Force transients and minimum cross-bridge models in muscular contraction
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DOI: 10.1007/s10974-008-9131-3
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- Kawai, M. & Halvorson, H.R. J Muscle Res Cell Motil (2007) 28: 371. doi:10.1007/s10974-008-9131-3
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Abstract
Two- and three-state cross-bridge models are considered and examined with respect to their ability to predict three distinct phases of the force transients that occur in response to step change in muscle fiber length. Particular attention is paid to satisfying the Le Châtelier–Brown Principle. This analysis shows that the two-state model can account for phases 1 and 2 of a force transient, but is barely adequate to account for phase 3 (delayed force) unless a stretch results in a sudden increase in the number of cross-bridges in the detached state. The three-state model \(({\mathbf{A \rightarrow B\rightarrow C\rightarrow A}})\) makes it possible to account for all three phases if we assume that the \({\mathbf{A\rightarrow B}}\) transition is fast (corresponding to phase 2), the \({\mathbf{B \rightarrow C}}\) transition is of intermediate speed (corresponding to phase 3), and the \({\mathbf{C \rightarrow A}}\) transition is slow; in such a scenario, states A and C can support or generate force (high force states) but state B cannot (detached, or low-force state). This model involves at least one ratchet mechanism. In this model, force can be generated by either of two transitions: \({\mathbf{B\rightarrow A}}\) or \({\mathbf{B\rightarrow C}}\). To determine which of these is the major force-generating step that consumes ATP and transduces energy, we examine the effects of ATP, ADP, and phosphate (Pi) on force transients. In doing so, we demonstrate that the fast transition (phase 2) is associated with the nucleotide-binding step, and that the intermediate-speed transition (phase 3) is associated with the Pi-release step. To account for all the effects of ligands, it is necessary to expand the three-state model into a six-state model that includes three ligand-bound states. The slowest phase of a force transient (phase 4) cannot be explained by any of the models described unless an additional mechanism is introduced. Here we suggest a role of series compliance to account for this phase, and propose a model that correlates the slowest step of the cross-bridge cycle (transition \({\mathbf{C\rightarrow A}}\)) to: phase 4 of step analysis, the rate constant k_{tr} of the quick-release and restretch experiment, and the rate constant k_{act} for force development time course following Ca^{2+} activation.
Key words
Le ChâtelierStep analysisSinusoidal analysisTensionKineticsRate constantsTwo-state modelThree-state modelPhosphateATPADPIntroduction
Mathematical symbols used; (t) indicates time-dependent variable; Sch = Scheme
Symbol | Section | Definition |
---|---|---|
A(t) | 2 | Concentration of cross-bridges in A |
A_{1} | 2 | Steady-state concentration of A |
A_{C} | 5 | Cross-sectional area |
A_{T} | 2, 18 | Total concentration of myosin S1 |
α | 2, 6 | Rate constant of detachment \(({\mathbf{A \rightarrow B}})\), Sch 1–3 |
α′ | 2, 6 | Rate constant of attachment \(({\mathbf{B \rightarrow A}})\), Sch 1–3 |
B(t) | 2, 6 | Concentration of detached cross-bridges |
B_{1} | 2 | Steady-state concentration of B |
β | 6 | Rate constant of attachment \(({\mathbf{B \rightarrow C}})\), Sch 3 |
β′ | 6 | Rate constant of detachment \(({\mathbf{C \rightarrow B}})\), Sch 3 |
C(t) | 6 | Concentration of attached cross-bridges in C |
C_{1} | 6 | Steady-state concentration of C |
D | 12 | MgADP concentration |
δ, δx | 3 | Perturbation |
δl | 4 | Stretch applied to a cross-bridge (δl > 0). δl < 0 for release |
δW | 4 | Work performed on a cross-bridge by stretch (δW > 0). δW < 0 for release. Eq. 16 |
E_{α} | 4 | Activation energy of \(\alpha\,({\mathbf{A \rightarrow B}})\), Fig. 4A |
E_{α}^{′} | 4 | Activation energy of \(\alpha^{\prime}\,({\mathbf{B\rightarrow A}})\), Fig. 4B |
ɛ | 12 | \(\varepsilon\equiv \quad K_{1}S/(1+K_{0}D+K_{1}S)\), Eq. 49 |
η | 19 | Step size |
F(t) | 5 | Force time course |
F_{0} | 19 | Isometric force at steady sate |
F_{A}(t) | 5 | Force contribution by state A |
F_{G}(t) | 5 | Force contribution by state G |
ϕ | 4, 9 | Unitary force, force/cross-bridge |
G(t) | 5 | Concentration of strained cross-bridges, Sch 2 |
G_{1} | 5 | Steady-state concentration of G |
γ | 6 | Rate constant of \({\mathbf{C \rightarrow A}}\), Sch 3 |
γ′ | 6 | Rate constant of \({\mathbf{A\rightarrow C}}\), Sch 3 |
H | A2 | Reaction matrix, Eq. A12 |
J | 8, 18 | Turnover rate (ATPase), Eqs. 40, 58 |
K_{0} | 12 | Association constant of MgADP, Sch 6 |
K_{1} | 11 | Association constant of MgATP, Sch 5 |
k_{1b} | 16 | Rate constant of step 1b, Sch 9 |
k_{−1b} | 16 | Reversal rate constant of step 1b, Sch 9 |
k_{2} | 11 | Rate constant of step 2, Sch 5 |
k_{−2} | 11 | Reversal rate constant of step 2, Sch 5 |
