ﬁtting the passive curves (${\text{f}}^{1}({\stackrel{~}{\mathcal{l}}}^{1})$ and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}2}({\stackrel{~}{\mathcal{l}}}^{2})$) to simultaneously ﬁt the passive forcelength curve recorded by Herzog and Leonard, 2002, using a mixture of tension from titin and the ECM that is consistent with the data of Prado et al., 2005, all while maintaining the geometric relationship between $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{IgP}$ and $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PEVK}$ as measured by Trombitás et al., 1998b.
2.2. Fitting the tendon’s stiffness and damping
Similar to previous work (Millard et al., 2013), we model the forcelength relation of the tendon using a quintic Bézier spline (Appendix 2—figure 1A) that begins at $({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}},{\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}})=(1.0,0)$ (where $\stackrel{~}{\mathcal{l}}}^{\text{T}$ is tendon length normalized by $\mathcal{l}}_{\text{s}}^{\text{T}$, and $\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{T}$ is tension normalized by $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$), ends at $({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}},\phantom{\rule{thinmathspace}{0ex}}{\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}})=(1.0+{e}_{\text{toe}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}},{f}_{\text{toe}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}})$ with a normalized stiffness of $\stackrel{~}{k}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}$, and uses the constants ${f}_{\text{toe}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}}=2/3$ and $\stackrel{~}{k}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}}=1.375/{e}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}$ (given $30.0\phantom{\rule{thinmathspace}{0ex}}{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$ from Scott and Loeb, 1995, $e}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{T}$ is thus 4.58%). Using the experimental data of Netti et al., 1996, we have also constructed a curve to evaluate the damping coefficient of the tendon. The normalized tendon stiffness (termed storage modulus by Netti et al., 1996) and normalized tendon damping (termed loss modulus by Netti et al., 1996) both have a similar shape as the tendon is stretched from slack to $e}_{\text{o}}^{\text{T}$ (Appendix 2—figure 1B and C). The similarity in shape is likely not a coincidence.
The nonlinear forcelength characteristics (Appendix 2—figure 1A) of tendon originate from its microscopic structure. Tendon is composed of many ﬁber bundles with differing resting lengths (Netti et al., 1996). Initially, the tendon’s ﬁber bundles begin crimped, but gradually stretch as the tendon lengthens, until ﬁnally all ﬁber bundles are stretched and the tendon achieves its maximum stiffness (Appendix 2—figure 1B) and damping (Appendix 2—figure 1C; Netti et al., 1996). Accordingly, in Equation 23, we have described the normalized damping of the tendon as being equal to the normalized stiffness of the tendon scaled by a constant $U$. To estimate $U$, we have used the measurements of Netti et al., 1996; Figure 1B and C and have solved a leastsquares problem
to arrive at a value of $U=0.057$. The resulting damping model (Figure 1C) ﬁts the measurements of Netti et al., 1996 closely.
2.3. Fitting the CE’s Impedance
We can now calculate the normalized impedance of the XE using the viscoelastictendon model we have constructed and the measurements of Kirsch et al., 1994 of the impedance of the entire MTU. Since an MTU is structured with a CE in series with a tendon, the compliance of the MTU is given by
where $k}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ is the stiffness of the CE in the direction of the tendon. We can calculate $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ directly by ﬁtting a line to the stiffness vs tension plot that appears in Figure 12 of Kirsch et al., 1994 (0.8 mm, 0–35 Hz perturbation) and resulting in ${k}^{\text{M}}=$ 2.47 N/mm at a nominal force of 5 N. Here we use a nominal tension of 5 N so that we can later compare our model to the 5 N frequency response reported in Figure 3 of Kirsch et al., 1994. Since Kirsch et al., 1994 did not report the architectural properties of the experimental specimens, we assume that the architectural properties of the cat used in Kirsch et al.’s experiments are similar to the properties listed in Table 1. We evaluate the stiffness of the tendon model by stretching it until it develops the nominal tension of Kirsch et al.’s Figure 3 data (5 N), and then compute its derivative which amounts to ${k}^{\text{T}}=$ 16.9 N/mm. Finally, using Equation 33, we can solve for a value of ${k}_{\text{AT}}^{\text{M}}=$ 2.90 N/mm. Since the inverse of damping adds for damping elements in series
we can use a similar procedure to evaluate $\beta}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, the damping of the CE along the tendon. The value of $\beta}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ that best ﬁts the damping vs. tension plot that appears in Figure 12 of Kirsch et al., 1994 at a nominal tension of 5 N is 0.0198 Ns/mm. The tendon damping model we have just constructed develops 0.697 Ns/mm at a nominal load of 5 N. Using Equation 34, we arrive at ${\beta}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}=$ 0.020 Ns/mm. Due to the pennation model, the stiffness and damping values of the CE differ from those along the tendon.
The stiffness of the CE along the tendon is
which can be expanded to
Since we are using a constant thickness pennation model
and thus
which simpliﬁes to
Similarly, the constant thickness pennation model means that
which leads to
Recognizing that
we can solve for $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ in terms of $k}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ by solving Equation 36 for $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and substituting the values of Equation 39, and Equation 41. In this case, the values of $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ (4.37 N/mm) and $k}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ (4.37 N/mm) are the same to three signiﬁcant ﬁgures.
