Actomyosin-ADP States, Interhead Cooperativity, and the Force-Velocity Relation of Skeletal Muscle

Abstract

Despite intense efforts to elucidate the molecular mechanisms that determine the maximum shortening velocity and the shape of the force-velocity relationship in striated muscle, our understanding of these mechanisms remains incomplete. Here, this issue is addressed by means of a four-state cross-bridge model with significant explanatory power for both shortening and lengthening contractions. Exploration of the parameter space of the model suggests that an actomyosin-ADP state (AM^∗ADP) that is separated from the actual ADP release step by a strain-dependent isomerization is important for determining both the maximum shortening velocity and the shape of the force-velocity relationship. The model requires a velocity-dependent, cross-bridge attachment rate to account for certain experimental findings. Of interest, the velocity dependence for shortening contraction is similar to that for population of the AM^∗ADP state (with a velocity-independent attachment rate). This accords with the idea that attached myosin heads in the AM^∗ADP state position the partner heads for rapid attachment to the next site along actin, corresponding to the apparent increase in attachment rate in the model.

Introduction

Muscle contraction is caused by ATP-driven cycles of structural changes in billions of actomyosin (AM) cross-bridges, i.e., the globular motor domains (heads) of myosin II molecules (henceforth termed “myosin”) interacting with specific sites on the actin filaments (1,2). To elucidate how such a cross-bridge ensemble accounts for muscle contraction, statistical models are required (3) in which observable properties are inferred from the probability distributions of cross-bridge states. The first model of this type, proposed by Huxley (1), postulated that 1), there is an elastic element in the cross-bridge; 2), the cross-bridges generate force immediately upon formation; 3), there is a limited rate constant for cross-bridge formation; and 4), the rate constant of cross-bridge dissociation increases if the elastic force counteracts shortening. On the basis of these assumptions, the main features of the relationship between the applied external load and velocity (the force-velocity (P-V) relationship), as well as important energetic aspects of muscle contraction, have been predicted.

Although the majority of recent models (4–13) have incorporated key assumptions of the Huxley model (1), the growth of experimental information has prompted several additions and modifications. More chemical and mechanical states have been introduced, and the force-generating step has been separated from the attachment step (4). Some models also explicitly consider the role of the two heads of each myosin II molecule (7–9) or intermolecular cooperativity (7,8,10,14). In spite of these developments, however, there are still conflicting ideas about some issues, such as which kinetic step(s) in the AM ATP turnover cycle that determine(s) the maximum shortening velocity (V₀) (10,15–17), and what mechanisms underlie the double-hyperbolic shape of the force-velocity relationship (10,13,18). Moreover, several of the recent cross-bridge models are quite complex, with many cross-bridge states and parameters to be fitted. The associated loss of transparency compromises insight into real phenomena, particularly since several of the model states may be difficult to identify with those inferred from biochemical and ultrastructural studies (7,8,12). Thus, there is a need for simpler models that capture the most essential aspects of key phenomena, and in which the states can be readily identified with those directly observed in experimental studies.

A recently proposed model (6) has the potential to meet these requirements. This model differs from most other models for muscle contraction (7,8,10,13) by the explicit inclusion of two cross-bridge states with ADP at the active site (but see West et al. (19) and Siththanandan et al. (20)). One of these states (denoted AM^∗ADP) is separated from the actual ADP-release step by a strain-dependent isomerization (5,6,16,21). Since a similar state has been implicated as being critical for the processivity of nonmuscular myosins (e.g., myosin V (22)), it is of interest to consider the possibility that processivity may also occur with myosin II (6,13,23–26), and that the AM^∗ADP state may play a role in this connection.

In this work, the model of Albet-Torres et al. (6) was updated with recent estimates (27,28) of power-stroke length and cross-bridge stiffness. Additionally, the free-energy profiles of the two AM ADP states were slightly changed to better fit the experimental data (5,6,16,21). After verifying that the model faithfully simulates a range of chemomechanical events in steady-state shortening and lengthening contractions, a systematic exploration of the parameter space was undertaken, with the main focus on the shortening part of the P-V relation. This exploration corroborates a hypothesis that several biochemical steps in sequence influence V₀. One of these steps is the strain-dependent isomerization between the two AM ADP states. Additionally, critical aspects of cross-bridge function are identified as a potential basis for the double-hyperbolic shape of the P-V relationship. The requirement for a velocity-dependent attachment rate in the model is also considered in relation to the AM^∗ADP state and the possible processive action of the two myosin heads. Finally, a number of testable model predictions are highlighted.