K_{2} | 12 | Equilibrium constant of step 2, \(K_{2}\equiv k_{2}/k_{-2}\) |
k_{4} | 13 | Rate constant of step 4, Sch 7 |
k_{−4} | 13 | Reversal rate constant of step 4, Sch 7 |
K_{4} | 15 | Equilibrium constant of step 4, \(K_{4}\equiv k_{4}/k_{-4}\) |
K_{5} | 13 | Association constant of Pi, Sch 7 |
k_{6} | 14, 18 | Rate constant of step 6, Sch 8 |
k_{−6} | 14 | Reversal rate constant of step 6, Sch 8 |
K_{6} | 14 | Equilibrium constant of step 6, \(K_{6}\equiv k_{6}/k_{-6}\) |
K_{α} | 7 | Equilibrium constant of \({\mathbf{A\leftrightarrow B}}\). \(K_{\alpha}\equiv\alpha/\alpha^{\prime}\) |
K_{β} | 7 | Equilibrium constant of \({\mathbf{B\leftrightarrow C}}\). \(K_{\beta}\equiv\beta/\beta^{\prime}\) |
k_{B} | 4 | Boltzmann’s constant, k_{B} = 1.381 × 10^{−23} JK^{−1} |
l_{0} | 5 | Half sarcomere length |
λ | 2 | Apparent rate constant, \(\lambda\equiv\alpha+\alpha^{\prime}\), Eq. 4 |
λ_{2} | 7, 11 | Apparent rate constant of phase 2. λ_{2} = 2πc. Eqs. 32, 44, 50 |
λ_{3} | 7, 13 | Apparent rate constant of phase 3. λ_{3} = 2πb. Eqs. 37, 53, 54 |
λ_{4} | 19 | Rate constant of force development, Eq. 63 |
M | 8, A2 | Eqs. 39, A16 |
μ | 5 | Perturbed rate constant α by stretch δl. \(\mu\equiv\alpha+\delta\alpha\) |
N_{A} | 5 | Avogadro’s number, N_{A} = 6.022 × 10^{23}/mole |
ν | 19 | Number of cross-bridge cycles in 1 sec |
P | 13 | Phosphate concentration |
2πa | 19 | Rate constant of process A (phase 4) |
2πb | 10 | Rate constant of process B (phase 3) |
2πc | 10 | Rate constant of process C (phase 2) |
q | 17 | Series compliance of half sarcomere |
R | 15 | Gas constant, \(R{\equiv}k_{B}N_{A}=8.314\,\hbox{JK}^{-1}\hbox{mol}^{-1}\) |
ρ | 5 | Cross-bridge stiffness |
ρ′ | 19 | Stiffness of half sarcomere |
S | 11 | Substrate (MgATP) concentration |
σ | 12 | \(\sigma\equiv K_{1} SK_{2}/[1+K_{0}D+K_{1}S(1+K_{2})]\), Eq. 52 |
S_{P} | 17 | Parallel stiffness of half sarcomere |
t | 2 | Time |
T | 4 | Absolute temperature |
T_{0} | 15 | Tension supported by X_{0} (AM.ADP) |
T_{1} | 15 | Tension supported by X_{1} (AM), Eq. 55 |
T_{1b} | 15 | Tension supported by X_{1b} (AM\(^{\dagger}\)ATP) |
T_{2} | 15 | Tension supported by X_{2} (AM*ATP), Eq. 55 |
T_{34} | 15 | Tension supported by X_{34} (Det), T_{34} = 0 |
T_{5} | 15 | Tension generated/supported by X_{5} (AM*ADP.P) |
T_{6} | 15 | Tension generated/supported by X_{6} (AM*ADP) |
T_{C} | 15 | Tension of standard activation, C=control |
U | A2 | 3x3 eigen matrix consisting of 3 eigen (column) vectors. Eq. A20 |
X_{0}(t) | 12 | Probability of cross-bridges at AM.ADP |
X_{1}(t) | 11 | Probability of cross-bridges at AM |
X_{1b}(t) | 11 | Probability of cross-bridges at AM\(^{\dagger}\)ATP |
X_{2}(t) | 11 | Probability of cross-bridges at AM*ATP |
X_{34}(t) | 14 | Probability of cross-bridges at Det state |
X_{5}(t) | 13 | Probability of cross-bridges at AM*ADP.Pi |
X_{6}(t) | 13 | Probability of cross-bridges at AM*ADP |
X_{a}(t) | 12 | Probability of strongly attached cross-bridges, Eq. 57, \(X_{a}\equiv X_{0}+X_{1}+X_{2}+X_{5}+X_{6}\) |
Y_{a} | 17 | Stiffness of cross-bridges in half sarcomere when all are strongly attached |
\(\zeta\) | 21 | \(\zeta\equiv K_{5}P/(1+K_{5}P)\), Eq. 73 |
The two-state model
Thus, the first-order reaction of Scheme 1 results in a time course with one exponential process whose rate constant is λ, which is the sum of the forward rate constant α and the reverse rate constant α′ (Eq. 4). λ is termed the “apparent” rate constant, indicating that this is the rate constant experimentally observed as in Eq. 5 or 6. In contrast α and α′ are termed the “intrinsic” or “fundamental” rate constants (Gutfreund 1995). Although the adjective “intrinsic”, “fundamental” or “apparent” is often left out, which of these would apply is usually self evident. A_{0} is an integration constant that is determined by the initial condition, and it is termed the amplitude (or magnitude) of the exponential process in Eqs. 5 and 6.
As time approaches infinity (\(t\rightarrow\infty\)), Scheme 1 achieves an equilibrium, resulting in dA(t)/dt = 0 (in Eqs. 1 and 3). From Eqs. 5 and 6, it becomes clear that A(∞) = A_{1} and B(∞) = B_{1}. Thus, \(K\equiv\alpha/\alpha^{\prime}=B_{1}/A_{1}\) is the equilibrium constant, and A_{1} and B_{1} are the equilibrium concentrations of individual molecular species. K = B_{1}/A_{1} is the well-known law of mass action.
Perturbation
The changes described in Eqs. 8–10 are assumed to occur simultaneous to perturbation δx, with δα being the sensitivity of α, δα′ the sensitivity of α′, and δλ the sensitivity of λ, to perturbation δx. Because δx is a small quantity, all other δs are also small quantities.
Conclusion 1. An experimenter will observe the same rate constant λ whether starting a new reaction as in Scheme 1 (Eq. 5), or applying a small perturbation (of any kind) during equilibrium and measuring the transient as in Eq. 14. Although not shown explicitly, this conclusion holds true for any multiple-state model.
