Remarks on Muscle Contraction Mechanism II. Isometric Tension Transient and Isotonic Velocity Transient

Abstract

Mitsui and Ohshima (2008) criticized the power-stroke model for muscle contraction and proposed a new model. In the new model, about 41% of the myosin heads are bound to actin filaments, and each bound head forms a complex MA₃ with three actin molecules A1, A2 and A3 forming the crossbridge. The complex translates along the actin filament cooperating with each other. The new model well explained the experimental data on the steady filament sliding. As an extension of the study, the isometric tension transient and isotonic velocity transient are investigated. Statistical ensemble of crossbridges is introduced, and variation of the binding probability of myosin head to A1 is considered. When the binding probability to A1 is zero, the Hill-type force-velocity relation is resulted in. When the binding probability to A1 becomes finite, the deviation from the Hill-type force-velocity relation takes place, as observed by Edman (1988). The characteristics of the isometric tension transient observed by Ford, Huxley and Simmons (1977) and of the isotonic velocity transient observed by Civan and Podolsky (1966) are theoretically reproduced. Ratios of the extensibility are estimated as 0.22 for the crossbridge, 0.26 for the myosin filament and 0.52 for the actin filament, in consistency with the values determined by X-ray diffraction by Wakabayashi et al. (1994).

1. Introduction

In 1999, Mitsui [1] criticized the power-stroke model on the muscle contraction mechanism and proposed a new model. In 2008, Mitsui and Ohshima [2] refined the new model and discussed the steady filament sliding in detail demonstrating that the calculation results were in good agreement with experimental observations. They also outlined the discussion on the isometric tension transient and the isotonic velocity transient given in [1] by citing some calculation results. Thereafter, however, a few readers of [1] commented that it was very difficult to understand the discussion on the transient phenomena in [1], since there was no detailed explanation on the molecular processes upon which the theoretical treatment was based. In the present paper, we have largely revised that part of [1], trying to make the discussion more readable.

Now the article [2] is regarded as Part I of Remarks series, in which the basic ideas of our model are introduced and steady muscle behaviors are discussed. In the present paper (Remarks II), non-steady muscle behaviors are discussed. We are preparing an article as Remarks III, in which discussion will be done on more recent experimental studies as cited in the articles [3,4].

The basic ideas of our model introduced in [2] are summarized as follows. A simple thermodynamic relation is derived, which indicates that there is an inconsistency in the power stroke model or swinging lever model. Our model is proposed to avoid this difficulty. It is assumed that a myosin head forms a complex with three actin molecules when it attaches to an actin filament. Here it should be noted that Andreeva et al. [5] found the evidence that the crossbridge can interact with more than 1 actin monomer. The complex corresponds to the crossbridge. According to the X-ray diffraction studies [6–9], the intensity ratio of the [1, 0] and [1, 1] equatorial reflections increases only minimally as the shortening velocity increases, indicating that the total number of myosin heads in the vicinity of the actin filament decreases only slightly. Taking this fact into account, it is assumed that about 41% of the myosin heads forms the crossbridges at any sliding velocity. Then mutual cooperativity takes place among the crossbridges in filament sliding, so that energy dissipation becomes reasonable magnitude (the order of kT) for one step of the crossbridge movement. Calculation based upon the model well reproduce the force-velocity relation given by Hill [10] and the energy liberation rate vs. force relation given by Hill [11].

Since the present study is based upon ideas that are quite different from the power stroke model and others, the basic ideas of our study are explained in some detail in Section 2. In Section 3, discussion is done of how variation in the crossbridge binding affects the muscle tension. In Section 4, discussion is done on the difference between the molecular processes in Phases 1 of the isometric tension transient and of the isotonic velocity transient, and extensibility ratios for the crossbridge, the myosin filament and the actin filament are estimated. Time course of the isometric tension transient is studied in Section 5. Time course of the isotonic velocity transient is studied in Section 6. The deviation from the Hill-type force-velocity relation is derived in Section 7. Obtained results are summarized and discussed in Section 8.