Materials and Methods

Equations 1–5 below were solved numerically using the fourth-order Runge-Kutta-Fehlberg method implemented in Simnon (v. 1.3; SSPA, Gothenburg, Sweden). Statistical analyses, including regression analysis, were performed using GraphPad Prism (v. 4.0 and 5.0; GraphPad Software, San Diego, CA).

Theory

The following kinetic scheme (Scheme 1) accounts for several results from transient and steady-state kinetics studies of AM in solution (2,29,30):

Here, the lower and upper rows represent myosin (M) and AM states, respectively, with substrate (ATP) or products (ADP and inorganic phosphate (P_i)) at the active site. The quantities k₂, k₄, etc., are rate constants, whereas K₁, K₃, etc., are equilibrium constants. The gray highlight indicates the key AM^∗ADP state mentioned in the Introduction.

The simulations of the P-V relationship were performed on the basis of a simplified four-state or, for some purposes, five-state model (Scheme 2):

Here, the low-force state A₀ can be identified with the AM ADP P_i state in Scheme 1, whereas A₁ (gray highlight) can be identified with the AM^∗ADP state (see above). The AM ADP, AM, and AM ATP states in Scheme 1 are lumped together into state A₂ (6–8), and the M ATP and M ADP P_i states are lumped together into state A₃. These simplifying assumptions, together with the idea that the myosin heads act as independent force generators, are discussed and justified in the Supporting Material. The free-energy profiles of all states are shown in Fig. 1 A.

Figure 1.

Free-energy profiles and rate functions. (A) The free energy of the detached state A₃ is set to 20 kT (38,60) for all values of x (see text). The bent double arrow indicates that the lever arm is disordered in this state. Schematic illustrations of the equilibrium lever arm positions for the different states are given. (B) Rate functions for transitions between attached states (k₀₁(x), full black line; k₁₀(x), dashed black line; k₁₂(x), dark gray line; k₂₁(x), dashed dark gray line) and the detachment rate function (k₂₃(x), light gray line). (C) Attachment rate functions (k₃₀(x) for the standard simulations (full black line) in all figures and for some specific simulations in Fig. 3 (dark and light gray lines). Reversal of attachment rate functions (k₀₃(x) is shown by dashed lines.

The reaction within parentheses in Scheme 2 is included only in a part of the Results (see the section, “A velocity-dependent attachment rate improves the fit to certain data”) to represent flux through a kinetic pathway simulating sequential attachment of the two partner heads of one myosin molecule. The subscript 0 indicates that the A₀² state corresponds to attachment of the second head into state A₀. It is assumed that the trailing head positions the leading head favorably for rapid attachment into this state at the next site along the actin filament (7,9,31). The rate process governed by the constant k₀₀ in Scheme 2 is included to allow determination of the flux via the path involving A₀².

The rate constants for transitions between states (Fig. 1, B and C) depend on the variable x, i.e., the distance between the nearest appropriately oriented binding site on the actin filament and the cross-bridge. The variable is defined so that x = 0 nm when the free energy of the state A₂ is at its minimum. The free-energy difference (Fig. 1 A) between two pairs of neighboring states in Scheme 2 determines the ratio of the forward and backward rate constants for the transitions between these states at any value of x (3). The rate constants for the forward and reverse transitions between states A₁ and A₀² do not comply explicitly with this rule, since the geometric and energetic situation is not clear. Therefore, the rate functions for these transitions are tentatively chosen to have the same shapes as k₃₀(x) (Gaussian; Fig. 1 C, black) and k₀₃(x), but shifted by 6.3 nm to the left along the x axis (to be centered around the free-energy minimum of the A₁ state). Moreover, the amplitude of these shifted rate functions was increased 7 times, compared to the values of k₃₀(x) and k₀₃(x) under isometric conditions. This gave an average value for the attachment rate of ∼200 s⁻¹ in the range −1 nm < x < 3 nm, in reasonably good agreement with the rate of regeneration (repriming) of the power-stroke after rapid releases (32). The shift of 6.3 nm is slightly larger than the distance (5.5 nm) between two subsequent sites along an actin protofilament, indicating that the cross-bridge stiffness should, ideally, be slightly increased and the power-stroke length slightly reduced in the model. However, such optimizations are beyond the scope of this study.