Tension transients and delayed tension
The expanded two-state model
According to the above analysis, the length change can be seen to induce an extra state G,which decays with a unique rate constant (μ). Steiger and Abbott (1981) used this line of modeling on a four-state model (with two attached and two detached states after Lymn and Taylor 1971), to generate one with six states. Although we believe that Scheme 2 and its theory represent one of the most rigorous systems for treating the stretched (or released) cross-bridges, they may not allow for practical analysis of experimental results because the apparent rate constants λ and μ are similar and, hence, difficult to distinguish from one another. In addition, modeling must include one extra cross-bridge state per attached cross-bridge state, which adds to the complexity of the algebra required when multiple attached states are considered. For these reasons, we do not use this line of approach in the following discussions.
The three-state model
This is a cyclic scheme with the forward reaction consisting of the outer clockwise cycle: \({\mathbf{A\rightarrow B\rightarrow C\rightarrow A}}\), for which the respective rate constants are α,β and γ. The reversal reaction consists of the inner counter-clockwise cycle: \({\mathbf{A\rightarrow C\rightarrow B\rightarrow A}}\), with the respective rate constants γ′, β′ and α′. In Scheme 3, we assume that A and C are attached states, which generate and/or support force, and that B is a detached state, which does not support force. Because the analysis of Scheme 3 is very complex, we will discuss a simple case here. The complete analysis of Scheme 3 can be found in the Appendix 2.
Three-state model, a simple case analysis (α + α′ ≫ β + β′ ≫ γ + γ′)
Conclusion 2. The apparent rate constant of a fast step (λ_{2}) is the sum of the forward rate constant and the reversal rate constant of this step (Eq. 32), and it is not influenced by either the rate or equilibrium constants of slower steps.
Conclusion 3. The apparent rate constant of a medium-speed step (λ_{3}) is a weighted sum of the intrinsic rate constants β and β′ of step 2; although λ_{3} is influenced by the equilibrium constant of the fast steps, it is not influenced by the rate or equilibrium constants of the slow steps.
Thus, the three-state model (Scheme 3) has 2 exponential processes with 2 apparent rate constants, λ_{2} and λ_{3} (λ_{2} > λ_{3}) (generally, an N-state model has (N−1) exponential processes). Each apparent rate constant is the weighted sum (linear combination) of the forward and reversal rate constants of the respective steps (Eqs. 32 and 37). The time courses A(t), B(t) and C(t) take the form described in Eqs. A21-A23 of Appendix 2. Step 3 occurs slowly, which is important for calculation of the turnover rate. In perturbation analysis, the rate constants of the fast steps (1 and 2) can be observed (Eqs. 32 and Eq. 37), whereas the rate constant of the slowest step (3) cannot be observed. However, because of the presence of in-series compliance, the forward rate constant of step 3 can be observed as discussed later in Section 19.
Steady state
In the case of the cross-bridge cycle, J is the same as the ATP hydrolysis rate. The equilibrium is a special case (J = 0) of the steady state. If αA_{1}−α′B_{1} or βB_{1}−β′C_{1} is evaluated in place of Eq. 40, both become 0 because of the approximations we used. In this case, B_{1}/A_{1} = α/α′ = K_{α}, and C_{1}/B_{1} = β/β′ = K_{β}. Hence, step 1 and step 2 can be approximated by the equilibrium. However, although C_{1}/A_{1} = αβ/α′β′ = K_{α}K_{β}, generally A_{1}/C_{1}≠γ/γ′. Hence, step 3 does not contribute to the equilibrium of the reaction \(\hbox{C}\leftrightarrow\hbox{A}\) (step 3) directly, but rather steps 1 and 2 (reactions \(\hbox{A}\leftrightarrow\hbox{B}\leftrightarrow\hbox{C}\)) determine the equilibrium.
Conclusion 4. At steady state, the fast steps of the cyclic reaction can be approximated by the equilibrium, whereas the slowest (rate-limiting) step cannot be approximated by the equilibrium.
The mechanisms of tension transients that satisfy the Le Châtelier–Brown Principle
The next thing that happens is an increase in the slower step 2 \(({\mathbf{B\rightarrow C}})\), because its rate is βB, and the number (B) of cross-bridges in state B is now larger than steady-state (δB_{1} > 0: Eq. 43). The stretch also increases the elastic energy in state C, which in turn reduces the activation energy E_{β}^{′}. Hence β′ is also accelerated (or E_{β} is elevated and β is diminished). Consequently, the reversal reaction of step 2 \(({\mathbf{C \rightarrow B}})\) is accelerated. The net reaction is the sum of those in both directions, with the rate constant λ_{3} (Eq. 37). When the rate βB\(({\mathbf{B\rightarrow C}})\) is greater than the rate β′C\(({\mathbf{C\rightarrow B}})\), we observe a delayed rise in tension (phase 3, Figs. 2 and 3A); when the rates are equal, we observe a plateau (phase 3, Fig. 1A); when βB is less than β′C, we do not observe the delayed rise in tension. The above-mentioned ratchet mechanism is important here, because the projected rate β′C is large and comparable to βB. In addition, there may be a ratchet mechanism in step 2 to decelerate the reversal reaction \({\mathbf{C\rightarrow B}}\). Because the amplitude (magnitude B in sinusoidal analysis) of phase 3 is smaller in frog semitendinosus fibers than that in rabbit psoas fibers under the same activating condition (Kawai and Brandt 1980), phase 3 in frog fibers shows a plateau in step analysis (Fig. 1A). The amplitude is sizable in rabbit psoas fibers. Hence a delayed rise in tension is clearly visible (Fig. 2). We have also noticed that the presence of exponential process B (phase 3) depends on the goodness of the fiber preparation; if it is poor, phase 3 magnitude is less, or it even disappears.
