Muscle wobbling mass dynamics: eigenfrequency dependencies on activity, impact strength, and ground material

In legged locomotion, muscles undergo damped oscillations in response to the leg contacting the ground (an impact). How muscle oscillates varies depending on the impact situation. We used a custom-made frame in which we clamped an isolated rat muscle (M. gastrocnemius medialis and lateralis: GAS) and dropped it from three different heights and onto two different ground materials. In fully activated GAS, the dominant eigenfrequencies were 163 Hz, 265 Hz, and 399 Hz, which were signficantly higher (p < 0.05) compared to the dominant eigenfrequencies in passive GAS: 139 Hz, 215 Hz, and 286 Hz. In general, neither changing the falling height nor ground material led to any significant eigenfrequency changes in active nor passive GAS, respectively. To trace the eigenfrequency values back to GAS stiffness values, we developed a 3DoF model. The model-predicted GAS muscle eigenfrequencies matched well with the experimental values and deviated by − 3.8%, 9.0%, and 4.3% from the passive GAS eigenfrequencies and by − 1.8%, 13.3%, and − 1.5% from the active GAS eigenfrequencies. Differences between the frequencies found for active and passive muscle impact situations are dominantly due to the attachment of myosin heads to actin.


Supplementary Text S1
Eigenfrequencies (3dof -fixated proximally and distally) Equations of motion: Equations of motion (matrix): We start with introducing some symbols and expressions that will be useful later.
The symbol ω represents an angular eigenfrequency of the clamped 3DoF model oscillating, and we abbreviate λ = ω 2 .The characteristic equation determining the angular eigenfrequencies of the 3DoF model, with outer mass and stiffness symmetry, but central parameters different from peripheral ones (m C = m S , k C = k S ; physiologically: k C > k S ), requires the determinant of the (three equations of motion coupled in one dimension) to vanish: Applying the rule of Sarrus, this yields a cubic equation in terms of λ, which writes as a the product of a term linear in λ and another quadratic one in λ.
The product to vanish is fulfilled by either which yields the square of one angular eigenfrequency here eventually given in terms of the basic model parameters m S , m C , k S , and k C .The remaining two angular eigenfrequencies are determined by otherwise the second factor of the left hand side of Eq.S9 vanishing, i.e. by solving the corresponding quadratic equation for λ, which eventually yields If full mass symmetry is assumed (m S = m C = m), the squares of the angular eigenfrequencies according to Eq. S11 and Eq.S13, respectively, write Correspondingly, the eigenfrequencies are That only the k SSC stiffness is needed to determine freq 2 (Eq.S16) is because of the distalproximal symmetry (Eq.S8), which makes the corresponding eigenvector (V 2 ) of x S,p , x C , and x S,d (Eq.S5) for freq 2 for λ 2 = k SSC m S (Eq.S11) equal to: with Similarly, the eigenvectors for λ 1,3 (Gaussian elimination); and when using m S =m C =0.00063 kg (Eq.7 in main text) and either passive or active values for 3 in main text), and λ 1,3 (Eq.S13).

Supplementary Text S2
Estimating spring stiffnesses values from eigenfrequencies If instead ω = ω 2 1 = freq 1 • 2 • π is used to to estimate the local stiffness of the suspended 3DoF system, and not the other way around, then by rewriting Eq.S13 where m S,p = m S,p = m S = m C (case1): From this (Eq.S27), the inferred spring stiffness k SC = k SC2 2 (using the quadratic formula) is equal to: where Herein, the spring stiffness k SSC for case1, using Eq.S16 (for ω 2 2 ), is In the simpler 3DoF model variation, where m S,p = m S,d = m C = m GAS 3 = m (case2), the inferred stiffnesses k SC and k SSC (Eq.S28 and Eq.S30, respectively) then are where and respectively.

Supplementary Text S3
Stiffness comparisons to literature In a previous work that examined the wobbling characteristics of a muscle 1,2 , the stiffness of the tendon-aponeurosis-complex (k T AC ), was inferred assuming the muscle was a simple spring mass system that is fixated proximally and has a lumped mass attached distally: With both k M T C and k CE (Eq.S34) calculated using the dynamic force change in response to TD (∆F = m GAS • a COM ) and either the measured displacement of the COM (∆L M T C ) or the fibre material elongation in response to TD, respectively, k T AC is equal to Here, however, the GAS is instead treated as a 3DoF symmetrical spring-mass system with both ends of the springs fixated, and because of the distal-proximal symmetry in our 3DoF model (see 'GAS 3DoF model: spring stiffness' and 'GAS 3DoF model: mass' in main text), the displacement of x C (Eq. S2) is assuming that, firstly, m S,p = m C = m S,d (Eq.7 in main text), secondly, that the displacement of the first mass is half of the second mass, and, thirdly, that ∆L M T C = x C is the same as in Eq.S35, because it is a measured quantity for the centre of mass displacement after TD.Accordingly, By comparing Eq.S34 and Eq.S39, and remembering that L CE (used to calculate k CE ) spans across GAS' COM, then  ).As a result thereof, the k T AC (Supplementary Eq.S34) changes (dashed black lines), whereas k M T C remains constant (dashed, black lines).The right y-axis gives the 3DoF estimated eigenfrequencies in GAS: freq 1 (dotted, grey lines), freq 2 (solid, grey lines), and freq 3 (dashed, grey lines) illustrate the dependencies of the eigenfrequencies on the experimentally determined stiffness values (Supplementary Eq.S16) due to the lumped, regional stiffnesses k CE and k T AC (in accordance with the 1DoF model used for extracting these values from our experiments 1,2 ) changing with the fibrematerial-only length L CE in various GAS specimens, while assuming full mass symmetry m = m GAS 3 and a given value of Young's modulus of CE, 1.3 MPa 2 .The "X" values are from Supplementary Table S4.

− ω 2 •
[m] = 0 .(S6) Deduction of eigenfequencies of 3DoF model, assuming outer mass symmetry m S,p = m S,d = m S yet inequality of central mass m C = m S ; the same for the stiffnesses k C = k S ; see Fig. 1 in manuscript

1
|GAS stiffnesses and eigenfrequencies as a function of the fibre length region.Depending on the lines they touch, the "X" symbols are the k CE , k T AC and k M T C in either active (a) or passive (b) GAS.Here, the fibre material stiffness k CE in both passive and active muscle (dotted, black lines) scales linearly with the change in fibre region length (k CE • L CE L CE,0