Muscle properties during steady-state contraction are derived from the state probabilities a₀ ≡ a₀(x, v), a₁ ≡ a₁(x, v), a₂ ≡ a₂(x, v), a₃ ≡ a₃(x, v), and a₀²≡ a₀²(x, v) for the states A₀, A₁, A₂, A₃, and A₀², respectively (3). These state probabilities are obtained by solving the following nonlinear ordinary differential equations for different velocities, v (positive for shortening):

$$\frac{d{a}_0}{dx}=-\left(k_{30}\left(x,v\right)a_3+k_{10}\left(x\right)a_1-\left(k_{03}\left(x,v\right)+k_{01}\left(x\right)\right)a_0\right)/v$$ (1)

$$\frac{d{a}_1}{dx}=-\left(k_{01}\left(x\right)a_0+k_{21}\left(x\right)a_2-\left(k_{10}\left(x\right)+k_{12}\left(x\right)\right)a_1\right)/v$$ (2)

$$\frac{d{a}_2}{dx}=-\left(k_{12}\left(x\right)a_1-\left(k_{21}\left(x\right)+k_{23}\left(x\right)\right)a_2\right)/v$$ (3)

$$\frac{d{a}_3}{dx}=-\left(k_{23}\left(x\right)a_2+\left(k_{03}\left(x,v\right)a_0-k_{30}\left(x,v\right)\right)a_3\right)/v$$ (4)

$$\frac{d{a}_{0_0}^2}{dx}=-\left(7k_{30}\left(x+6.3\right)a_1-\left(7k_{03}\left(x+6.3\right)-k_{00}\right)a_0^2\right)/v\text{.}$$ (5)

The cross-bridge stiffness and positions of the free-energy minima of different states were fixed on the basis of recent experiments (27,28), putting stringent first constraints onto the model. The rate functions for transitions between the states A₀ and A₁ (Fig. 1 B) were similar to those in Edman et al. (13), and the maximum value of the rate function for cross-bridge attachment (k₃₀(x); Fig. 1 C) was chosen to account for the maximum power output during shortening. The free-energy difference between states A₀ and A₁ was adjusted to account for both the maximum efficiency of muscle and the average isometric force per attached cross-bridge. The free-energy difference between states A₁ and A₂, on the other hand, was constrained to ensure a low population of state A₁ when ADP was added to AM in solution or skinned fibers (6,21). This is different from the previous model (6). Finally, V₀ was fitted by adjusting the rate functions (k₂₃(x)) and k₂₁(x). For details on the choice of rate functions, see the Supporting Material (6,13).

Results

Simulation of steady-state data

It is shown in Fig. 2, A and B (black lines), that the model with rate functions as in Fig. 1 B (all) and C (black) fits the experimental P-V data (purple symbols) well for both shortening (positive velocity) and slow lengthening contractions (negative velocity), with a continuous change in slope at the maximum isometric force (P₀; open symbol in Fig. 2 B). Fitting the experimental data for fast lengthening contractions (inset, Fig. 2 B) requires modifications of the model (see below). The model relation between the number of attached cross-bridges and force (Fig. 2 C; black) fits the experimental data (purple) well. Here, the latter are based on the force level at which the cross-bridge ensemble was forcibly detached by a rapid stretch (33). The increased number of attached cross-bridges in the model for slow lengthening is in semiquantitative agreement with the increase in fiber stiffness up to ∼20% under these conditions (34). However, the interpretation of the stiffness data depends critically on the characteristics of the filament compliance, e.g., whether it is Hookean (35) or non-Hookean (36,37).

Figure 2.

Simulation of physiological data. (A) Experimental P-V data (purple; measured from Fig. 2 in Edman et al. (13)) compared to P-V data simulated with a velocity-independent attachment rate function with a maximum value of 178 s⁻¹ (black) or 67 s⁻¹ (green) or with a velocity-dependent attachment rate function (orange, inset). (B) The low-velocity parts of the P-V relation for shortening (positive velocity) and lengthening (negative velocity). Force was normalized to P₀ (open square) in each case. Experimental force-velocity data were obtained from Fig. 2 of Edman et al. (13) (purple circles) and Fig. 7 of Edman (18) (purple triangles). Inset: PV relation for high lengthening velocities. Experimental data (purple squares) were measured from Fig. 11 of Lombardi and Piazzesi (34). (C) Simulated fraction of attached cross-bridges under physiological conditions (full lines) plotted against force during shortening and lengthening, and compared with experimental data (purple circles) measured from Fig. 3 of Bagni et al. (33). Data were normalized to those at P₀ (open square). Velocity-independent (black and green) and velocity-dependent (orange) attachment rate function, as in A. Dashed line: Data simulated with an increased free energy (by 1.1. kT) of state A₂, accounting for the effects of 1 mM amrinone on shortening velocity. (D) Relationship between the simulated ATP turnover rate (left vertical axis) efficiency (right axis) and velocity. Simulation of physiological data: solid lines, 1 mM amrinone; dashed lines, see above. Same color code in A–D.