A mirror image of the above events occurs on release (δl < 0) of the active muscle fibers. On release, force drops (phase 1, Figs. 1B and 3B) due to simultaneous release of the combined elasticity of series compliance and the attached cross-bridges. This results in a decrease in the elastic energy stored in state A (Eq. 16), which in turn increases E_{α} (Eq. 17) and reduces α (Eq. 20) (or reduces E_{α}′ and increases α′) to satisfy the Le Châtelier–Brown Principle. Thus, the rate αA decreases (or α′B increases). Because there is a continuous supply of A via \({\mathbf{B\rightarrow A}}\) and \({\mathbf{C\rightarrow A}}\) transitions, the net effect is a transient increase (accumulation) of state A, and a transient decrease of state B. Overall, this results in an exponential rise of force (phase 2, Fig. 1B), at a rate constant λ_{2} (Eq. 32). Step 2 is slower and does not take place yet. The result over time is a temporary decrease in the cross-bridges of the B state and, as seen from Eq. 43, this decrease is substantial. The next event is the slower step 2. The rate of its forward reaction \(\beta B\,({\mathbf{B\rightarrow C}})\) is now reduced because B is reduced. The elastic energy stored in state C decreases with the release of length; hence the activation energy is increased and the rate constant β′ reduced, resulting overall in a decline of the rate of the reversal reaction \(\beta^{\prime}C\,({\mathbf{C\rightarrow B}})\). Again, the net reaction is the sum of those in both directions, with rate constant λ_{3}. When rate \(\beta^{\prime}C\,({\mathbf{C\rightarrow B}})\) is greater than rate \(\beta B\,({\mathbf{B\rightarrow C}})\), we observe a delayed drop in tension (phase 3, Fig. 3B); when the rates are equal, we observe a plateau (phase 3, Fig. 1B); and when rate \(\beta^{\prime}C\,({\mathbf{C\rightarrow B}})\) is weaker than rate \(\beta B\,({\mathbf{B\rightarrow C}})\), we do not observe a delayed drop in tension. Recently, Burton et al. (2006) presented a case in which the polarity of phase 3 and 4 was in the same direction when a large (1%) release in rabbit psoas fiber was tested. Thus, polarity of phase 3 may be determined by the experimental conditions.
In all cases, the transitions satisfy the Le Châtelier–Brown Principle. From the above discussion, it is clear that phase 2 represents \({\mathbf{A\leftrightarrow B}}\) interconversion (step 1), with the fast apparent rate constant of λ_{2}, and that phase 3 represents \({\mathbf{B\leftrightarrow C}}\) interconversion (step 2), with the medium apparent rate constant λ_{3} as indicated in Scheme 3. What it is important to notice here is that both phase 2 of stretch and phase 2 of release arise from the same elementary step of the cross-bridge cycle, because tension transients on stretch and release are almost in mirror image to each other (Figs. 1 and 3). Similarly, phase 3 of stretch and phase 3 of release arise from the same elementary step. Due to energetic considerations (Eqs. 16–21), however, the resulting rate constants for stretch and release are asymmetric. This is referred to as “non-linearity” of the rate constants (Huxley and Simmons 1971), and accounts for the fact that tension transients are not in perfect mirror image on stretch and release, as shown in Fig. 1A and 1B. From the above discussions, it can be concluded that tension develops (increases) by two mechanisms: through the fast \({\mathbf{B\rightarrow A}}\) transition (phase 2) that occurs on release (counter-clockwise cycle in Scheme 3; Fig. 1B), and through the medium-speed \({\mathbf{B\rightarrow C}}\) transition (phase 3) that occurs on stretch (clockwise cycle; Figs. 2 and 3A).
Conclusion 5. The three-state model (Scheme 3) can explain phase 2 and phase 3 of force transients without violating the Le Châtelier–Brown Principle, provided that step 3 is the rate-limiting step, the middle state (B) is a low- or no-force state, and there is a ratchet mechanism to increase the \({\mathbf{A\rightarrow B}}\) transition or to diminish the \({\mathbf{C\rightarrow B}}\) transition in response to stretch. Both states A and C must bear force, and step 1 must be faster than step 2.
In performing perturbation experiments in muscle fibers and probing the elementary steps of the cross-bridge cycle, it is important to keep the length change small. We typically use a sinusoidal length change with a 0.25% peak-to-peak amplitude, which corresponds to 1.9 nm (or ±0.95 nm) at the cross-bridge level when 40% series compliance is considered. Because this is smaller than the step size (4–11 nm: Molloy et al. 1995; Guilford et al. 1997; Kitamura et al. 1999), the elementary steps can be probed. A 10% length change, by contrast, corresponds to 75 nm at the cross-bridge level, and the cross-bridge would have to cycle many (7 to 19) times to compensate for this large length change. The time course of such an experiment would be limited by the slowest step of the cycle. As is seen in Eq. 43, even a small length change has a strong effect on cross-bridge occupancy, and therefore results in a large change in force.
Effects of MgATP and phosphate (Pi)
The next question is whether, in Scheme 3, cross-bridges cycle in the clockwise direction or in the counter-clockwise direction to consume ATP and to perform work. To answer this, we need to look into the effects the substrate (MgATP) and product (phosphate and MgADP) concentrations have on the apparent rate constants.
In rabbit psoas fibers, an increase in the the MgATP concentration was shown to accelerate the rate constants of phases 2 and 3 in sinusoidal length-change experiments (Kawai 1978). In sinusoidal analysis, the apparent rate constant of phase 2 is termed 2πc, and that of phase 3 is termed 2πb (see Scheme 3). Because the temporal resolving power (signal-to-noise ratio) is greater in sinusoidal length-change experiments than in step length-change experiments, we adapted this technique. According to conclusions 2 and 3 above, the effect of ATP implies that the MgATP binding is associated with a fast reaction step whose rate constant is comparable to 2πc. We have also found that an increase in the Pi concentration accelerates the rate constant of phase 3 (2πb), but that Pi does not significantly affect the rate constant of phase 2 (2πc) (Kawai 1986). In light of conclusion 3, this observation implies that the Pi binding step is coupled to a medium-speed step with a rate constant comparable to 2πb. If we fit these results to Scheme 3, then it follows that ATP binding is associated with step 1, whereas Pi binding is associated with step 2. These possibilities are considered quantitatively in the following Sections 11–13.