2. Basic Ideas for Discussion of the Transient Phenomena

2.1. Deviation from the Hill-Type Force-Velocity Relation

In the present study, the deviation from the hyperbolic force-velocity relation shown in Figure 1 has very important implication. The red line is our calculation results reported in [2] which agrees with the empirical hyperbolic force-velocity relation proposed by Hill [10]. The curve will be called Hill-type below. Edman [12], however, reported that carefully measured velocity deviated from the Hill-type relation as given by the black circles in Figure 1, which he called a double-hyperbolic force-velocity relation. In Figure 1, the tension T at which the experimental data start to deviate from the Hill-type curve is indicated as T_dev and the tension T at which T becomes 0 is denoted as T_0obs. The following values are determined by the data presented in Figure 6A of [12].

$$T_{\text{dev}}/T_0=0.66$$

(1)

$$T_{0\text{obs}}/T_0=0.88$$

(2)

2.2. MA₃ Complex and U*₁₂ Transition

In our model, about 41% of the myosin heads are bound to actin filaments (cf. Equation 3–1–1 in [2]), and each bound head forms a complex MA₃ with three actin molecules. The complex MA₃ translates along the actin filament changing the partner actin molecules. Figure 2 illustrates the step motion of MA₃. The translation of the crossbridge along the actin filament is made possible by the structural change of MA₃ induced by the force f_J which exerts on the junction J between the crossbridge and the actin filament.

Figure 3 shows the distribution of potential of force exerted on the myosin head in MA₃. There are three potential wells on the actin filament corresponding to the three binding sites A1, A2 and A3. The wells are also as called A1, A2 and A3. Figure 3(a) shows the potential distribution considered in [2], where a myosin head exists solely in well A2. It is assumed that the potential barrier U* depends upon the force f_J by U* = U*₀ − af_J (Equation 3–5–2 in [2])). The kinetics of the myosin head will be discussed based upon Eyring’s theory of the rate process (cf. [13]) as was done by Huxley and Simmons [14] and in [2]. Then the probability that the myosin head moves from well 2 to 3 across the potential barrier U* becomes proportional to exp(−U*/kT). Calculation was done in [2] based upon the model in Figure 3(a). An example of calculation results is cited by the red curve in Figure 1. Since the calculation well explains the experimental data for T < T_dev. this model seems close to reality when T < T_dev. The deviation from the Hill-type relation when T > T_dev suggests that the model in Figure 3(a) should be modified, and the model in Figure 3(b) becomes the object to be considered in the present study. It is assumed that myosin heads start to move over the potential barrier U*₁₂ when T becomes larger than T_dev, and heads are distributed in A1 and A2 when T > T_dev. The potential peak between A1 and A2 will be called U*₁₂ and the potential barrier for backward movement from A2 to A1 will be denoted as U*_12b and the barrier for forward movement from A1 to A2 as U*_12f. In the followings, transitions of these types will be called U*₁₂ transition.

The displacement of the myosin head from A2 to A1 was briefly mentioned in Section 3.6 of [2] in term of “pull-up transition”. In the article [1], the isometric tension transient was discussed in relation with the U*₁₂ transition by the statistical mechanics. Although the present study is based upon the same idea, we have found that such statistical-mechanical approach as in [1] makes the discussion very complex. We shall discuss the problem in a different manner as described below.