The maximum thermodynamic efficiency occurs at 0.1 V₀ (Fig. 2 D, black, Efficiency) with a numerical value within the experimental range for frog muscle (0.37–0.55 (38)). The predicted ATP turnover rate in isometric contraction (∼1 s⁻¹) and a marked decrease in this rate for slow lengthening contractions (not shown) are also in agreement with experimental data (29,39–41) (Table S2). The predicted increase in ATP turnover rate during shortening (Fig. 2 D, black) is slightly larger than that found experimentally (29,39,40), and a decline in ATP turnover rate in rapidly shortening frog muscle (29,42,43) is not reproduced (see further below). The average steady-state force (∼6 pN) and strain (∼2 nm) per attached cross-bridge, the fraction of attached cross-bridges (7% at V₀ and 18% at P₀), and the maximum isometric force per cross-sectional area (∼0.2 N mm⁻²) are all in reasonable agreement with experimental data (13,18,27,30,44–46) (Table S2).

P-V relation and the attachment rate function

Changes in the shape of the attachment rate function (Fig. 1 C) markedly affect (Fig. 3, A and B) the high-force region of the P-V relation. The part of the high-force region with a low value of the derivative |dV/dP| (phase 1, Fig. 3 B) is associated with a substantial loss (with increased velocity) of the cross-bridges in state A₀ that produce the highest force (largest x; Fig. 3 C, shown in another way in Fig. S2). This has to do with the very low attachment rate constant at the right edge of the Gaussian rate function (Fig. 1 C, black curve). When the velocity increases further (up to 150 nm s⁻¹), this effect is counterbalanced by an increase in force due to the increased population of force-producing cross-bridges in states A₁ and A₂ (Fig. 3 C). This leads to increased |dV/dP|, corresponding to phase 2 of the high-force region of the P-V relation (Fig. 3 B). For certain shapes of the attachment rate function (Fig. 1 C, dark gray), the reduced force due to A₀ cross-bridges with increased velocity is counterbalanced by increased force due to cross-bridges in states A₁ and A₂ (Fig. 3 D), to the extent that phase 1 is significantly reduced in amplitude (Fig. 3, A and B, dark gray). Finally, if the loss of force due to cross-bridges in state A₀ is just canceled out by the increase due to bridges in states A₁and A₂ (Fig. 3 E), both phase 1 and phase 2 may virtually disappear (Fig. 3, A and B, light gray). In the case illustrated here, this is attributed to a wide and flat attachment rate function (Fig. 1 C, light gray) giving a relatively high attachment rate also at the right edge of the attachment range.

Figure 3.

Effect of the attachment rate function on the P-V relationship. (A) P-V relationships modeled with attachment rate functions corresponding to the gray tones in Fig. 1C. Force in N mm⁻² of fiber cross-section. (B) The high-force region of the P-V relationships in A with force normalized to P₀ for each curve. (C) Distributions over x of the fraction of cross-bridges in states A₀, A₁, and A₂, corresponding to the black lines and symbols in A and B. The distributions refer to different velocities: 1 nm hs⁻¹ s⁻¹ (full line), 40 nm hs⁻¹ s⁻¹ (dashed line), and 150 nm hs⁻¹ s⁻¹ (dotted line). (D) Cross-bridge distributions as in C but corresponding to the dark gray lines and symbols in A and B. (E) Cross-bridge distributions as in C and D but corresponding to the light gray lines and symbols in A and B.