Substrate-binding step with a conformational change
Kinetic constants of the cross-bridge cycle of Scheme 8
Animal and muscle | K_{0}, mM^{−1} | K_{1}, mM^{−1} | k_{1b}, s^{−1} | k_{−1b}, s^{−1} | k_{2}, s^{−1} | k_{−2}, s^{−1} | k_{4}, s^{−1} | k_{−4}, s^{−1} | K_{5}, mM^{−1} | Perturbation | Source |
---|---|---|---|---|---|---|---|---|---|---|---|
Rabbit psoas | 2.8 | 1.4 | 1530^{a} | 1610^{a} | 440 | 100 | 56 | 129 | 0.069 | Length | Kawai and Halvorson (1991) |
Rabbit psoas | – | – | – | – | – | – | 79.2 | 114.7 | 0.27 | Pi | Dantzig et al. (1992) |
Rabbit psoas, 15°C | – | – | – | – | – | – | 27 | 115 | 0.164 | Pi | Walker et al. (1992) |
Rabbit psoas, 12°C | – | – | – | – | – | – | ∼15^{b} | ∼36^{b} | 0.255 | Pressure | Fortune et al. (1991) |
Rabbit AM, type IIB | 5 | 0.84 | – | – | 526 | 328 | 143 | 81 | 0.26 | Length | Galler et al. (2005) |
Rabbit AM, type IID | 18 | 4.9 | – | – | 352 | 121 | 58 | 63 | 0.16 | Length | Galler et al. (2005) |
Rabbit EDL and soleus, type IIA | – | 8.7 | – | – | 198 | 51 | 13.6 | 13.6 | 0.18 | Length | Galler et al. (2005) |
Drosophila melanogaster, indirect flight, 15°C | – | 0.19 | – | – | 3698 | 8 | 1778 | 11 | – | Length | Swank et al. (2006) |
Lethocerus colossicus,indirect flight | – | 0.7 | – | – | 900 | 180 | – | 150^{c} | 0.13^{c} | Length | Marcussen and Kawai (1990) |
Rabbit soleus, type I | 18 | 1.2 | 90 | 100 | 21 | 14.1 | 5.7 | 4.5 | 0.18 | Length | Wang and Kawai (1997) |
Ferret cardiac | – | 0.99 | 270 | 280 | 48 | 14 | 11 | 107 | 0.060 | Length | Kawai et al. (1993) |
Porcine cardiac | 80 | 10.6 | – | – | 13.0 | 9.1 | 3.2 | 10.5 | 0.104 | Length | Zhao and Kawai (1996) |
Bovine cardiac, 25°C | – | 9.1 | – | – | 26.6 | 12.1 | 7.1 | 12.6 | 0.14 | Length | Fujita et al. (2002) |
From the above discussion, it is clear that X_{1} and X_{2} correspond to state A in Scheme 3, and that X_{3} corresponds to state B. This assignment cannot be transposed because that would require that the slow rate constant, λ_{3}(= 2πb), decreases as S increases, yet 2πb was found to increase concomitant with an increase in tHe MgATP concentration (Kawai 1978; Kawai and Zhao 1993). This assignment demonstrates that a cross-bridge cycles in the clockwise direction in Scheme 3. With regard to the ratchet mechanism discussed in Section 9, we have experimental evidence that K_{1} (ATP binding) becomes larger as isometric tension is decreased by various experimental manipulations (Zhao et al. 1996), indicating that cross-bridges detach more readily as load is reduced. This mechanism is consistent with the Fenn effect (Fenn 1923).
ADP as a competitive inhibitor of ATP
Therefore, Scheme 6 includes an exponential process with an apparent rate constant λ_{2}, as defined by Eq. 50. λ_{2} decreases hyperbolically as D is increased. Fitting the data from the ADP study in rabbit psoas fibers at 20°C to Eq. 50 led us to deduce that K_{0} = 2.8 mM^{−1} (Kawai and Halvorson 1989) (Table 2). In comparison, solution studies of S1 performed at 20–23°C yielded K_{0} = 3.7 mM^{−1} (Bagshaw et al. 1974). K_{0} is least important among all kinetic constants, however, because D is minimal (0.01–0.02 mM) in the presence of CP/CK (Meyers et al. 1985). Hence X_{0} < 0.01 and few cross-bridges are distributed in the AMD state.
The above discussion makes it clear that X_{0}, X_{1} and X_{2} correspond to state A in Scheme 3, and that X_{3} corresponds to state B in Scheme 3. Specifically, X_{1} corresponds to AM (A = actin. M = myosin), to which no nucleotide is bound. X_{2} corresponds to AM*ATP, and X_{0} to AM.ADP. X_{3} corresponds to both the weakly attached AM.ATP and the detached M.ATP. Recently, Palmer et al. (2007) showed a cross-bridge model in which 2πc corresponds to the detachment step (\(\mathbf{X}_{2}\leftrightarrow\mathbf{X}_{3}\) interconversion), based on the stochastic attachment and detachment cycle of cross-bridges.
Phosphate release step with a conformational change
The above discussion makes it clear that X_{4} corresponds to state B, and X_{5} and X_{6} to state C, of Scheme 3. This assignment cannot be transposed, because from that it would follow that an increase in the Pi concentration results in an increase in force (state C), which is opposite to what was observed (Cooke and Pate 1985; Kawai 1986). A transposition of this assignment would also contradict to the previous assignment of the ATP binding step (cross-bridges cycle in the clockwise direction in Scheme 3), because Pi must be released from the cycle. Consequently, X_{4} corresponds to the detached M.ADP.Pi state and the weakly attached AM.ADP.Pi state; X_{5} corresponds to the strongly attached AM*ADP.Pi state, and X_{6} corresponds to the AM*ADP state. It must be pointed out here that the AM*ADP state (X_{6}) is completely distinct from the AM.ADP state (X_{0}), which is consistent to solution studies (Sleep and Hutton 1980).
