Crossbridge and filament compliance in muscle: implications for tension generation and lever arm swing

Abstract

The stiffness of myosin heads attached to actin is a crucial parameter in determining the kinetics and mechanics of the crossbridge cycle. It has been claimed that the stiffness of myosin heads in the anterior tibialis muscle of the common frog (Rana temporaria) is as high as 3.3 pN/nm, substantially higher than its value in rabbit muscle (~1.7 pN/nm). However, the crossbridge stiffness measurement has a large error since the contribution of crossbridges to half-sarcomere compliance is obtained by subtracting from the half-sarcomere compliance the contributions of the thick and thin filaments, each with a substantial error. Calculation of its value for isometric contraction also depends on the fraction of heads that are attached, for which there is no consensus. Surprisingly, the stiffness of the myosin head from the edible frog, Rana esculenta, determined in the same manner, is only 60% of that in Rana temporaria. In our view it is unlikely that the value of such a crucial parameter could differ so substantially between two frog species. Since the means of the myosin head stiffness in these two species are not significantly different, we suggest that the best estimate of the stiffness of the myosin heads for frog muscle is the average of these data, a value similar to that for rabbit muscle. This would allow both frog and rabbit muscles to operate the same low-cooperativity mechanism for the crossbridge cycle with only one or two tension-generating steps. We review evidence that much of the compliance of the myosin head is located in the pliant region where the lever arm emerges from the converter and propose that tension generation (“tensing”) caused by the rotation and movement of the converter is a separate event from the passive swinging of the lever arm in its working stroke in which the strain energy stored in the pliant region is used to do work.

Appendices

Appendix 1: Density of myosin heads in myofibrils and fibres

A cross-section through a myofibril joining the centres of three neighbouring thick filaments at the vertices of an equilateral triangle has an area of ${\frac{\sqrt 3 }{4}}d^{2}$, where d nm is the centre-to-centre spacing between the thick filaments. The cross-section includes half a thick filament and one thin filament. So there are ${\frac{{2.10^{6} }}{{\sqrt 3 d^{2} }}}$ thick filaments per μm² cross-sectional area of myofibrils. In each half-thick filament there are 49 crowns of heads, each comprising 3 myosin molecules and therefore 6 myosin heads. So there are 294 myosin heads per half-thick filament. Thus there are ${\frac{{2.294.10^{6} }}{{\sqrt 3 d^{2} }}}$ myosin heads in a half sarcomere per μm² cross-sectional area of myofibrils.

For frog muscle at s = 2.1 μm, d = 43 (Matsubara and Elliott 1972). Hence there are 1.84 × 10⁵ myosin heads in a half sarcomere per μm² of myofibrils. In this fast muscle, the myofibrils occupy a fraction 0.83 of the cross-sectional area (Mobley and Eisenberg 1975), so the total number of myosin heads in a half-sarcomere per μm² cross-sectional area of fibre is 1.53 × 10⁵.

For skinned relaxed rabbit psoas fibres at s = 2.3–2.4 μm d _1,0 = 42 nm (Brenner and Yu 1991) and hence d = 48 nm. Hence assuming that the myofibrils in this muscle also occupy a fraction 0.83 of the cross-sectional area of a fibre, the number of myosin heads in a half-sarcomere per μm² cross-sectional area of fibre is 1.22 × 10⁵. In human soleus muscle, because of the high numbers of mitochondria, the myofibrils occupy perhaps only a fraction ~0.5 of the cross-sectional area of the fibre, so the number of myosin heads in a half-sarcomere per μm² cross-sectional area of fibre would be lower, ~0.74 × 10⁵.

Appendix 2: Temperature dependence of the standard Gibbs (free) energy of the tension-generating step

Decostre et al. (2005) and Linari et al. (2007) have claimed that the increased fall in standard Gibbs (free) energy for the tension-generating step for a rise in absolute temperature from t ₁ to t ₂, $\Updelta G_{{t_{2} - t_{1} }}$, is equal to the increase in mechanical energy stored in the crossbridge compliant element and hence consider it gives information on the crossbridge stiffness. We think this is incorrect. To demonstrate this we shall consider for simplicity the case for a single tension-generating step and, for this purpose only, follow Decostre et al. in assuming that this step is in equilibrium. We shall also for simplicity ignore filament compliance.

Consider the tension-generating step in muscle where the heads are tethered to the thick filament backbone. We shall suppose for simplicity that before this step heads have zero strain, but that after they have executed this step, they have strain l, the stroke distance. Then when the tension-generating step occurs, work will be done on the crossbridge compliant element equal to $\kappa l^{2} /2$ per molecule where κ is the stiffness of the myosin heads.