We can use a similar process to transform $\beta}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ into $\beta}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ using the pennation model by noting that
which expands to a much smaller expression
than Equation 36 since $\alpha$ does not depend on $v}^{\text{M}$, and thus $\mathrm{\partial}\alpha /\mathrm{\partial}{v}^{\text{M}}=0$. By taking a time derivative of Equation 40, we arrive at
which allows us to solve for
By recognizing that
and using Equation 44 and Equation 46 we can evaluate $\beta}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ in terms of $\beta}_{\text{AT}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$. Similar to $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$, the value of $\beta}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ (0.020 Ns/mm) is close to $\beta}_{\text{AT}}^{\text{M}$ (0.020 Ns/mm). When this same procedure is applied to the stiffness and damping coefficients extracted from the gain and phase proﬁles from Figure 3 of Kirsch et al., 1994, the values of $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and $\beta}^{\text{M}$ differ (4.37 N/mm and 0.0090 Ns/mm) from the results produced using the data of Figure 12 (2.90 N/mm and 0.020 Ns/mm). Likely these differences arise because we have been forced to use a ﬁxed maximum isometric force for all specimens when, in reality, this property varies substantially. We now turn our attention to ﬁtting the titin and ECM elements, since we cannot determine how much of $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and $\beta}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ are due to the XE until the titin and ECM elements have been ﬁtted.
2.4. Fitting the forcelength curves of titin’s segments
The nonlinear forcelength curves used to describe titin (${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ in series), and the ECM (${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$) must satisfy three conditions: the total forcelength curve produced by the sum of the ECM and titin must match the observed passiveforcelength relation (Herzog and Leonard, 2002); the proportion of titin’s contribution relative to the ECM must be within measured bounds (Prado et al., 2005); and ﬁnally the stiffness of the ${\mathbf{f}}^{2}({\stackrel{~}{\mathcal{l}}}^{2})$ must be a linear scaling of ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ to match the observations of Trombitás et al., 1998b.
First, we ﬁt the passive forcelength curve to the data of Herzog and Leonard, 2002 to serve as a reference. The curve $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ begins at the normalized length and force coordinates of ($1+{e}_{0}^{\text{PE}}$,0) with a slope of 0, ends at ($1+{e}_{1}^{\text{PE}}$,1.0) with a slope of ${k}_{1}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}=2/({e}_{1}^{\text{PE}}{e}_{0}^{\text{PE}})$, and is linearly extrapolated outside of this region. We solve for the $e}_{0}^{\text{PE}$ and $e}_{1}^{\text{PE}$ such that
the squared differences between $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ and the passive forcelength data of Herzog and Leonard, 2002 (Figure 2A shows both the data and the ﬁtted $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ curve) are minimized. While $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ is not used directly in the model, it serves as a useful reference for constructing the ECM and titin forcelength curves. We assume that the ECM forcelength curve is a linear scaling of $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$
where $\text{P}$ is a constant. In this work, we set $\text{P}$ to 56% which is the average ECM contribution that Prado et al., 2005 measured across 5 different rabbit skeletal muscles (see Appendix 9, Note 14). The remaining fraction, $1\text{P}$, of the forcelength curve comes from titin.
In mammalian skeletal muscle, titin has three elastic segments (Prado et al., 2005) connected in series: the proximal Ig segment, the PEVK segment, and the distal Ig segment that is between the PEVK region and the myosin ﬁlament (Figure 1A). Trombitás et al., 1998b labelled the PEVK region of titin with antibodies allowing them to measure the distance between the Zline and the proximal Ig/PEVK boundary ($}^{\text{Z}}{\mathcal{l}}^{\text{IgP/PEVK}$), and the distance between the Zline and the PEVK/distal Ig boundary ($}^{\text{Z}}{\mathcal{l}}^{\text{PEVK/IgD}$), while the passive sarcomere was stretched from 2.35–4.46 μm. By ﬁtting functions to the data of Trombitás et al., 1998b, we can predict the length of any of titin’s segments under the following assumptions: the T12 segment is rigid (Figure 1A), the distal Ig segment that overlaps with myosin is rigid (Figure 1A), and that during passive stretching the tension throughout the titin ﬁlament is uniform. Since the sarcomeres in the experiments of Trombitás et al., 1998b were passively stretched it is reasonable to assume that tension throughout the free part of the titin ﬁlament is uniform because the bond between titin and actin depends on calcium (Dutta et al., 2018; Kellermayer and Granzier, 1996) and crossbridge attachment (Leonard et al., 2010).
We begin by digitizing the data of Figure 5 of Trombitás et al., 1998b and using the leastsquares method to ﬁt lines to $}^{\mathrm{Z}}{\mathcal{l}}^{\mathrm{I}\mathrm{g}\mathrm{P}/\mathrm{P}\mathrm{E}\mathrm{V}\mathrm{K}$ and $}^{\text{Z}}{\mathcal{l}}^{\text{PEVK/IgD}$ (where the superscripts mean $\displaystyle {}^{\text{from}}{\mathcal{l}}^{\text{to}}$ and so $}^{\text{Z}}{\mathcal{l}}^{\text{I}gP/PEVK$ is the distance from the Zline to the border of the IgP/PEVK segments). From these lines of best ﬁt we can evaluate the normalized length of the proximal Ig segment
the normalized length of the PEVK segment
and the normalized length of the distal Ig segment
as a function of sarcomere length. Next, we extract the coefficients for linear functions that evaluate the lengths of
and
given $\stackrel{~}{\mathcal{l}}}^{\text{M}$. The coefficients that best ﬁt the data from Trombitás et al., 1998b appear in Appendix 2—table 1.