2.3. Definition of Crossbridge Shortening Y

In the following discussion, a parameter y is frequently used to represent shortening (which is positive for negative length change) of the crossbridge as in [2]. Unfortunately, the symbol y was used to represent elongation of the crossbridge in [1]. In the present paper, as an extension of the discussion of [2], y represents shortening. Figure 4 is to make the definition of y clear. K and J, respectively, are ends of the crossbridge on the myosin and actin filaments. The symbol x is defined as the projection of the vector from K to J on the myosin filament. Since the positions J are set on the right-handed red dotted line indicated as z in Figure 4, x is determined by the position of K. As shown in Figure 4(a), x is denoted as x_eq when the tilting angle of the myosin head neck domain is at the equilibrium angle θ_Equation. Then y is defined by

$$y=x-x_{\text{eq}}$$

(3)

The red dotted line indicated as x_eq in Figure 4 indicates the origin of y. Thus y = 0 in Figure 4(a), y < 0 in (b) and y > 0 in (c). As discussed in [2], the crossbridge with shortening y exerts the force p(y) on the myosin filament. The stiffnesses for forward and backward forces are denoted as κ_f and κ_b, and, according to Equation 3–4–6 in [2], p(y) is given by

$$p(y)=-{\kappa }_{\text{f}}y\hspace{0.17em}\text{for}\hspace{0.17em}y<0$$

(4a)

$$p(y)=-{\kappa }_{\text{b}}y\hspace{0.17em}\text{for}\hspace{0.17em}y>0$$

(4b)

Values of κ_f and κ_b are given in Appendix.

2.4. Time Constant of the U* Transition

To discuss the transition phenomena, it is important to have an idea about the mean time interval of occurrence of the U* transition. In the steady filament sliding, the mean time interval is equal to the mean time τ_step in which a myosin head moves from one actin molecule to the neighboring one. This τ_step is given by

$${\tau }_{\text{step}}=L/v$$

(5)

where L is the distance between the centers of actin molecules 2 and 3 in Figure 2 (the value of L is given in Appendix) and v is the sliding velocity. Values of τ_step are calculated by using v given by the red curve in Figure 1, and shown as a function of the relative tension in Figure 5, which indicates that τ_step is about 2 ms at T/T₀ = 0, becomes 10 ms around T/T₀ = 0.5 and increases to 1 s around T/T₀ = 1. This result means that U* transition generally does not contribute to the early processes in the transition phenomena.

2.5. Statistical Ensemble of Crossbridges and Crossbridge Binding Probability ρ

In discussion of the steady filament sliding [2], the crossbridge shortening y repeats the cycle from y_c-L to y_c through filament sliding and from y_c to y_c-L through the U* transition (cf, Figure 7 in [2]), where y_c is the parameter which decreases as the tension T increases (as shown in Figure 8 given later.) In the following discussion, this temporal scheme is replaced by a spatial scheme. In our model, the ratio r = (number of myosin heads bound to actin filaments)/(total number of myosin heads) is constant independent of tension (r = 0.41, cf. Appendix). Accordingly, we can consider an ensemble of bound crossbridges of definite number independent of the tension.

Suppose that all complexes MA₃ in right half sarcomeres are collected and superposed putting A2 at the same position. This ensemble can be characterized by the ratio ρ(y)dy = (number of crossbridges having shrinkage between y and y + dy)/(total number of crossbridges). By definition, integration of ρ(y) is 1. Actually ρ(y) is a smooth function. The mean value of y in the ρ distribution is denoted as <y> and y_c is defined by

$$y_{\text{c}}=<y>+(L/2)$$

(6)

Then ρ(y) is expected to be large between y_c-L and y_c. In the following discussion, ρ(y) is approximated to be constant (1/L) between y_c-L and y_c and zero outside of the region. This will be called rectangular ρ approximation. Equivalent approximation was used in the temporal scheme in [2] for studies of the steady filament sliding.

The force-velocity relation deviates from Hill-type for T > T_dev as shown in Figure 1. The origin of the deviation is considered as follows. In Figure 3, the potential of force exerted on the myosin head in the MA₃ complex is shown. The potential at A2 is lower than that at A1, and all the myosin heads are present in well A2 in the case of T < T_dev as shown in Figure 3(a). On the other hand, as seen in Figure 4(b), the backward force exerted on the myosin head becomes stronger as y negatively increases, so that the myosin heads having negatively large y tend to bind to A1 through the U*₁₂ transition. Accordingly, the distribution of the myosin heads becomes as symbolically shown in Figure 3(b) for T > T_dev. This effect is regarded as the origin of the deviation of force-velocity relation from the Hill-type.