In muscle the change in standard¹⁰ Gibbs (free) energy per molecule accompanying the tension-generating step is made up of two components: firstly that due to the change in conformational state (which would be the same for actomyosin in solution), and secondly this work term (Huxley and Simmons 1971a, b; Eisenberg and Hill 1978). These are tightly coupled i.e. if there is a change in conformational state there is necessarily work done on the compliant element. So in muscle at temperature t ₁, the change in standard Gibbs energy between the pre-tensing state and the post-tensing state, $\Updelta G_{{m,t_{1} }}^{0}$, is equal to that in solution, $\Updelta G_{{s,t_{1} }}^{0}$, plus the work done in the conversion between these two states.

$$ \Updelta G_{{m,t_{1} }}^{0} = \Updelta G_{{s,t_{1} }}^{0} + \kappa l^{2} /2 $$

(6a)

Similarly at the higher temperature t ₂, the change in standard Gibbs energy for the conversion between the pre-tensing and post-tensing states, $\Updelta G_{{m,t_{2} }}^{0}$, is equal to that in solution at this higher temperature, $\Updelta G_{{s,t_{2} }}^{0}$, plus the work done.

$$ \Updelta G_{{m,t_{2} }}^{0} = \Updelta G_{{s,t_{2} }}^{0} + \kappa l^{2} /2 $$

(6b)

Although the fall in standard Gibbs energy in solution is greater at the higher temperature, in muscle the work done for the conversion between the pre-tensing and post-tensing states at the two temperatures is the same. So the increase with rise of temperature for the fall in Gibbs standard energy for the conversion between these two states in muscle, $\Updelta G_{{t_{2} - t_{1} }}$, is the same as in solution and is simply

$$ \Updelta G_{{t_{2} - t_{1} }} = \Updelta G_{{s,t_{1} }}^{0} - \Updelta G_{{s,t_{2} }}^{0} = (t_{2} - t_{1} )\Updelta S_{s} $$

(7)

where ΔS _s is the entropy change per molecule for the conformational change between the two states in actomyosin in solution. Note this expression for $\Updelta G_{{t_{2} - t_{1} }}$ carries no work term and therefore gives no information about the crossbridge stiffness.

We now derive an expression for the increase in fall of standard Gibbs energy at the higher temperature in terms of the average strain in the crossbridges at the two temperatures. The greater strain in the crossbridges at the higher temperature arises because the rise in temperature increases the equilibrium constant of the tension-generating step. If $o_{{pre,t_{1} }}$ and $o_{{post,t_{1} }}$ are the fractions of the heads in the pre- and post-tensing states at temperature t ₁, the equilibrium constant for the tension-generating step in muscle at this temperature,$K_{{m,t_{1} }}$, is given by

$$ K_{{m,t_{1} }} = {\frac{{o_{{post,t_{1} }} }}{{o_{{pre,t_{1} }} }}} $$

(8a)

$$ {\text{Hence}}\,{\frac{{o_{{post,t_{1} }} }}{{o_{{post,t_{1} }} + o_{{pre,t_{1} }} }}} = {\frac{{K_{{m,t_{1} }} }}{{1 + K_{{m,t_{1} }} }}} $$

(8b)

The average strain in the attached heads at this temperature is given by

$$ s_{{t_{1} }} = {\frac{{l\,o_{post} }}{{o_{post} + o_{pre} }}} = {\frac{{l\,K_{{m,t_{1} }} }}{{1 + K_{{m,t_{1} }} }}} $$

(9a)

$$ {\text{Hence}}\,{\frac{1}{{K_{{m,t_{1} }} }}} = {\frac{l}{{s_{{t_{1} }} }}} - 1 $$

(9b)

But the equilibrium constant for the tension-generating step in muscle at this temperature is also given by $\Updelta G_{{m,t_{1} }}^{0} = - k_{B} t_{1} \ln K_{{m,t_{1} }}$.

$$ {\text{Hence }}\Updelta G_{{m,t_{1} }}^{0} = k_{B} t_{1} \ln \left[ {{\frac{l}{{s_{{t_{1} }} }}} - 1} \right] $$

(10a)

Similarly, at the higher temperature

$$ \Updelta G_{{m,t_{2} }}^{0} = k_{B} t_{2} \ln \left[ {{\frac{l}{{s_{{t_{2} }} }}} - 1} \right] $$

(10b)

Hence the increase in the standard Gibbs energy fall of the tension-generating step when the temperature is raised from t ₁ to t ₂ is

$$ \Updelta G_{{t_{2} - t_{1} }} = k_{B} t_{1} \ln \left[ {{\frac{l}{{s_{{t_{1} }} }}} - 1} \right] - k_{B} t_{2} \ln \left[ {{\frac{l}{{s_{{t_{2} }} }}} - 1} \right] $$

(11)

This equation, linking the increase in fall of standard Gibbs energy when the temperature is raised, to the change in strain is clearly very different from the equation proposed by Decostre et al. (2005). Equation 11 does not contain κ and again shows that $\Updelta G_{{t_{2} - t_{1} }}$ gives no information on the crossbridge stiffness.