These functions can be scaled to ﬁt a titin ﬁlament of a differing geometry. Many of the experiments simulated in this work used cat soleus. Although the lengths of titin ﬁlament segments in cat soleus have not been measured, we assume that it is a scaled version of a human soleus titin ﬁlament (68 proximal Ig domains, 2174 PEVK residues, and 22 distal Ig domains Trombitás et al., 1998b) since both muscles contain predominately slowtwitch ﬁbers: slow twitch ﬁbers tend to express longer, more compliant titin ﬁlaments (Prado et al., 2005). Since the optimal sarcomere length in cat skeletal muscle is shorter than in human skeletal muscle ($2.43$ μm vs. $2.73$ μm from Rassier et al., 1999) the coefficients for Equations 53–55 differ slightly (see the feline soleus column in Appendix 2—table 1). In addition, by assuming that the titin ﬁlament of cat skeletal muscle is a scaled version of the titin ﬁlament found in human skeletal muscle, we have implicitly assumed that the cat’s skeletal muscle titin ﬁlament has 60.5 proximal Ig domains, 1934.7 PEVK residues, and 19.6 distal Ig domains. Although a fraction of a domain does not make physical sense, we have not rounded to the nearest domain and residue to preserve the sarcomere lengthbased scaling.
In contrast, the rabbit psoas ﬁbril used in the simulation of Leonard et al., 2010 has a known titin geometry (50 proximal Ig domains, 800 PEVK residues, and 22 distal Ig domains as measured by Prado et al., 2005) which differs substantially from the isoform of titin expressed in the human soleus. To create the rabbit psoas titin length functions ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\text{IgP}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$, ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PEVK}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$, and ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\text{IgD}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$, we begin by scaling the human soleus PEVK length function ${\stackrel{~}{\mathcal{l}}}_{\text{H}}^{\text{PEVK}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$ by the relative proportion of PEVK residues of 800/2174. The length of the two Ig segments
is what remains from the halfsarcomere once the rigid lengths of titin (0.100 μm for $\text{L}}^{\text{T12}$ and 0.8150 μm for $\text{L}}^{\text{IgD}$ Higuchi et al., 1995) and the PEVK segment length have been subtracted away. The function that describes ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\text{IgP}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$ and ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\text{IgD}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$ can then be formed by scaling ${\stackrel{~}{\mathcal{l}}}_{\text{R}}^{\text{Ig}}({\stackrel{~}{\mathcal{l}}}^{\text{M}})$ by the proportion of Ig domains in each segment
which produce the coefficients that appear in the rabbit psoas column in Appendix 2—table 1. While we have applied this approach to extend the results of Trombitás et al., 1998b to a rabbit psoas, in principle this approach can be applied to any isoform of titin provided that its geometry is known, and the Ig domains and PEVK residues in the target titin behave similarly to those in human soleus titin.
The only detail that remains is to establish the shape of the IgP, PEVK, and IgD forcelength curves. Studies of individual titin ﬁlaments, and of its segments, make it clear that titin is mechanically complex. For example, the tandem Ig segments (the IgD and IgP segments) are composed of many folded domains (titin from human soleus has two Ig segments that together have nearly 100 domains Trombitás et al., 1998b). Each domain appears to be a simple nonlinear spring until it unfolds and elongates by nearly 25 nm in the process (DuVall et al., 2013). Unfolding events appear to happen individually during lengthening experiments (DuVall et al., 2013), with each unfolding event occurring at a slightly greater tension than the last, giving an Ig segment a forcelength curve that is sawtoothed. Although detailed models of titin exist that can simulate the folding and unfolding of individual Ig domains, this level of detail comes at a cost of a state for each Ig domain which can add up to nearly a hundred extra states (SchappacherTilp et al., 2015) in total.
Active and passive lengthening experiments at the sarcomerelevel hide the complexity that is apparent when studying individual titin ﬁlaments. The experiments of Leonard et al., 2010 show that the sarcomeres in a ﬁlament (from rabbit psoas) mechanically fail when stretched passively to an average length of $2.86{\mathcal{l}}_{\text{o}}^{\text{M}}$, but can reach $3.38{\mathcal{l}}_{\text{o}}^{\text{M}}$ when actively lengthened. Leonard et al., 2010 showed that titin was the ﬁlament bearing these large forces since the sarcomeres were incapable of developing active or passive tension when the experiment was repeated after the titin ﬁlaments were chemically cut. It is worth noting that the forces measured by Leonard et al., 2010 contain none of the complex sawtooth pattern indicative of unfolding events even though 72 of these events would occur as each proximal and distal Ig domain fully unfolded and reached its maximal length (see Appendix 9, Note 15). Although we cannot be sure how many unfolding events occurred during the experiments of Leonard et al., 2010 due to sarcomere nonhomogeneity (Johnston et al., 2016), it is likely that many Ig unfolding events occurred since the average sarcomere length at failure $3.38{\mathcal{l}}_{\text{o}}^{\text{M}}$ was longer than the maximum length of 2.4–2.7 $\mathcal{l}}_{\text{o}}^{\text{M}$ that would be predicted from the geometry of rabbit psoas titin (see Appendix 9, Note 16).