Symbols y_0obs and y_cdev are defined as y_c’s at T_0obs and T_dev, respectively, on the Hill-type force-velocity relation (cf. Figure 8 given later.) As mentioned above, y_c decreases with increasing T, and thus T < T_dev corresponds to y_c > y_cdev.

Figure 6 shows examples of the rectangular ρ distribution for the steady filament sliding. Figure 6(a) is for fast sliding and (b) is for relatively slow sliding in the case of y_c > y_cdev or T < T_dev. Figure 6(c) is for slow sliding in the case of y_c < y_cdev or T > T_dev. In Figures 6(a1), (b1) and (c1), the upper horizontal line is an axis of the crossbridge shortening y and the black thick segment on it shows the regions of finite ρ(y) of length L. The lower horizontal lines correspond to the actin filament and A1, A2 and A3 represent actin molecules. The crossbridges having negative y produce positive stress p(y) according to Equation 4a. The red segments with their ends at A2 are examples of such crossbridges. The crossbridges having positive y produce negative stress p(y) according to Equation 4b. The blue segments with their ends at A2 or A1 are examples of such crossbridges.

Figures 6(a2) and (b2) show the ρ distributions as functions of y, ρ(y) by the rectangular approximation, where ρ(y) is equal to (1/L) between y_c-L and y_c, and equal to zero outside of the region. Crossbridges in the red area produce positive stress, and crossbridges in the blue area produce negative stress. As the edge y_c shifts to the left from (a2) to (b2), the red region increases and the crossbridge ensemble produce more stress and the filament sliding becomes slow (Note κ_f > κ_b in Appendix).

Figure 6(c) corresponds to the case of T > T_dev and thus y_c < y_cdev. In (c1), the blue dotted segment symbolically indicates that a portion of crossbridges of small y bind to A1 and produces negative tension, while the red dotted segment indicates that the rest of crossbridges having the same small y still bind to A2 and produce positive tension. In (c2), the violet triangle symbolically indicates that the crossbridges in this triangle are bound to A1. As shown in (c3), the violet triangle is replaced by the violet rectangle containing the same number of the crossbridges, to make calculation easy. In the following calculation, it is assumed that all the crossbridges in the violet rectangle are bound to A1 and produce negative tension.

Now let us calculate values of y_cdev and y_c0obs. By definition, y_cdev and y_c0obs are y_c’s for T_0obs and T_dev in the Hill-type force-velocity relation. The mean tension per one crossbridge is denoted as p in [2], which can be obtained by integration of the tension p(y). Calculation is done by using p(y) = −κ_fy/L (Equation 4a) in the red region and p(y) = −κ_by/L (Equation 4b) in the blue region in Figure 6 (a2) or (b2). Result of the integration is

$$p=\{-(-{\kappa }_{\text{f}}/2){(y_{\text{c}}-L)}^2+(-{\kappa }_{\text{b}}/2)y_{\text{c}}^2\}/L$$

(7)

This relation leads us to the Hill-type force-velocity relation in cooperation with the U* transition [2]. The isometric tension in the Hill-type relation is denoted as p₀ and T₀ following [2]. Then, by using p/p₀ = T/T₀ (Equation 3–1–2 in [2]), the relative tension T/T₀ is given by

$$T/T_0=\{-(-{\kappa }_{\text{f}}/2){(y_{\text{c}}-L)}^2+(-{\kappa }_{\text{b}}/2)y_{\text{c}}^2\}/({\mathit{\text{Lp}}}_0)$$

(8)

This equation gives relations between T_dev and y_cdev and between T_0obs and y_c0obs. Combining Equations 1, 2 and 8 gives

$$y_{\text{cdev}}=1.60\hspace{0.17em}\text{nm},$$

(9)

$$y_{\text{c}0\text{obs}}=1.03\hspace{0.17em}\text{nm}.$$

(10)

In Figure 6, y_cdev and y_c0obs are indicated by the vertical dotted red lines.

