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 linear-time-invariant (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 of these mechanical properties. In this work, we present the viscoelastic-crossbridge active-titin (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 Hill-type muscle model. The VEXAT model more accurately captures the impedance of biological muscle and its responses to long active stretches than a Hill-type model and can still reproduce the force-velocity and force-length 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.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
A number of spelling and typesetting errors have been corrected: 1. Spelling: changed 'de Groote' to 'De Groote' 2. Spelling: changed 'Kirch' to 'Kirsch' 3. Spelling: changed 'et al' to 'et al.' 4. Spelling: changed 'twich' to 'twitch' 5. Math typesetting: Put '<' in the math environment to correctly typeset a greater than symbol rather than an inverted question mark (affects Tables 4D and 5D)
1 Small in the context of an LTI system is larger than the short-range of Rack and Westbury’s [6] short-range-stiffness: the response of an LTI system can include both length and velocity dependence, while Rack and Westbury’s [6] short-range ends where velocity dependence begins.
2 A Matlab implementation of the model and all simulated experiments are available from https://github.com/mjhmilla/Millard2023VexatMuscle under the branch elife2023.
3 The term rheological is used because the model includes a component that deforms with plastic flow in response to an applied force.
4 a change of ±4 mm to a typical cat soleus with an [19]
5 for a muscle with a maximum shortening velocity of 180 mm/s [18]
6 Although activation normally refers to the presence of Ca2+ ions in the sarcomere, Ca2+ 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 Ca2+ during lengthening [8].
7 Which means that the second derivative of the curve is continuous.
8 For readers who require an activation model with continuity to the second-derivative, the model of De Groote et al. [64] is recommended.
9 Note that we have used the symbols D, and not β, because the D terms damp the acceleration of actin-myosin movement and as such cannot be interpreted as a viscous damping term. In contrast, viscous damping terms are indicated using the β symbol.
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 35-137 N [71]. With such a low mass and a high maximum isometric force, the cubic centimeter of muscle would have to be accelerated at an incredible 3,500-13,700 m/s2 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.
11 The impedance (z) of two serially connected components (z1 and z2) is given by 1/z = 1/z1 + 1/z2, or z = (z1 z2)/(z1 + z2)
12 Kirsch et al. [5] note on page 765 a VAF of 88-99% for the medial gastrocnemius, and 8-10% lower for the soleus.
13 For brevity we will refer to the -3 dB frequency of the perturbation waveform rather than the entire bandwidth
14 See the elife2023 branch of https://github.com/mjhmilla/Millard2021ImpedanceMuscle
15 See main_ActinMyosinAndTitinStiffness.m in the elife2023 branch of accompanying code repository for details.
16 Figure 8 of Prado et al. [58] shows titin’s contribution ranging from values ranging from (24%-57%) which means that the ECM’s contribution ranges from (43%-76%)
17 Referred to as contour lengths in a worm-like chain model [65]
18 Rabbit psoas titin [58] attaches at the Z-line 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 half-myosin filament which can also be considered rigid. If the Ig domains were all unfolded (adding around 25 nm [88]) and each PEVK residue could reach a maximum length of between 0.32nm [65] (see Fig. 5: 700nm/2174 residues is 0.32 nm per residue) to 0.38 nm [90] (see pg. 254), two titins in series would reach a length of 2(100nm + 72(25nm) + 800(0.32nm-0.38nm) + 800 nm) = 5192-6008nm. Since rabbit sarcomeres have an of 2.2µm a sarcomere could be stretched to a length between 5192-6008nm, or , before the contour lengths of the tandem Ig and PEVK segments is reached.