Since we are interested in a computationally efficient model that is accurate at the whole muscle level, we model titin as a multisegmented nonlinear spring but omit the states needed to simulate the folding and unfolding of Ig domains. Simulations of active lengthening using our titin model will exhibit the enhanced force development that appears in experiments (Herzog and Leonard, 2002; Leonard et al., 2010), but will lack the nonlinear sawtooth forcelength proﬁle that is measured when individual titin ﬁlaments are lengthened (DuVall et al., 2013). To have the broadest possible application, we will ﬁt titin’s forcelength curves to provide reasonable results for both moderate (Herzog and Leonard, 2002) and large active stretches (Leonard et al., 2010). Depending on the application, it may be more appropriate to use a stiffer forcelength curve for the Ig segment if normalized sarcomere lengths stays within 1.5 $\mathcal{l}}_{\text{o}}^{\text{M}$ and no unfolding events occur as was done by Trombitás et al., 1998b.
To ensure that the serially connected forcelength curves of ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ closely reproduce $(1\text{P}){\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}})$, we are going to use affine transformations of $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ to describe ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$. The total stiffness of the halfsarcomere titin model is given by
which is formed by the series connection of ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$
Since each of titin’s segments is exposed to the same tension in the experiment of Trombitás et al., 1998b, the slopes of the lines that Equations 53–55 describe are directly proportional to the relative compliance (inverse of stiffness) between of each of titin’s segments. Using this fact, we can deﬁne the normalized stretch rates of the proximal titin segment
and the distal titin segment
which are proportional to the compliance of two titin segments in our model. When both the ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ curves are beyond the toe region the stiffness is a constant and is given by
and
Dividing Equation 63 by Equation 64 eliminates the unknown $\mathrm{\Delta}\tilde{f}$ and results in an expression that relates the ratio of the terminal linear stiffness of ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$
to the relative changes in Equation 61 and Equation 62. Substituting Equation 65, and Equation 60 into Equation 59 yields the expression
which can be simpliﬁed to
and this expression can be evaluated using the terminal stiffness of titin $\stackrel{~}{k}}^{\text{Ti*}$ and the coefficients listed in Appendix 2—table 1. Substituting Equation 67 into Equation 65 yields
The curves ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ can now be formed by scaling and shifting the total forcelength curve of titin $(1\text{P}){\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}$. By construction, titin’s forcelength curve develops a tension of $(1\text{P})$, and has reached its terminal stiffness, when the CE reaches a length $\stackrel{~}{\mathcal{l}}}^{\text{M}\ast$ such that ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}\ast})=1$. Using Equation 5355 and the appropriate coefficients in Appendix 2—table 1, we can evaluate the normalized length developed by the $\mathcal{l}}^{\text{1}$ segment
and $\mathcal{l}}^{\text{2}$ segment
at a CE length of $\stackrel{~}{\mathcal{l}}}^{\text{M}\ast$. The ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ curve is formed by shifting and scaling the $(1\text{P}){\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}$ curve so that it develops a normalized tension of $(1\text{P})$ and a stiffness of $\stackrel{~}{k}}_{\text{toe}}^{\text{1}$ at a length of $\stackrel{~}{\mathcal{l}}}_{\text{toe}}^{\text{1}$. Similarly, the ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ curve is made by shifting and scaling the $(1\text{P}){\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}}$ curve to develop a normalized tension of $(1\text{P})$ and a stiffness of $\stackrel{~}{k}}_{\text{toe}}^{\text{2}$ at a length of $\stackrel{~}{\mathcal{l}}}_{\text{toe}}^{\text{2}$.
By construction, the spring network formed by the ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$, ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$, and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ curves follows the ﬁtted $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ curve (Figure 3A) such that the ECM curve makes up 56% of the contribution. When the CE is active and $\stackrel{~}{\mathcal{l}}}^{\text{1}$ is effectively ﬁxed in place, the distal segment of titin contributes higher forces since $\stackrel{~}{\mathcal{l}}}^{\text{2}$ undergoes higher strains (Figure 3A). Finally, when the experiment of Trombitás et al., 1998b are simulated the movements of the IgP/PEVK and PEVK/IgD boundaries in the titin model closely follow the data (Figure 3C).
The process we have used to ﬁt the ECM and titin’s segments makes use of data within modest normalized CE lengths (2.35–4.46 μm, or 0.86–1.63 $\mathcal{l}}_{\text{o}}^{\text{M}$ Trombitás et al., 1998b). Scenarios in which the CE reaches extremely long lengths, such as during injury or during the experiment of Leonard et al., 2010, require ﬁtting titin’s forcelength curve beyond the typical ranges observed invivo. The WLC model has been used successfully to model the forcelength relation of individual titin segments (Trombitás et al., 1998b) at extreme lengths. In this work, we consider two different extensions to ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\text{2}}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$: a linear extrapolation, and the WLC model. Since the ﬁtted $\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{PE}$ curve is linearly extrapolated, so too are the ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$, ${\mathbf{f}}^{1}({\stackrel{~}{\mathcal{l}}}^{1})$, and ${\mathbf{f}}^{2}({\stackrel{~}{\mathcal{l}}}^{2})$ curves by default. Applying the WLC to our titin curves requires a bit more effort.