3. Effect of U*₁₂ Transition on the Tension

Equation 8 leads us to the Hill-type force-velocity relation. The stress T and y_c related by Equation 8 is denoted as T^H and y_c^H, which are in agreement with the observation when y_c^H > y_cdev. The observed T and y_c are denoted as T* and y_c* when y_c^H < y_cdev.

Figure 7 illustrates the ρ distributions in various cases of y_c^H < y_cdev. Figure 7(a1) shows the ρ distribution by Equation 8 having the edge y_c^H. This ρ distribution gives T^H even though y_c < y_cdev. It is an imaginary state in which the U*₁₂ transition is absent. Actually, however, the U*₁₂ transition takes place and the violet area appears causing decrease of the red area as shown in (a2), where the width of the violet area is denoted as x(y_c). Naturally T in (a2) is smaller than T in (a1). Then y_c in (a2) is changed into y_c* in (a3) to make T the same as T in (a1), where the width of the violet region is denoted as x. The change of the blue area from y_c to y_c* causes an increase of the tension while the change of the violet area from x(y_c) to x causes an decrease of the tension. If the effect of the former change is larger than that of the latter change, the tension in (a2) will increase and can be the same as in (a1). Below we shall assume that such tension adjustment actually takes place and discuss the relation between y_c* and T*.

Since the tension is the same in Figures 7(a1) and (a3), the blue area in (a1) is the same as the sum of the blue area and violet area in (a3). Figures 7(b1)–(b3) show the case of y* = 0. Similarly to (a1) and (a3), the blue area in (b1) is the same as the violet area in (b3). The width x of the violet area in (b3) is denoted as x₀. Figure 7 (c) shows the case that y_c* becomes negative. The brown area means that crossbridges in this area bind to A1 and produce positive stress. The width of the violet area is indicated as x. Extrapolating the change from x in (a3) to x₀ in (b3), x in (c) is assumed to be larger than x₀. Then sum of the red area and brown area in (c) is smaller than red area in (b3) and thus the tension produced in (c) is smaller than the tension in (b3). Then the stress produced in (b3) is the maximum that the muscle machine can produce, and should be equal to T_0obs. Accordingly, y_c^H in (b1) should be equal to y_c0obs. Since the red area in (b1) is the same as the red area in (b3), y_c^H in (b1) is equal to x₀ in (b3). Thus we have

$$x_0=y_{\text{c}0\text{obs}}$$

(11)

The tension T* deviates from T^H at y_c = y_c* = y_cdev, and reaches its maximum at y_c* = 0. To express such characteristics of T* vs. y* relation, the following set of equations are used:

$$T^{*}/T_0=-{\mathit{\text{ay}}}_c^{{*}^2}+b$$

(12)

$$a=(T_{0\text{obs}}-T_{\text{dev}})/(T_0{y}_{\text{cdev}}^2)=0.086\hspace{0.17em}{(\text{nm})}^{-2}$$

(13)

$$b=T_{0\text{obs}}/T_0=0.88$$

(14)

In Figure 8, the blue curve illustrates the T*/T₀ vs. y_c* relation given by this set of equations, and the red curve shows the Hill-type T^H/T₀ vs. y_c^H relation. The blue curve deviates from the red curve at y_c = y_c* = y_cdev and exhibits its maximum at y_c* = 0 where T^*/T₀ = T_0obs/T₀.