Equations 53 or 54 has been used extensively to analyze cross-bridge kinetics in rabbit skeletal muscle fibers (Fortune et al. 1991; Kawai and Halvorson 1991; Dantzig et al. 1992; Millar and Homsher 1992; Walker et al. 1992; Wang and Kawai 1997; Galler et al. 2005). With sinusoidal analysis, k_{4} = 56 s^{−1}, k_{−4} = 129 s^{−1}, and K_{5} = 0.069 mM^{−1} were deduced for rabbit psoas fibers at 20°C (Kawai and Halvorson 1991). These values compare with those from caged Pi experiments (k_{4} = 79.2 s^{−1}, k_{−4} = 114.7 s^{−1}, K_{5} = 0.27 mM^{−1}; Dantzig et al. 1992, 20°C) as well as pressure release experiments (k_{4} ∼ 15 s^{−1}, k_{−4} ∼ 36 s^{−1}, K_{5} = 0.255 mM^{−1} at 12°C: deduced from Fortune et al. 1991; see note b of Table 2 legend), and are summarized in Table 2. If the data of the pressure-release experiment are extrapolated to 20°C using the Q_{10} values reported for k_{4} (6.8), k_{−4} (1.6), and K_{5} (1.1) (Zhao and Kawai 1994), then k_{4} = 69 s^{−1}, k_{−4} = 52 s^{−1}, and K_{5} = 0.275 mM^{−1} will result. If the data of the caged Pi experiment by Walker et al. (1992) (Table 2) are extrapolated to 20°C using the same Q_{10} values, then k_{4} = 70 s^{−1}, k_{−4} = 145 s^{−1}, and K_{5} = 0.172 mM^{−1}. As seen with these numbers, approximate agreement was achieved based on three different techniques from four laboratories. K_{5} deduced from sinusoidal analysis is somewhat (2.5–4 ×) smaller, primarily because Fortune et al. (1991), Dantzig et al. (1992), and Walker et al. (1992) fitted the rate constant results to Eq. 53, whereas Kawai and Halvorson (1991) fitted the Pi dependence data of both rate constant (Eq. 54) and amplitude simultaneously. In slow-twitch fibers of rabbit soleus, Millar and Homsher (1992) found: k_{4} = 1.96 s^{−1} and the second order Pi binding constant K_{5}k_{−4} = 1.994 mM^{−1} s^{−1} when using caged Pi at 20°C. These numbers compare well to those of Wang and Kawai (1997) in Table 2: k_{4} = 5.7 s^{−1}, k_{−4} = 4.5 s^{−1}, K_{5} = 0.18 mM^{−1}, hence K_{5}k_{−4} = 0.81 mM^{−1} s^{−1}. k_{4} is usually larger in our case, because of the inclusion of σ (Eq. 54) that accounts for fast equilibria to the left of Scheme 7. Because Pi released by the photolysis of caged Pi is limited to 2–3 mM, it is difficult to probe the saturation phase using this technique (K_{5} = 0.18 mM^{−1} indicates that half saturation occurs at 5.5 mM Pi). Although Eq. 54 demonstrates a sensitivity to S through σ, this equation has not been used to deduce K_{1} or K_{2} because of uncertainties about the ATP cleavage step (step 3), which skinned-fiber studies have not been able to resolve (see Section 14).
Conclusion 6. When the effects of ATP, ADP and Pi are considered, the only model consistent to the Le Châtelier–Brown Principle involves assigning phase 2 to steps surrounding the ATP binding step, and assigning phase 3 to steps surrounding the Pi release step. Consequently, cross-bridges cycle in the clockwise direction in Scheme 3, consuming ATP, generating force, and releasing Pi and ADP. Therefore, the transient force increase in phase 2 that was seen by Huxley and Simmons (1971) following step-length release is not the result of energy transduction (that requires ATP), but of a transient accumulation of the number of cross-bridges in the A state (which includes both the AM and AM*ATP species). Phase 2 was not correctly assigned because Huxley and Simmons (1971) used intact fibers, and hence did not examine the effects of ATP or Pi concentration.
Combined cross-bridge model
Note that the steps are renumbered. These are called “elementary steps” or “fundamental steps” (Gutfreund 1995). In this scheme, states X_{3} and X_{4} are equated and represented as X_{34} for simplicity. In reality, \({{\mathbf{X}}_{3}}\leftrightarrow{{\mathbf{X}}_{4}}\) (step 3) corresponds to the ATP cleavage step (Lymn and Taylor 1971; Bagshaw et al. 1974; Xu et al. 2003), which is difficult to resolve in skinned fiber studies as they depend on strongly attached cross-bridges. Therefore, cross-bridge states before and after the cleavage of ATP are merged together and referred to as the Det (detached) state. The Det state includes both the weakly attached states (AM.ATP, AM.ADP.Pi) and the truly detached states (M.ATP, M.ADP.Pi). Because weakly attached states are based on ionic interactions (Brenner et al. 1982), at 200 mM ionic strength the majority of Det consists of truly detached states. An accounting of step 3 is one remaining problem that should be solved by skinned fiber experiments in the coming years. What we know already is that step 3 is not very much slower than 2πb (λ_{3}), because if it were slower, then the rate constant 2πb would not be sensitive to the ATP concentration. Therefore, step 3 cannot be rate-limiting for the entire cross-bridge cycle as once suggested (Stein et al. 1985) based on four-state model of Lymn and Taylor (1971). Based on solution studies, it is known that in fast-twitch muscles at 20°C, k_{3} and k_{−3} are on the order of 30–120 s^{−1} (Bagshaw et al. 1974; Xu et al. 2003). The fact that this is similar to the rate constants of step 4 complicates the problem.
Step 6 is the slowest step in the cycle, and can be considered the “rate-limiting” step. If k_{6} + k_{−6} is similar to or faster than k_{4} + k_{−4}, then it follows that an increase in [MgATP] reduces 2πb and an increase in [MgADP] increases 2πb, both of which are contrary to the experimental evidence (Kawai 1978, 1986). Therefore, k_{6} + k_{−6} < k_{4} + k_{−4}, which makes step 6 the slowest in the cross-bridge cycle. The product of step 6 can be either X_{0} (as shown in Scheme 8) or can be converted directly to X_{1} by ADP release. Experiments in solution carried out by Sleep and Hutton (1980) indicated that K_{6}∼50 (i.e., k_{−6} ≪ k_{6}), and that reversal of the X_{0} state to the X_{6} state is possible (shown in a dashed arrow in Scheme 8). That is, X_{0} is likely to be an intermediate product of the hydrolysis pathway, and a part of the cross-bridge cycle.