We have modiﬁed the WLC to include a slack length $\stackrel{~}{\text{L}}}_{\text{S}}^{\text{W}$ (the superscript $\text{W}$ means WLC) so that the WLC model can made to be continuous with ${\mathbf{f}}^{\text{1}}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}2}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}2})$. The normalized force developed by our WLC model is given by
where $\text{B}$ is a scaling factor and the normalized segment length $\stackrel{~}{\mathcal{l}}}^{\text{W}$ is deﬁned as
where $\stackrel{~}{\text{L}}}_{\text{S}}^{\text{W}$ is the slack length, and $\text{L}}_{\text{C}}^{\text{W}$ is the contour length of the segment. To extend the ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}1}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$ curve to follow the WLC model, we ﬁrst note the normalized contour length of the $\mathcal{l}}^{\text{1}$ segment
by counting the number of proximal Ig domains ($\text{N}}^{\text{IgP}$), the number of PEVK residues ($\text{QN}}^{\text{PEVK}$) associated with $\mathcal{l}}^{\text{1}$ and by scaling each by the maximum contour length of each Ig domain (25 nm DuVall et al., 2013), and each PEVK residue (between 0.32 Trombitás et al., 1998b and 0.38 nm Cantor and Schimmel, 1980 see pg. 254). This contour length deﬁnes the maximum length of the segment, when all of the Ig domains and PEVK residues have been unfolded. Similarly, the contour length of $\stackrel{~}{\text{L}}}_{\text{C}}^{2W$ is given by
Next, we deﬁne the slack length by linearly extrapolating backwards from the ﬁnal ﬁtted force $(1\text{P})$
and similarly
We can now solve for $\text{B}$ in Equation 71 so that ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}1}({\stackrel{~}{\mathcal{l}}}^{1})$ and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}2}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$ are continuous with each respective WLC extrapolation. However, we do not use the WLC model directly because it contains a singularity which is problematic during numerical simulation. Instead, we add an additional Bézier segment to ﬁt the WLC extension that spans between forces of $(1\text{P})$ and twice the normalized failure force ($2\times 5.14{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}$) noted by Leonard et al., 2010. To ﬁt the shape of the ﬁnal Bézier segment, we adjust the locations of the internal control points to minimize the squared differences between the modiﬁed WLC model and the ﬁnal Bézier curve (Figure 10A). The ﬁnal result is a set of curves (${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}1}({\stackrel{~}{\mathcal{l}}}^{\text{1}})$,${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}2}({\stackrel{~}{\mathcal{l}}}^{\text{2}})$, and ${\mathbf{f}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}}({\stackrel{~}{\mathcal{l}}}^{\phantom{\rule{thinmathspace}{0ex}}\text{ECM}})$) which, between forces 0 and $(1\text{P})$, will reproduce $\mathbf{f}}^{\text{}\text{PE}$, the measurements of Trombitás et al., 1998b, and do so with titinECM balance similar to the measurements of Prado et al., 2005. For forces beyond $(1\text{P})$, the curve will follow the segmentspeciﬁc WLC model up to twice the expected failure tension noted by Leonard et al., 2010.
2.5. Fitting the XE’s impedance
With the passive curves established, we can return to the problem of identifying the normalized maximum stiffness $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ and damping $\stackrel{~}{\beta}}_{\text{o}}^{\text{X}$ of the lumped XE element. Just prior to discussing titin, we had evaluated the impedance of the cat soleus CE from Figure 12 of Kirsch et al., 1994 to be ${k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}=$ 2.90 N/mm and ${\beta}^{\text{M}}=$ 0.020 Ns/mm at a nominal active tension of 5 N. The normalized stiffness $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ can be found by taking the partial derivative of Equation 15 with respect to $\stackrel{~}{\mathcal{l}}}^{\text{M}$
By noting that all of our chosen state variables in Equation 13 are independent and by making use of the kinematic relationships in Equation 9 and Equation 10 we can reduce Equation 77 to
and solve for $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$
When using to the data from Figure 12 in Kirsch et al., 1994, we end up with $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=49.1{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ for the elastictendon model, and $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=41.8{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ for the rigidtendon model. When this procedure is repeated for Figure 3 of Kirsch et al., 1994 (from a different specimen), we are left with $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=74.5{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ for the elastictendon model and $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=59.1{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/{\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ for the rigidtendon model. The value for $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ is much larger than $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ because the $a$ needed to generate 5 N is only 0.231. Similarly, we can form the expression for the normalized damping of the CE by taking the partial derivative of Equation 15 with respect to $\stackrel{~}{v}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$
As with $k}^{\text{}\text{M}$, the expression for $\stackrel{~}{\beta}}^{\text{M}$ can be reduced to
which evaluates to ${\stackrel{~}{\beta}}_{\text{o}}^{\text{X}}=0.347{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/({\mathcal{l}}_{\text{o}}^{\text{M}}/s)$ for both the elastic and rigidtendon models using the data from Figure 12 of Kirsch et al., 1994. The damping coefficients of the elastic and rigidtendon models is similar because the damping coefficient of the musculotendon is dominated by the damping coefficient of CE. When the data from Figure 3 of Kirsch et al., 1994 is used, the damping coefficients of the elastic and rigidtendon models are ${\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=0.155{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/({\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/s)$ and ${\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}}=0.153{f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/({\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}}/s)$ respectively.
