Myosin Cross-Bridge Behaviour in Contracting Muscle—The T1 Curve of Huxley and Simmons (1971) Revisited

Figure A1. (a) Radial projection of a portion of the sarcomere’s A-band. The myosin filament (represented by black circles at the resting positions of the myosin heads on the surface of the backbone) is surrounded by six actin filaments (the actin filaments on the far right and far left represent the same filament). The white actins are equivalent, and so are the light blue ones. The red circles represent actin monomers to which a head is attached. Blue circles represent inaccessible monomers on particular filaments, because a head is already attached there on an equivalent filament. (b) Cross-section of the sarcomere A-band. A myosin filament (black circle in the middle) is surrounded by six other myosin filaments. In between there are actin filaments. Monomers with attached heads and equivalent filaments are colour-coded as above. (c) One-dimensional Model obtained from (a). Black circles represent the myosin backbone and red circles the actin monomers with attached heads. Equivalent actin filaments have been merged together.

Figure A2. Diagram of the simplified model of the sarcomere used to derive Equation (A1) in the appendix. ℓA and ℓM represent the thin and thick filament lengths respectively. ζ is the overlap region length. The cross bridges are modelled as a ‘rubber band’ spanning the overlap region and joining the thin and thick filaments (Figure adapted from Figure 15 in Ford et al. [4]).

Figure A3. Discrete model approximating the continuous model by Ford et al. [4]. The values chosen for the thin filament (a), thick filament (m) and head stiffness (h) are given in the text.

Figure A4. Cross-bridge longitudinal displacement as a function of distance along the overlap region for the continuous model of Ford et al. [4] (grey) and the discrete model of Figure A3 (black line).

Figure A5. Tension curves along the myosin filament (blue) and actin filaments (orange) calculated from (a) Model 2 and (b) Model (3). The tension curves were calculated with 30% head attachment (first column), 50% head attachment (middle column) and 100% head attachment (third column). After applying a force of 480 pN to the last monomer in the actin filament and calculating the new monomer positions, we obtained the tension of the springs between adjacent monomers using Hooke’s law, and we plotted these values as a function of spring’s position along the half-sarcomere. The position 0 corresponds to the centre of the M-band. In all cases, km = 720 pN/Å for the myosin filament, ka = 2280 pN/Å for the actin filament and kh = 0.8 pN/Å for the heads. Note that the analysis of Ford et al. [4] went part way towards this in their Figure 4 (lower left).

Figure A6. Tension through the half sarcomere, averaged over several possible initial configurations, using Model 3 for (a) 100% attachment and (b) 50% attachment. The tension along the myosin filament is depicted in blue, along one set of equivalent actins in light green and along the other set of equivalent actins in dark green. Their sum, corresponding to the tension along all actins, is depicted in orange. The summed tensions (actin plus myosin) along the A-band are depicted in red. In both cases km = 720 pN/Å for the myosin filament, ka = 2280 pN/Å for the actin filament and kh = 0.8 pN/Å for the heads.

Figure 1. Previous observations: (a,b) representation of the observations of Huxley and Simmons [2] (as improved by [3,4]) showing the tension transient (b) in an active frog muscle fibre after a rapid shortening step (a) of about 6 nm (60 Å) per half sarcomere and the point where the T1 tension was recorded (b). (c) The T1 and T2 plots from experiments as in (a), but for different shortening steps (filament displacement per half sarcomere) and shown at two different sarcomere lengths—solid lines full overlap, dashed lines 3.1 µm. Po is the isometric tension at the tension plateau. (d) Representation of the results from Huxley et al. [5] showing the change in positions of the M15 (M15′ on the opposite side) and A13 (A13′) meridional X-ray diffraction peaks from myosin and actin filaments respectively at rest (solid lines) and full activation (dashed lines). The peaks shift towards the middle (longer spacings) and increase in intensity in X-ray patterns from active muscle. Wakabayashi et al. [6] obtained similar results. Figures redrawn from (a,b,c) [2] and (d) [6]. For details, see text.

Figure 2. Illustrations of the three different models of increasing complexity used in the calculations in this paper. (a,b) Model 1 allows a simple and approximate evaluation of filament and cross-bridge stiffnesses to show roughly what they must be: (a) at the isometric force plateau Po before release, and (b) after release to zero tension. In (a): MB: edge of the myosin filament bare zone; MAI: the end of the myosin filament at the A-I junction; AB: the tip of the actin filament away from the Z-band; AAI: the actin filament position at the A-I junction. (c) Model 2 mimics cross-bridges at levels m1, m2 etc., attaching to actin positions a1, a2 etc., at 144.9 Å intervals with appropriate weighting to allow for binding to actin by only a fraction of the available heads at any one time. (d) Model 3 represents, in a simplified way, the kind of result from assessing myosin head labelling realistically in three dimensions (3D) using the program MusLABEL [16]. For details see text.

Figure 3. Results from Model 2 using optimised searching for good fits as measured by a goodness of fit factor Chi. The lower Chi is the better the fit. The best results were obtained with a cross-bridge stiffness kh around 0.45 to 0.8 pN/Å. (a) 30% attachment, (b) 50% attachment, (c) 30% attachment, with the T1 intercept unconstrained. In (a) and (b) the T1 intercept was constrained to be 40 Å. In (c) the value of the T1 intercept was unconstrained, but it still came out at around 40 Å. Note the non-linear scale for kh.

Figure 4. (a) Example of the computed X-ray profiles from calculations using Model 3. The purple curve is calculated from a resting muscle, the black curve from a muscle subject to a Po = 480 pN external force and the red curve from a muscle subjected to a 2Po external force. (b) Black curve: Changes of the T1 intercept as a function of the head stiffness kh, with the filament stiffnesses held constant. Purple curve: observed value of the Table 1 intercept (taken as 40Å). (c) M15 and A13 peak Positions as a function of head stiffness as calculated from the simulated X-ray diffraction from Model 3 after subjecting them to a Po and 2Po external force. (d) Black: M15 active to rest intensity ratio as a function of stiffness as calculated from Model 3; Purple-experimental measurement. (e) Black: A13 active to rest intensity ratio as a function of stiffness as calculated from Model 3; Purple-experimental measurement. (f) Black: M3 active to rest intensity ratio as a function of stiffness as calculated from Model 3; Purple: upper limit as found experimentally; Red: lower experimental limit. (g) Variation of the positions of various half sarcomere features as a function of head stiffness: Purple:…

Figure 5. (a) T1 plots showing the position of the T1 intercept on the x axis for zero tension as a function of the myosin head stiffness in Model 3 for a muscle at full overlap. (b) Similar to (a) but here for different percentage cross-bridge attachments to actin at full overlap and also what happens at 3.1 µm sarcomere length (tension 0.39 Po) using Model 3. (c) The observations of Ford et al. [4] plotted here assuming purely linear elasticities and assuming that tension varies linearly with overlap, shown for comparison with (a) and (b).

Figure 6. The variation of myosin filament stiffness (km; blue) and actin filament stiffness (ka; orange) with tension Po assuming that the X-ray spacing changes in the M15 and A13 peaks are as observed by Huxley et al. [5] and Wakabayashi et al. [6]. The grey curve shows the myosin filament stiffness if the same attachment number is achieved with a different set of labelling parameters in MusLABEL.