An examination of Scheme 8 reveals that collision complex formation is followed by a conformational change in three places: (1) step 1 (ATP binding) to step 2, (2) the reversal of step 5 (Pi binding) to step 4, and (3) the reversal of step 0 (ADP binding) to step 6. The conformational change presumably accompanies a change in the force on a cross-bridge and secures ligand binding. This is consistent with the suggestion from Huxley (1980), that chemical reaction (collision complex formation) and mechanical reaction (conformational change) alternate in the cross-bridge cycle. Because these sequential reactions are difficult to resolve, they are frequently merged together. Therefore, the “Pi release step” may include both steps 4 and 5. In fact, the strongly attached AM*ADP.Pi state was not recognized in solution studies. Hence, in solution systems the conformational change (step 4) must have been merged together and referred to the “Pi release step”. Similarly, the “ADP release step” may include both steps 6 and 0. It has been often stated that the ADP release step is slow, but what that really means is that the preceding conformational change (step 6) is slow. The ADP release step itself (step 0) is a reversal of collision complex formation and, hence, cannot be slow. Similarly, “ATP binding” may include steps 1 and 2.
For the steady-state calculations, step 0 through step 5 can be approximated by equilibrium, and X_{i} can be calculated from five equilibrium constants (K_{i}) as formulated by Eq. 18 of Kawai and Halvorson (1991), except when the Pi concentration becomes small (such as < 2 mM). In this case, X_{6} becomes large, increasing the turnover rate k_{6}X_{6}, which contributes significantly to the steady-state probability of all Xs, hence calculation of X_{i} needs k_{6} (Section 18) and all other rate constants in Scheme 8. The rate-limiting step (step 6) cannot be approximated by equilibrium, as discussed in Sections 8 and 18 and in conclusion 4.
Force generation step, isometric tension, and work performance
Solution studies have shown that the Pi release step accompanies a large reduction in free energy (\({\Updelta}G^{\circ}\)), amounting to ∼1/2 the energy generated by ATP hydrolysis (White and Taylor 1976). Hence, this step is almost irreversible (Sleep and Hutton 1980) (note: \({\Updelta}G^{\circ}=-RT\) log K, where R is the gas constant and K is the equilibrium constant), requiring 10–100 M Pi for reversal (Taylor 1979). This is why White and Taylor (1976) suggested that the Pi release step is associated with the force generation step. Skinned-fiber studies, in contrast, demonstrated that even at mM levels, Pi can diminish isometric tension (Cooke and Pate 1985; Kawai 1986), which indicates that the Pi release step is reversible. This reversibility has been demonstrated through ^{18}O exchange experiments (Webb et al. 1986), and by equilibrium constant measurements showing that K_{4} = 0.43 (step 4), and \(1/K_{5}P\, \cong\,1.8\) (step 5 at 8 mM Pi) based on values listed in Table 2 for rabbit psoas fibers (Kawai and Halvorson 1991). Therefore, \(\Updelta G_{4}^{\circ}=+2.0\) kJ/mole and \({\Updelta}G_{5}^{\circ}=-1.5\) kJ/mole, and their absolute values are less than RT (= 2.44 kJ/mole), hence there is not much change in the free energy in steps 4 or 5. It is likely that in skinned fibers, the free energy of ATP hydrolysis is stored in the contractile apparatus as elastic (mechanical) energy, which can be sensed as force, and can be readily reversed to chemical energy as the Pi concentration is elevated in the mM range. In the solution system, in contrast, a great deal of the liberated free energy is lost as heat. Hence, such a reaction cannot be reversed.
X_{5} (AM*ADP.Pi) is a collision complex between P and X_{6} (AM*ADP). Tension supported (or generated) by X_{5} and X_{6} must be the same, i.e., T_{5} = T_{6} because the conformation of a protein (or proteins) cannot change instantly as the collision complex is formed. If the conformation is the same, the tension must be the same. Because T_{34} = 0, force must be generated at step 4, with the transition of X_{34} to X_{5}. Experimental evidence in support of this insight comes from an evaluation of force as the function of the Pi concentration (Fig. 5) (Kawai and Halvorson 1991; Kawai and Zhao 1993; Wang and Kawai 1997; Ranatunga 1999; Tesi et al. 2002); force does not suddenly decrease as Pi is bound to cross-bridges (Dantzig et al. 1992; Takagi et al. 2004). If additional force is generated upon ADP release, as reported in the case of smooth muscles (Rosenfeld et al. 2001), then it follows that T_{6} < T_{0}, T_{0} < T_{1}, or T_{6} < T_{1}. However, there is no evidence for this in skeletal muscle fiber studies as shown in Fig. 5.
An examination of Fig. 5 shows that forces supported by AM.ADP and AM are not significantly different (T_{0}≈T_{1}). This observation demonstrates that AM.ADP is indeed the collision complex. However, force supported by AM*ATP is significantly less than that supported by AM, indicating that a conformational change may occur by the time AM*ATP is formed. This supposition is strengthened by the finding of yet another state \(\hbox{AM}^{\dagger}\hbox{ATP} ({\mathbf{X}}_{1b})\) between the AM and AM*ATP states, as discussed in Section 16 below. In this case, the tension data as a function of S fit well to the model if we assume that T_{1} = T_{1b} (Kawai and Zhao 1994; Wang and Kawai 1997).
Exponential process D and one extra cross-bridge state
Stiffness
ATP hydrolysis rate and the rate constant of step 6
Series compliance, force development, and the slowest step in the cross-bridge cycle
This system has one exponential process with an apparent rate constant of λ_{4} and amplitude F_{0}. Note that F(0) = 0 is assumed. Equation 65 would be the form of a force time course in which a fiber is suddenly activated by Ca^{2+}, or in which the length of the active fiber is suddenly released to zero load and restretched to its original length (resulting in force redevelopment like that seen by k_{tr} measurement). The exponential function (Eq. 65) may be a good approximation for describing the time course data. In fact, available data are consistent with the exponential function (e.g. Regnier et al. 1995; Stehle et al. 2002; Piroddi et al. 2003).