P-V relation and free-energy profiles

Small changes in the free-energy profiles of cross-bridge states may markedly affect the P-V relationship (Fig. 4 and examples in Fig. S3). For instance, the different phases of the high-force region (inset, Fig. 4 A) are changed by differences in the relative free-energy levels (inset, Fig. 4 B) of states A₁ and A₂. In Fig. 4, A and B, the free energy of state A₂ is either increased (light blue) or reduced (green) compared to the free energy of state A₁. It can be seen that the nonhyperbolic deviation of the P-V relation at high loads becomes less conspicuous with the increased free energy of state A₂. As with changes in the attachment rate function, the changes in the free-energy difference between the A₁ and A₂ states affect the high-force region by affecting the balance between forces due to cross-bridges in states A₀ and (A₁ + A₂). The reduction in V₀ in response to the increased free energy of state A₂ is caused by the associated downward shift of the rate function k₁₂(x) (Fig. 4 B) and the increased population of negatively strained cross-bridges in state A₁.

Figure 4.

Effects of altered free-energy profiles on the P-V relationship. (A) Effects of increase in free energy of state A₂ by 1.6 kT (light blue) or decrease by 1.4 kT (green). Inset: The high-force region with force normalized to P₀ in each case. (B) Rate function k₁₂(x) for the P-V curves in colors corresponding to those in A. Inset: Free-energy diagrams corresponding to the P-V data in A. (C) Effects on the P-V relationship of increased free energy of states A₁ and A₂ by 3 kT (blue) or reduced free energy of both states by 3 kT (red). Inset: Free-energy diagrams corresponding to P-V data in the main panel C. (D) Plots of velocity against working stroke distance, h, as defined in the inset. Blue, black, and red solid circles correspond to the P-V relationships and free-energy diagrams in C. Green circles refer to further changes in the free energy of states A₁ and A₂ but with other parameter values similar to those in C. The black squares and triangles refer to the same free-energy diagrams and parameter values as in C except that the numerical value of the rate function k₂₃(x) is altered. For the black squares, the numerical value of k₂₃(x) is doubled for |x| < 5 nm and its minimum value is increased to 1200 s⁻¹, compared to 600 s⁻¹ for the simulations in C. For the black triangles, the numerical value of k₂₃(x) is halved for |x| < 5 nm, with a minimum value of 300 s⁻¹. Full straight lines are regression lines, and curved dashed lines are 95% confidence limits. The Edman equation (18) is fitted to the force-velocity data in A and C.

Of interest, the increased free energy of state A₂ compared to state A₁ (Fig. 4, A and B; light blue) reproduces the characteristic effects of the drug amrinone on the P-V relation (47,48), i.e., a reduction of V₀, increase of P₀, and reduced deviation from hyperbola at high loads (6). A similar combination of effects is not reproduced by any other tested changes in parameter values (Figs. 3 and 4; Fig. S1, Fig. S2, and Fig. S3). An increase in the free energy of state A₂ (by 1.1. kT, sufficient to reduce V₀ to a similar extent as 1 mM amrinone) is also predicted to produce other changes. This includes an altered relationship between force and the number of attached cross-bridges during shortening (dashed line in Fig. 2 C), and between velocity on the one hand, and efficiency and ATP turnover rate on the other (Fig. 2 D).

The increased free-energy difference (inset in Fig. 4 C, red) between the A₀ state on the one hand, and the A₁ and A₂ states on the other, does not affect the attachment rate function. Nor is the detachment rate function k₂₃(x) changed, provided that it is limited by the rate constant k₆ (Scheme 1 (6), Supporting Material). Under these conditions, the number of attached cross-bridges during isometric contraction is unaltered, but the average force per cross-bridge is increased. This is reminiscent of the effect of increased temperature (35,49,50), although the latter condition is likely to be associated with several other changes in rate constants and free-energy profiles. In further similarity to experimental effects of increased temperature, both V₀ and P₀ are markedly increased by a larger free-energy difference between state A₀ and states A₁/A₂. In contrast, the overall shape of the P-V relation in the model is only slightly modified (Fig. 4 C). This is in accordance with experimental findings in frog (18) (but not mammalian (51)) muscle. In general terms, the changes in the P-V relationship in Fig. 4 C are due to the altered average distance, h, over which a cross-bridge produces force in the shortening direction. If h is defined as in Fig. 4 D, a plot of V₀ for the data in Fig. 4 C against h is approximately linear with an intercept on the ordinate, not significantly different from zero (Fig. 4 D). Accordingly, h can be identified with the power-stroke length in the relationship V₀ ∝ h k_off (52), where k_off is an apparent dissociation constant for cross-bridges with negative tension. The latter constant is proportional to the slope of lines like those in Fig. 4 D (52). As can be seen here, this slope (and thereby k_off) is affected by doubling and reduction to one-half of the magnitude of the detachment rate function k₂₃(x) (in the range of −4 nm < x < 4 nm).