The dimensionless parameters $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ and $\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ can be used to approximate the properties of other MTUs given $f}_{\text{o}}^{\text{}\text{M}$ and $\mathcal{l}}_{\text{o}}^{\text{M}$. The stiffness and damping of the lumped crossbridge element will scale linearly with $f}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ and inversely with $\mathcal{l}}_{\text{o}}^{\text{M}$ provided the impedance properties of individual crossbridges, and the maximum number of crossbridges attached per sarcomere, is similar between a feline’s skeletal muscle sarcomeres and those of the target MTU. This approximation is rough, however, since the values for $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ and $\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ (Appendix 2—table 2) have a relative error of 41% and 76% when fit to Figure 3 and Figure 12 from Kirsch et al., 1994. In addition, when simulated, the stiffness and damping of the LTI system of best ﬁt may differ from $\stackrel{~}{k}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ and $\stackrel{~}{\beta}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{X}$ at low frequencies because the movement of the attachment point has been ignored in Equation 79 and Equation 81. This approximation explains why the VEXAT’s stiffness proﬁle (Figure 7A, C) is below the data of Kirsch et al., 1994, despite having used this data to ﬁt the $k}^{\text{}\text{M}$ and $\stackrel{~}{\beta}}^{\text{M}$ terms in Equation 79 and Equation 81. The accuracy of this approximation, however, improves at higher frequencies (Figure 6E and F) because the attachment point’s movements become increasingly limited due to the time constant $k}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ in Equation 16. Unfortunately, this is a tradeoff due to the formulation of Equation 16: the VEXAT model can ﬁt the data of Kirsch et al., 1994 at low frequencies, or high frequencies, but not both simultaneously.
Appendix 3
Model initialization
Solving for an initial state is challenging since we are given $a$, $\mathcal{l}}^{\text{P}$, and $v}^{\text{P}$ and must solve for $v}^{\text{S}$, $\mathcal{l}}^{\text{S}$, and $\mathcal{l}}^{\text{1}$ for a rigidtendon model, and additionally $\mathcal{l}}^{\text{M}$ if an elastictendon model is used. The type of solution that we look for is one that produces no force or state transients soon after a simulation begins in which activation and path velocity is well approximated as constant. Our preliminary simulations found that satisfactory solutions were found by iterating over both $\stackrel{~}{\mathcal{l}}}^{\text{M}$ and $\stackrel{~}{v}}^{\text{M}$ using a nested bisection search that looks for values which approximately satisﬁes Equation 22, result in small values for $\dot{\stackrel{~}{v}}}^{\text{S}$ from Equation 16, and begins with balanced forces between the two segment titin model in Equation 20.
In the outer loop, we iterate over values of $\stackrel{~}{\mathcal{l}}}^{\text{M}$. Given $a$, $\mathcal{l}}^{\text{P}$, $v}^{\text{P}$, and a candidate value of $\stackrel{~}{\mathcal{l}}}^{\text{M}$, we can immediately solve for $\alpha$ and $\mathcal{l}}^{\text{T}$ using the pennation model. We can numerically solve for the value of another state, $\mathcal{l}}^{\text{1}$, using the kinematic relationship between $\mathcal{l}}^{\text{M}$ and $\mathcal{l}}^{\text{1}$ and by assuming that the two titin segments are in a force equilibrium
In the inner loop, we iterate over values of $\stackrel{~}{v}}^{\text{M}$ between $0$ and ${v}^{\text{P}}\mathrm{cos}\alpha$ (we ignore solutions in which the sign of $v}^{\text{M}$ and $v}^{\text{T}$ differ) to ﬁnd the value of $\stackrel{~}{v}}^{\text{M}$ that best satisﬁes Equation 22. Prior to evaluating Equation 22, we need to set both $\stackrel{~}{v}}^{\text{X}$ and $\stackrel{~}{\mathcal{l}}}^{\text{X}$. Here we choose a value for $\stackrel{~}{v}}^{\text{X}$ that will ensure that the XE is not producing transient forces
and we use ﬁxedpoint iteration to solve for $\stackrel{~}{\mathcal{l}}}^{\text{X}$ such that Equation 16 evaluates to zero. Now the value of $\stackrel{~}{v}}^{\text{S}$ can be directly evaluated using the candidate value of $\stackrel{~}{v}}^{\text{M}$, the ﬁrst derivative of Equation 9, and the fact that we have set $\stackrel{~}{v}}^{\text{X}$ to zero. Finally, the error of this speciﬁc combination of $\stackrel{~}{\mathcal{l}}}^{\text{M}$ and $\stackrel{~}{v}}^{\text{M}$ is evaluated using Equation 22, where the best solution leads to the lowest absolute value for of $\stackrel{~}{f}}^{\phantom{\rule{thinmathspace}{0ex}}\u03f5$ in Equation 22. If a rigidtendon model is being initialized the procedure is simpler because the inner loop iterating over $\stackrel{~}{v}}^{\text{M}$ is unnecessary: given that $v}^{\text{P}$ and $\stackrel{~}{v}}^{\text{X}$ are zero, the velocities $\stackrel{~}{v}}^{\text{M}$ and $\stackrel{~}{v}}^{\text{S}$ can be directly solved using the ﬁrst derivative of Equation 9. While in principle any root solving method can be used to solve this problem, we have chosen to use the bisection method to avoid local minima.