Up to here, T/T₀ is used for the relative tension, where T₀ is the maximum tension in the extended Hill-type force-velocity relation. The experimentally observed maximum tension, however, is T_0obs where T_0obs = 0.88T₀ (Equation 2). Hereafter discussion is concerned with experimental data and relative tension is defined as T/T_0obs. In our model T/T₀ = p/p₀ (Equation 3–1–2 in [2]), and the force p corresponding to T_0obs is denoted as p_0obs. Then

$$p_{0\text{obs}}=0.88\hspace{0.17em}{p}_0$$

(15)

and

$$T/T_{0\text{obs}}=p/p_{0\text{obs}}$$

(16)

If we put y_c = y_c0obs in Equation 7 and use x₀ = y_c0obs (Equation 11), we get theoretical expression of p_0obs as

$$p_{0\text{obs}}=\{-(-{\kappa }_{\text{f}}/2){(-L+x_0)}^2+(-{\kappa }_{\text{b}}/2)x_0^2\}\hspace{0.17em}/L$$

(17)

4. Phases 1 in the Isometric Tension Transient and Isotonic Velocity Transient, and Extensibility Ratios for the Crossbridge, Myosin Filament and Actin Filament

Huxley [15] divided the transient responses to the sudden reduction of length or of load into four Phases. In our model, however, there is no exact correspondence between molecular processes of the four Phases in the tension transient and those in the velocity transient. To avoid confusion, we use the terms, Phase Tn in the tension transient and Phase Vn in the velocity transient, where n = 1, 2, 3, 4.

Phase T1 is simultaneous decrease of tension in the isometric tension transient, while Phase V1 is simultaneous shortening of muscle in the velocity transient. The length change per half sarcomere is denoted as ΔL_hs in both transients. The experimentally measured isometric tension is T_0obs and the relative tension is defined by T/T_0obs. The relative load used in the paper by Civan and Podolsky [8] is expressed as the relative tension T/T_0obs below. Figure 9 shows the experimental results on Phases T1 and V1, by circles for the isometric tension transient cited from the paper by Ford et al. [16] and by squares for the isotonic velocity transient cited from the paper by Civan and Podolsky [17].

At first sight, it was puzzling to see that distributions of the experimental data are quite different for the two cases, since the structural changes seem to be purely elastic both in Phases T1 and V1. Then it was reminded that the length change of sarcomere is a sum of those of the crossbridge, myosin filament and actin filament, which are proportional to each other (cf. the review by Irving [18]). This means that there are three elastic components, and it is a possibility that they have different response times in elastic changes from each other.

In this connection, the experimental data reported by Julian and Sollins [19] seem important. They measured tension changes of single frog skeletal muscle fiber at increasing speed of step shortening. Their Figures 2 and 4 show experimental data on relative force vs. ΔL_hs relation at different speeds of shortening. The distribution of open circles in their Figure 2 (the length change period of about 1 ms) is similar to that of the data by Civan and Podolsky (1∼2 ms) cited in our Figure 9. The distribution of filled triangles in Figure 4 of [9] (length change period of 0.4 ms) is close to the data by Ford et al. (the length change period of 0.2 ms) cited in our Figure 9. These facts seem to indicate that there are two kinds of elastic process in the tension response: The fast one occurs within about 0.4 ms and the slow one occurs between about 0.4 and 1 ms after the length change. The fast process seems responsible to the change in Phase T1 and combination of the fast and slow processes seems responsible to Phase V1.

It seems plausible that elastic response of crossbridge and myosin filament almost simultaneously occurs since they belong to the same molecule and elastic response of actin filament occurs with some delay. Thus it is assumed that crossbridge and myosin filament are responsible to the change in Phase T1 and that all three components are responsible to the change in Phase V1. Now changes of y_c in Phases T1 and V1 should be different from each other even when the length change ΔL_hs is the same. Calculation is done on this assumption by using the rectangular ρ distribution shown in Figure 10.