A velocity-dependent attachment rate improves the fit to certain data

The time course of the force development during an isometric tetanus (13) suggests a markedly lower cross-bridge attachment rate than required to account for the maximal power-output during shortening. This lower “isometric” attachment rate also accounts for the P-V data for very slow shortening and lengthening (green lines in Fig. 2 B). Of interest, a similar low attachment rate, at velocities close to V₀, would explain the decrease of energy output at high velocities (29,42,43). However, the number of attached cross-bridges at intermediate shortening velocities (Fig. 2 C, green) is not reproduced. To approximately fit all the data for shortening, the amplitude of the Gaussian attachment rate function was increased from 67 s⁻¹ in isometric contraction to 148 s⁻¹ at a velocity of 600 nm half-sarcomere (hs)⁻¹ s⁻¹ and then decreased back to 67 s⁻¹ at a velocity of 2500 nm hs⁻¹ s⁻¹ (inset, Fig. 2 A). It can be seen in Fig. 2, A and B (orange), that the velocity dependence of k₃₀(x, v) only slightly affects the predicted P-V relation compared to the situation with a high attachment rate at all velocities (Fig. 2, A and B, black). The relationship between force and the number of attached cross-bridges is slightly shifted (Fig. 2 C, orange) to better fit the experimental data, and the ATP turnover rate is reduced at the highest shortening velocities (Fig. 2 D, orange) as found experimentally. The quite small effects attributed to the velocity dependence of k₃₀(x, v) (compared to a velocity-independent high value of k₃₀(x)) on the shape of the P-V relation were eliminated by minor adjustments of other parameter values (not shown). The velocity dependence was therefore not considered in the sections entitled “P-V relation and the attachment rate function” and “P-V relation and free-energy profiles” above.

An increased attachment rate with increased velocity can also account for the P-V data at fast lengthening, as illustrated in the inset of Fig. 2 B (orange). Here, the attachment rate is increased from a maximum value of 67 s⁻¹ in isometric contraction to 335 s⁻¹ at a lengthening velocity of ≥600 nm hs⁻¹ s⁻¹.

To accommodate a velocity-dependent attachment rate during shortening, without changing other rate functions or free-energy profiles, it is possible that 1), the two heads of a given myosin molecule behave identically and independently subsequent to actin attachment; and 2), the first attached head positions the second head for rapid attachment during shortening against intermediate loads. If this explanation is correct, there should be a mechanism to make the attachment rate of the second head most probable at intermediate velocities (300–600 nm hs⁻¹). Indeed, a relevant mechanism has been proposed (9), with the aim to account for the regeneration of the power-stroke after a length step (as defined previously (32)). In this mechanism, the second head is postulated to attach more rapidly to the next site in the target zone only if the first head is in an appropriate, intermediate state of attachment. The latter state may here be identified with the state A₁ (AM^∗ADP). It is therefore interesting to note that the population of this state, in a range (−1 < x < 3.5 nm) around the minimum of the free energy of state A₁ (Fig. 1 A), varies with velocity (Fig. 5, black open symbols) similarly to the attachment rate (inset, Fig. 2 A).

Figure 5.

Population of state A₁ and processivity. The population of state A₁, integrated over −1 < x < 3.5 nm, in arbitrary units for different velocities (open symbols) together with the ratio between the flux from state A₁ to A₀² and the total flux (solid symbols; fraction processive). Black symbols represent the standard simulation parameters but with the attachment rate function at the isometric value (green lines in Fig. 2). Gray symbols represent predictions if the free energy diagram of state A₂ is changed to account for the amrinone induced reduction in V₀ (see also Fig. 4B).

To investigate this issue further, a state A₀² (Scheme 2) was introduced to simulate flux through a path corresponding to rapid attachment of the second head to the next site along the actin protofilament. It is essential to emphasize that only the flux through the two alternative paths—A₁ to A₂ (one head attached) or A₁ to A₀² (attachment also of second head)—as a myosin head pair passed an actin target zone during shortening was considered. The goal in these simulations was not to predict observable muscle properties. This would have required several more states and assumptions. Of importance, however, the simulations showed that the flux via state A₀² (corresponding to sequential binding of the two heads) varied with velocity, as illustrated in Fig. 5 (black solid symbols). This can be interpreted to mean that, at a velocity of 300–700 nm hs⁻¹, more than 50% of the first attachments of a myosin head led to attachment also of the second head further along the actin filament. At both lower and higher velocities, this fraction was smaller, in approximate accordance with the apparent velocity dependence required for the attachment rate (inset, Fig. 2 A). The simulations (Fig. 5, gray) also suggest that the type of change in free-energy diagrams introduced in Fig. 4 B to reproduce the amrinone effects would cause a larger fraction of the first attached heads to be in state A₁. This would increase the attachment probability for the second head (Fig. 5, gray solid symbols) corresponding to increased processivity.

