There is no experimental evidence for non-linear myofilament elasticity in skeletal muscle

Information on the stiffness of the actin and myosin filaments is a key parameter in the study of muscle contraction. For example, the number of motors attached to actin in each half-sarcomere of a muscle fibre can be obtained from measurements of the half-sarcomere stiffness, provided that the compliances of the actin and myosin filaments are known (Piazzesi et al., 2007). The stress—strain relationship in myofilaments has generally been considered to be linear (Kojima et al., 1994; Wakabayashi et al., 1994; Reconditi et al., 2004; Brunello et al., 2006), and data suggesting that the myofilament stiffness can vary with force could also find alternative explanations (Huxley et al., 1994; Higuchi et al., 1995).

The argument has been recently reinvestigated by Edman (Edman, 2009), who concluded that myofilament stiffness exhibits a strong non-Hookean behaviour. In Edman's paper (Edman, 2009), stiffness is measured by the change in force in response to 2-4 kHz length oscillations imposed on intact fibres isolated from the tibialis anterior muscle of Rana temporaria (2-2.5°C). Measurements are performed at two different sarcomere lengths (sl) in isometric contraction and during isotonic shortening at different pre-set loads. The changes in stiffness with sarcomere length for different loads are estimated from the ratio of the fibre stiffness at sl=2.6 μm (S_2.6) to that at sl=2.2 μm (S_2.2; the condition for full overlap between actin and myosin filaments). The pre-set loads are in the range of 0.4-0.7 the isometric tetanic force (T₀) at sl=2.2 μm. The elements considered to contribute to the half-sarcomere stiffness are the array of actin-attached cross-bridges and the portions of both myosin and actin filaments beyond the overlap region. The stiffness of the cross-bridge array is proportional to the number of attached cross-bridges, which depends solely and linearly on the overall level of force, independent of how force is modulated, either with a different degree of overlap (Gordon et al., 1966) or with isotonic shortening at different loads. This latter assumption has been recently found valid in the force range 0.5-1T₀ (Piazzesi et al., 2007).

The contribution of myofilaments to the half-sarcomere compliance is assumed proportional to the length of their non-overlapping regions. The compliance per unit length is considered the same for both actin and myosin. This is an arbitrary assumption but is acceptable as a first approximation for the purpose of determining whether or not the myofilament stiffness is Hookean. Other acceptable approximations are that the changes in the length of the fibre are used for estimating the changes in the half-sarcomere length and that a frequency of length oscillations in the range of 2-4 kHz is assumed sufficiently high to estimate the instantaneous stiffness, ignoring the effect of quick force recovery.

The conclusion that the myofilaments are non-Hookean is based on the results of the linear fit on the S_2.6/S_2.2—T data. In my reproduction of data from fig. 3 of Edman (Edman, 2009), the linear regression gives a best fit with m (the slope)=0.066±0.109 (best estimate ± s.e.m.) and q (the ordinate intercept)=0.861±0.067; R²=0.03. The mean of the squared residuals is 1.4×10⁻³. This result is taken by Edman as evidence of a strong unlinearity of the stress—strain relation of the myofilaments (Edman, 2009).

Equation 1 in Edman (Edman, 2009) can be rewritten as S_2.6/S_2.2=(T+S/F₀)/(T×1.61/0.81+S/F₀), where S is the stiffness of the myofilaments at sl=2.2 μm and F₀ is the stiffness of the cross-bridges for T=T₀.

Under the hypothesis that S/F₀ is independent of force, the above equation represents an hyperbola in the variables S_2.6/S_2.2 and T. The best hyperbolic fit to the S_2.6/S_2.2—T data gives a S/F₀ estimate of 5.24±0.74. The mean of the squared residuals is 1.7×10⁻³, which is practically the same as that obtained with the linear fit. In the range of forces used, the hyperbolic fit is very close to a straight line, as demonstrated by the linear fit through the values predicted by the hyperbolic fit that gives S_2.6/S_2.2=0.984-0.127T (R²=0.999), and lies well inside the 95% confidence limits of the linear fit to data, i.e. the bounds of the area that has a 95% chance of containing the true regression line (see Fig. 1). Thus, when uncertainties are taken into account, Edman's data are consistent with a linear stress—strain relation of myofilaments.

Edman reports a value of ~5 for S/F₀, the ratio of myofilament stiffness over the cross-bridge array stiffness, much higher than previous estimates, which are close to 1 (Huxley et al. 1994; Wakabayashi et al. 1994; Reconditi et al. 2004). This discrepancy is the consequence of having neglected the contribution to the half-sarcomere stiffness of the myofilaments in the overlap region. When S is not much smaller than F₀, the half-sarcomere compliance (C_hs) can be expressed with good approximation by the following equation: C_hs=c_A×(l_A−2/3ξ)+c_M×(l_M−2/3ξ)+1/fF₀, where c_A and c_M are the compliance per unit length of actin and myosin filament, respectively; l_A and l_M are the length of the actin and myosin filament in the half-sarcomere, respectively; ξ is the length of the overlap region; f is the fraction of cross-bridges attached to actin, relative to the fraction attached at T₀ [appendix A in Ford et al. (Ford et al., 1981)]. If, as in Edman (Edman, 2009), it is assumed that c_A=c_M=c, the contribution of myofilaments compliance C_f to C_hs can be expressed as: C_f=c×(l_A+l_M−4/3ξ)=c×(2×sl−(l_A+l_M))/3. With l_A=0.970 μm and l_M=0.775 μm (Edman, 2009), the contribution of the myofilaments to C_hs at 2.6 μm is 1.152/0.885 that at 2.2 μm. Thus, Edman's eqn 1 (Edman, 2009) can be rewritten as: S_2.6/S_2.2=(T+S/F₀)/(T×1.152/ 0.885+S/F₀).

The fit of the S_2.6/S_2.2—T plot with the above equation, in the hypothesis of S/F₀ independent of T, gives S/F₀=1.15±0.22, which is close to the values reported by the previous works quoted above.