Appendix 4
Evaluating a muscle model’s frequency response
To analyze the the frequency response of a muscle to length perturbation we begin by evaluating the length change
and force change
with respect to the nominal length ($\hat{\mathcal{l}}}^{\text{MT}$) and nominal force ($\hat{f}}^{\text{T}$). If we approximate the muscle’s response as a linear time invariant transformation $h(t)$ we can express
where ∗ is the convolution operator. Each of these signals can be transformed into the frequencydomain (Oppenheim et al., 1996) by taking the Fourier transform $\mathcal{F}(\cdot )$ of the timedomain signal, which produces a complex (with real and imaginary parts) signal. Since convolution in the timedomain corresponds to multiplication in the frequencydomain, we have
In Equation 87, we are interested in solving for $H(s)$. While it might be tempting to evaluate $H(s)$ as
the result will poorly estimate $H(s)$ because $Y(s)$ is only approximated by $H(s)X(s)$: $Y(s)$ may contain nonlinearities, nonstationary signals, and noise that cannot be described by $H(s)X(s)$.
Using crossspectral densities, Koopmans, 1995 (pg. 140) derived the estimator
that minimizes the squared errors between $Y(s)$ and its linear approximation of $H(s)X(s)$. The crossspectral density $G}_{xy$ between $x(t)$ and $y(t)$ is given by
the Fourier transform of the crosscorrelation $(\star )$ between $x(t)$ and $y(t)$. When the order of $x(t)$ and $y(t)$ are reversed in Equation 90 the result is $G}_{yx$, while $G}_{xx$ and $G}_{yy$ are produced by taking the Fourier transform of $x(t)\star x(t)$ and $y(t)\star y(t)$, respectively.
Although the estimator of Koopmans, 1995 is a great improvement over Equation 88, the accuracy of the estimate can be further improved using Welch’s method (Welch, 1967). Welch’s method (Welch, 1967) breaks up the time domain signal into $\text{K}$ segments, transforms each segment into the frequency domain, and returns the average across all segments. Using Welch’s method (Welch, 1967) with $\text{K}$ segments allows us to evaluate
which has a lower frequency resolution than Equation 89, but an improved accuracy in $H(s)$. Now we can evaluate the gain of $H(s)$ as
while the phase of $H(s)$ is given by
where $\mathbb{R}(H(s))$ and $\mathbb{I}(H(s))$ are the real and imaginary parts of $H(s)$, respectively.
The transfer function estimated in Equation 91 is meaningful only when $y(t)$ can be approximated as a linear timeinvariant function of $x(t)$. By evaluating the coherence (Koopmans, 1995) (pg. 137) between $x(t)$ and $y(t)$
we can determine the strength of the linear association between $X(s)$ and $Y(s)$ at each frequency. When ${C}_{xy}$ is close to 1 it means that $Y(s)$ is well approximated by $H(s)X(s)$. As $C}_{xy$ approaches 0, it means that the approximation of $Y(s)$ by $H(s)X(s)$ becomes poor.
Kirsch et al., 1994 analyzed a bandwidth that spanned from 4 Hz up to the cutoff frequency of the lowpass ﬁlter applied to the input signal $x(t)$ (15 Hz, 35 Hz, and 90 Hz). Unfortunately, we cannot use this bandwidth directly when analyzing model output because we have no guarantee that the simulated output is sufficiently linear in this range. Instead, to strike a balance between accuracy and consistency with Kirsch et al., 1994, we analyze the bandwidth that is common to the deﬁned range of Kirsch et al., 1994 and has the minimum acceptable $({C}_{xy}{)}^{2}$ of 0.67 that is pictured in Kirsch et al., 1994.
Appendix 5
Simulation summary data of Kirsch et al., 1994
Appendix 6
Supplementary plots: Gain and phase response rigidtendon muscle models
Appendix 7
Supplementary plots: active lengthening on the descending limb
Appendix 8
Rabbit psoas ﬁbril model parameters
Appendix 9
Article footnotes
Note 1
Small in the context of an LTI system is larger than the shortrange as defined by Rack and Westbury, 1974: the response of an LTI system can include both length and velocity dependence, while the shortrange of Rack and Westbury, 1974 ends where velocity dependence begins.
Note 2
A Matlab implementation of the model and all simulated experiments are available from https://github.com/mjhmilla/Millard2023VexatMuscle under the branch elife2023. All of the code is available under the APACHE2 and MIT software licenses as indicated in the repository.