Discussion

General issues

To a large extent, the force level is determined by the population of state A₁ (AM^∗ADP), which explains the increase of P₀ when the difference in free energy between states A₀ and A₁ is increased (6,19). The unloaded shortening speed in the model is influenced by both k₁₂(x) and k₂₃(x) (Fig. 4). This provides a platform for understanding ambiguous experimental results (6,15,17) that suggest different steps in Scheme 1 (k₂, k₅, or k₆) as the main determinant of V₀ (6,17). In agreement with earlier work, the power-stroke length (h) also affects the sliding velocity according to the relationship V₀ ∝ k_off h (52). However, interpretation of the parameters k_off and h is not straightforward.

The double-hyperbolic shape of the P-V relationship has been verified and characterized in various muscle preparations (13,18,53,54). The results presented here suggest that the loss of those cross-bridges in state A₀ that develop the highest force is important for the large drop in force associated with increased velocity during phase 1 (Fig. 3 B) at loads close to the isometric. Moreover, the compensating increase in force due to cross-bridges in states A₁ and A₂ (at further increased velocity) is the basis for phase 2. Additionally, the two effects need to be appropriately balanced for the correct nonhyperbolic shape, and this balance critically depends on fundamental aspects of the cross-bridge function (Figs. 3 and 4). This puts important constraints on cross-bridge models and, not least, on the evolutionary process. Thus, minor changes, e.g., as a result of a point mutation, might even cause a positive dV/dP and regions where dV/dP → ± ∞ (e.g., Fig. S1, Fig. S2, and Fig. S3). Such effects would be deleterious for normal muscle function and result in appreciable difficulties for regulation. The above explanation for the double-hyperbolic shape is consistent with the hypothesis of Edman et al. (13), but differs from that proposed by Duke (10).

The model also fits experimental data well for slow lengthening contractions, and there is a continuous change in slope of the relation around P₀ as found experimentally (18). The best fit for slow lengthening was obtained using the attachment rate that accounts for the rising phase of an isometric contraction, as opposed to using the rate that best reproduces the high-power output during shortening. When the low isometric attachment rate is used, the model also accounts for both the reduced ATP turnover rate and the increase in the number of attached cross-bridges for slow lengthening. At higher lengthening velocities, the situation is more complex. A substantially increased rate of cross-bridge attachment compared to that in an isometric contraction (34,55) is thus required to explain why the force continues to increase at lengthening velocities above 100 nm hs⁻¹s⁻¹.

The tension response to rapid length steps during an isometric tetanus (isometric tension transients) has been studied extensively (4,32,56) and may provide important insights into cross-bridge mechanisms. The model presented here gives a satisfactory fit to the T₁ and T₂ curves, i.e., plots of the extreme tension (T₁) and the tension level after rapid tension recovery (T₂) against the step size (Fig. S4). However, the tension transients have not been studied in detail, in part because of uncertainties in interpreting nonsteady-state experimental phenomena before the myofilament compliance has been fully characterized (35–37). In the effort to explain tension transients, cross-bridge models (7,8,12) have included states that have not been directly observed in ultrastructural and biochemical studies. Such additional states are necessary if force generation occurs according to a Kramers-like reaction mechanism (whereby global diffusion of parts of the myosin head strains the elastic cross-bridge element to reach the activation energy for a force-generating transition) (30). However, it has not been verified that force generation occurs by such a mechanism (30,57), and there are several pieces of evidence that point instead to an Eyring-like mechanism (whereby a sudden local chemical change around the nucleotide pocket leads to achievement of the activation energy; see Supporting Material). Under such conditions, one main force-generating transition is sufficient, as found in this work.

Efforts to explain the phosphate release mechanism have also prompted the introduction of additional states. However, these efforts have led to a variety of different models (20,49,56,58–60) and therefore are only considered in the Supporting Material.