Equation 68 is a hyperbolically increasing and saturating function with respect to P, which is similar to Eqs. 53 and 54. Thus, an increase in P results in the increase in λ_{4} via Eq. 63 and as reported (Regnier et al. 1995; Tesi et al. 2000). Similarly, an increase in S results in the increase in λ_{4} via \(S\rightarrow \sigma\rightarrow\lambda_{3}\rightarrow \nu_{0}\rightarrow \lambda_{4}\) (see Eqs. 52, 54, 66, 63, respectively).
Several colleagues in the field have suggested that exponential process A of sinusoidal analysis (and hence phase 4 of step analysis) may represent the slowest step of the cross-bridge cycle (step 3 in Scheme 3, and step 6 in Scheme 8). Until recently, we have discounted this possibility because the three-state model, as represented in Scheme 3, can have only two exponential processes (see Appendix 2 for an analysis of a general case) and, in principle, the rate constants of the slowest step cannot be detected by the perturbation analysis method. The same is true for Scheme 8. However, the introduction of series compliance changes this outlook significantly. As shown above, series compliance may introduce an extra exponential process with a slow rate constant, such as 2πa of process A, which is present in all three subtypes of fast-twitch muscle fibers examined (Galler et al. 2005). In contrast, the amplitude of process A (phase 4) is very small or even absent in the cases of insect indirect flight muscles (Fig. 3) (Thorson and White 1969; Abbott 1973; Pringle 1978; Marcussen and Kawai 1990; Swank et al. 2006), myocardium (Saeki et al. 1978, 1991; Zhao and Kawai 1996; Wannenburg et al. 2000; Fujita et al. 2002), and covalently but partially cross-linked rabbit psoas fibers (Tawada and Kawai 1990). In slow-twitch fibers, process A is present but its amplitude is small (Kawai and Schachat 1984; Wang and Kawai 1996, 1997). These observations suggest that the sarcomere structure is more rigid in these muscle preparations than in fast-twitch skeletal muscle fibers, minimizing the amplitude of process A. In fact, in insect muscle fibers, an extra C-filament is present to secure the connection between the thick filament and the Z-line, and to stabilize the sarcomere structure (White 1983). In cardiac muscles, there is a possibility that myosin binding protein C (MyBP-C) reaches the thin filament (Spirito et al. 1997; Squire et al. 2003), which may give extra stability to sarcomeres. In partially cross-linked rabbit psoas fibers, the extra covalent linkage (18% of cross-bridges) was created to stabilize the preparation, causing process A to disappear (Tawada and Kawai 1990).
Conclusion 7. The apparent rate constant of tension development (k_{act}, k_{tr}) or of process A of sinusoidal analysis (phase 4 of step analysis) appears to be proportionate to the rate constant of the slowest step of the cross-bridge cycle (k_{6}), with the proportionality constant ρ′η_{0}/F_{0} (Eq. 63). This proportionality exists because of the presence of in-series compliance in sarcomeres.
More realistic conditions
In the discussion in Sections 7–18, the condition (α + α′ ≫ β + β′ ≫ γ + γ′) was used for the sake of simplicity. However, what would happen if this condition were not met, and if the conditions were merely α + α′ > β + β′ > γ + γ′ as suggested from the kinetic constants listed in Table 2? The answer to this question is simply that the approximation becomes less accurate as these numbers get closer to each other, and that the analysis may become qualitative rather than quantitative. That is, there will be an increasing contribution of steps 4–5 to the apparent rate constant λ_{2} of phase 2, with the Pi concentration consequently having increasingly more influence on phase 2. The conclusions derived then become qualitative, and their usefulness depends on one’s own expectations. They include the assignment of phase 2 and its rate constant λ_{2} to step 2 of Scheme 8, and the assignment of phase 3 and its rate constant λ_{3} to step 4 of the same scheme. If, however, one is not satisfied with this approximation, then an exact derivation can be undertaken according to Appendix 2. Alternatively, one could use the method used to analyze the data from experiments in myocardium (Section 21), as this does not rely on approximation.
Cardiac muscle fibers
Martin and Barsotti (1994) used caged ATP on guinea pig heart muscle, and deduced the second order ATP binding constant (our K_{1}k_{2}) to be ∼39 mM^{−1} s^{−1} at 21°C. This number compares with the values we have calculated from Table 2: ferret myocardium 48 mM^{−1} s^{−1} (2°C, Kawai et al. 1993), porcine myocardium 138 mM^{−1} s^{−1} (20°C, Zhao and Kawai 1996), and bovine myocardium 240 mM^{−1} s^{−1} (25°C, Fujita et al. 2002). Araujo and Walker (1996) used caged Pi on rat ventricular myocytes, and found the second order Pi binding constant (our K_{5}k_{−4}) to be 3.1 mM^{−1} s^{−1} (15°C). This number compares well with our values calculated from Table 2 (same references): ferret myocardium 6.4 mM^{−1} s^{−1} (20°C), porcine myocardium 1.1 mM^{−1} s^{−1} (20°C), and bovine myocardium 1.8 mM^{−1} s^{−1} (25°C). Thus, even when very different techniques and different cardiac preparations are used, we are able to achieve approximate agreement.
Conclusion and future direction
As shown in this mini review, the three-state model is adequate to account for the three phases of force transients that are generated in response to a step length change. The six-state model is an extension of the three-state model that includes ligand (ATP, ADP, Pi)-bound states. In order to correlate each phase of a transient to individual elementary step of the cross-bridge cycle correctly, it is necessary to examine the effects of these ligand concentrations on the apparent rate constants and to fit their results to the model. Therefore, any future cross-bridge models must be able to predict the effects of these ligands, in particular those of ATP and Pi. When studying force transients, it is best not to make a large increase in force, because this may cause a delay in the time course to stretch series elastic elements. Since multiple cross-bridge cycles are needed to stretch series elastic elements, the time course may be limited by the slowest step (rate-limiting step) of the cross-bridge cycle (Section 19).
Acknowledgements
The authors are indebted to Dr. David W. Maughan of the University of Vermont (USA) for his critical reading of the manuscript and his constructive comments and useful suggestions. The authors are also grateful to Dr. Christine Blaumueller for her critical reading of the manuscript and creative suggestions. This work was supported by a grant from NIH HL70041 to MK. The contents of this work are solely the responsibility of the authors and do not necessarily represent the official view of the awarding organization.