Note 3
The term rheological is used because the model includes a component that deforms with plastic flow in response to an applied force.
Note 4
A change of ±4mm to a typical cat soleus with an $\mathcal{l}}_{\text{o}}^{\phantom{\rule{thinmathspace}{0ex}}\text{M}$ = 41.7 ± 1.3mm (Sacks and Roy, 1982).
Note 5
8 − 20 mm∕s ($v}_{\text{m}ax}^{\text{M}$) for a muscle with a maximum shortening velocity of 180 mm∕s (Forcinito et al., 1998).
Note 6
Although activation normally refers to the presence of Ca^{2+} ions in the sarcomere, Ca^{2+} ions alone are insufficient to cause titin to develop enhanced lengthening forces. In addition, crossbridge attachment appears to be necessary: when crossbridge attachment is inhibited titin is not able to develop enhanced forces in the presence of Ca^{2+} during lengthening (Leonard et al., 2010).
Note 7
Which means that the second derivative of the curve is continuous.
Note 8
For readers who require an activation model with continuity to the secondderivative, the model of De Groote et al., 2016 is recommended.
Note 9
Note that we have used the symbols D, and not β, because the D terms damp the acceleration of actinmyosin movement and as such cannot be interpreted as a viscous damping term. In contrast, viscous damping terms are indicated using the β symbol.
Note 10
Physically this assumption is equivalent to treating the CE and the tendon as massless. In general, this assumption is quite reasonable since a cubic centimeter of muscle has a mass of roughly 1.0 g but can generate tensions of between 35137 N (Buchanan, 1995). With such a low mass and a high maximum isometric force, the cubic centimeter of muscle would have to be accelerated at an incredible 350013,700 m/s^{2} before the inertial forces would be within 10% of the maximum isometric tension. Since everyday movements require comparatively tiny accelerations, ignoring inertial forces of muscle results in relatively small errors.
Note 11
The impedance ($z$) of two serially connected components ($z}_{1$ and $z}_{2$) is given by $1/z=1/{z}_{1}+1/{z}_{2}$, or $z=({z}_{1}{z}_{2})({z}_{1}+{z}_{2})$.
Note 12
Kirsch et al., 1994 note on page 765 a VAF of 8899% for the medial gastrocnemius, and 810% lower for the soleus.
Note 13
For brevity we will refer to the 3 dB frequency of the perturbation waveform rather than the entire bandwidth.
Note 14
Figure 8 of Prado et al., 2005 shows titin’s contribution ranging from values ranging from (24%57%) which means that the ECM’s contribution ranges from (43%76%).
Note 15
Referred to as contour lengths in a wormlike chain model (Trombitás et al., 1998b).
Note 16
Rabbit psoas titin (Prado et al., 2005) attaches at the Zline with a 100nm rigid segment that spans to T12 epitope, is followed by 50 Ig domains, 800 PEVK residues, and another 22 Ig domains until it attaches to the 800 nm halfmyosin filament which can also be considered rigid. If the Ig domains were all unfolded (adding around 25 nm; DuVall et al., 2013) and each PEVK residue could reach a maximum length of between 0.32nm (see Figure 5 of Trombitás et al., 1998a where the PEVK segment reaches a length of 700nm which is a strain of 0.32 nm for each of the 2174 residues) to 0.38 nm (Cantor and Schimmel, 1980, pg. 254), two titins in series would reach a length of 2(100nm + 72(25nm) + 800(0.32nm0.38nm) + 800 nm) = 51926008nm. Since rabbit sarcomeres have an $\mathcal{l}}_{\mathrm{o}}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{M}$ of 2.2 μm a sarcomere could be stretched to a length between 51926008nm, or 2.42.7 $\mathcal{l}}_{\mathrm{o}}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{M}$, before the contour lengths of the tandem Ig and PEVK segments is reached.
Data availability
All of the simulation results in this study can be generated using the code available from the elife2023 branch of https://github.com/mjhmilla/Millard2023VexatMuscle (copy archived at Millard, 2024). All of the code is publicly available either under the APACHE2 or MIT licenses as indicated in the file header and also by the licensing auditing tool https://api.reuse.software/. The code repository also includes a selection of manually digitized data sets from past papers in the 'experiment' folder, as well as raw experimental data in the 'HerzogLeonard2002' folder that WH has made publicly available.
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Article and author information
Author details
Funding
Deutsche Forschungsgemeinschaft (EXC 2075  390740016)
 Matthew Millard
Deutsche Forschungsgemeinschaft (MI 2109/11)
 Matthew Millard
Bayerische Staatsministerium für Wirtschaft, Landesentwicklung und Energie (Project X, grant no. 5140951)
 David W Franklin
Natural Sciences and Engineering Research Council of Canada (RGPIN202003920)
 Walter Herzog
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
Financial support is gratefully acknowledged from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC 2075 – 390740016) through the Stuttgart Center for Simulation Science (SimTech), from DFG grant no. MI 2109/1–1, the Lighthouse Initiative Geriatronics by StMWi Bayern (Project X, grant no. 5140951), and the Natural Sciences and Engineering Research Council of Canada (RGPIN2020–03920).
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© 2023, Millard et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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