Velocity-dependent attachment rate and interhead cooperativity

The need for a velocity-dependent attachment rate function to account for both the maximum power output during shortening and the rate of rise of isometric force exists in a range of models (1,12,61), even if it has seldom been made explicit (but see Albet-Torres et al. (6) and Edman et al. (13)). Additionally, a velocity-dependent attachment rate was also inferred from the experimental results of Piazzesi et al. (27), on slightly different grounds. The proposed velocity dependence makes physiological sense because it ensures a high-power output at intermediate velocities, which are most important in animal locomotion. At both very fast and very slow shortening, on the other hand, the power output is close to zero and a reduced number of cycling cross-bridges is desired to avoid futile ATP turnover. One interpretation of the velocity-dependent attachment rate is that the two heads of myosin II act cooperatively in sequence (processively) during shortening against intermediate loads, but not during very slow or very rapid shortening. A velocity dependence of rate functions is the only way to reproduce interhead cooperativity in models where the two myosin heads are assumed independent. Such velocity dependence is also required to account for contractile behavior during fast lengthening. Indeed, in this case, some experimental evidence suggests that attachment of the second head may be important (62).

The results in Fig. 5 provide mechanistic insight into how limited processivity can occur during shortening. The notion that the processivity is limited (on average to less than two sequential head attachments) is suggested by 1), geometrical constraints (31) with only three binding sites in each target zone on the actin filament, with, on average, one possibility for attachment also of the second head; 2), far from obligate attachment of the second head, according to simulations in Fig. 5; and 3), a <2 times larger absolute number of attached cross-bridges at v = 600 nm hs⁻¹ s⁻¹ (Fig. 2 C) with a velocity-dependent attachment rate than if the attachment rate were velocity-independent and equal to that in isometric contraction.

It was previously suggested (9) that sequential attachments of the two myosin heads may form the basis for regeneration of the power stroke after rapid length steps (32). Therefore, the rate function corresponding to attachment of the second head was chosen here to make its average value similar to that of the rate of regeneration of the power stroke. This clearly leads to a velocity dependence (Fig. 5) similar to that observed for the attachment rate function (inset, Fig. 2 A). Moreover, it would cause >50% of the myosin molecules to go through at least two sequential cycles, thereby accounting for the maximum power and the number of attached cross-bridges at intermediate loads/velocities. The key role for the A₁ (AM^∗ADP) state in mediating this type of processivity is consistent with suggestions (9) that the second head may attach more rapidly to the next site in the target zone only if the first head is in an appropriate, intermediate state of attachment. This is also broadly consistent with structural evidence (25). Moreover, in further accordance with this idea, the trailing head of the unambiguously processive myosin V motor seems to be in a long-lived state, corresponding to the AM^∗ADP state, when the leading head attaches to the next site along the actin filament (22).

To experimentally elucidate the proposed type of interhead cooperativity, it would be useful to examine the effects of amrinone on AM kinetics (6) because it prolongs the residence time of cross-bridges in state A₁ (9). One can predict that 1), the apparent velocity dependence of the attachment rate would increase in the presence of amrinone; and 2), regeneration of the power stroke would be affected (presumably resulting in more effective regeneration and a different dependence on the amplitude of the length step). Of interest, the first prediction accords with a previous study (6) in which an improved fit of the amrinone effects on the force-velocity relationship was obtained by assuming a stronger velocity dependence for the attachment rate function.

The issue of whether the two heads of myosin II act processively under any conditions in skeletal muscle has long been a topic of discussion (63,64). Although several studies (6,13,23–26) support the idea, it is not generally accepted (63,65) because it is difficult to obtain conclusive experimental results. The simulations presented here provide evidence that limited processivity would account for key experimental data, and also suggest a reasonable mechanism.

In some studies of cross-bridge models (7–9), the two myosin heads were considered explicitly. Although the velocity dependence of two-headed attachment and processivity were not considered in detail in those studies, it appears that the fraction of second-head attachments is highest during shortening at intermediate velocity. Moreover, the models (7–9) suggest that when the two heads are attached simultaneously, they exist in different states (corresponding to those represented by A₁ and A₀² states in Scheme 2). An alternative possibility that could account for the apparent velocity dependence of the attachment rate is that shortening at intermediate velocity increases the concentration of cross-bridges that are capable of de novo force generation (56,66). It remains to be clarified whether such a mechanism can account for a range of other phenomena in muscle contraction.