Figure 10(a) shows the ρ distribution at the isometric tension (the same as Figure 7 (b3)). Figure 10(b) shows the ρ distribution just after Phase T1 and (c) the one just after Phase V1 for the same length change ΔL_hs as (b). The violet areas in Figure 10 (b) and (c) are the same as (a) since U*₁₂ transition does not occur yet. Changes of the edge y_c in Phases T1 and V1 are denoted as Δy_cT1 and Δy_cV1. For the same ΔL_hs, the tension T in Phase T1 is smaller than T in Phase V1 in Figure 9. The edge Δy_cT1 in Figure 10 (b) is set larger than Δy_cV1 in (c) so as to make the red area in (b) smaller than that in (c), in accordance with the fact that the tension T in Phase T1 is smaller than T in Phase V1.

Since elastic changes of the crossbridge, myosin filament and actin filament are proportional to each other [10], the Δy_cT1 in (b) and Δy_cV1 in (c) are proportional to ΔL_hs, and expressed by

$$\Delta y_{\text{cT}1}=-\Delta L_{\text{hs}}/C_{\text{CBT}}$$

(18)

and

$$\Delta y_{\text{cV}1}=-\Delta L_{\text{hs}}/C_{\text{CBV}}$$

(19)

where C_CBT and C_CBV are constants.

Let the tensions in Figure 10(b) and (c) be denoted as T_T1 and T_V1, respectively. They are given by integration of −κ_b{y − (−L)} in the violet area, −κ_fy in the red area and −κ_by in the blue area. Thus we have

$$T_{\text{T}1}/T_{0\text{obs}}=[(-{\kappa }_{\text{b}}/2)\{{(x_0+\Delta y_{\text{cT}1})}^2-\Delta y_{\text{cT}1}^2\}-(-{\kappa }_{\text{f}}/2){(-L+x_0\Delta y_{\text{cT}1})}^2+(-{\kappa }_{\text{b}}/2)\Delta y_{\text{cT}1}^2]/(L{p}_{0\text{obs}})$$

(20)

and

$$T_{\text{V}1}/T_{0\text{obs}}=[(-{\kappa }_{\text{b}}/2)\{{({\text{x}}_0+\Delta y_{\text{cV}1})}^2-\Delta y_{\text{cV}1}^2\}-(-{\kappa }_{\text{f}}/2){(-L+x_0\hspace{0.17em}\Delta y_{\text{cV}1})}^2+(-{\kappa }_{\text{b}}/2)\Delta y_{\text{cV}1}^2]/(L{p}_{0\text{obs}})$$

(21)

Calculations were done for various trial values of C_CBT (Equation 18) and C_CBV (Equation 19). The best fit for the experimental data are obtained for the values

$$C_{\text{CBT}}=2.2$$

(22)

$$C_{\text{CBV}}=4.6$$

(23)

Figure 9 shows the calculation results for T_T1/T_0obs by the green curve and for T_V1/T_0obs by the brown curve. They are in good agreement with the experimental data.

Let us denote the ratios of extensibilities of the crossbridge, the myosin filament and the actin filament as r_CB, r_M, and r_A at the elastic equilibrium respectively. Shortenings of the three per half sarcomere are denoted, respectively, as Δy_CB, Δy_M and Δy_A, and the length change of half sarcomere as ΔL_hs. Then, at the elastic equilibrium, they are given by

$$\Delta y_{\text{CB}}=-r_{\text{CB}}\Delta L_{\text{hs}}$$

(24a)

$$\Delta y_{\text{M}}=-r_{\text{M}}\Delta L_{\text{hs}}$$

(24b)

$$\Delta y_{\text{A}}-r_{\text{A}}\Delta L_{\text{hs}}$$

(24c)

Naturally,

$$r_{\text{CB}}+r_{\text{M}}+r_{\text{A}}=1$$

(25)

On the above assumption, only the length changes of the crossbridge and the myosin filament contribute to Phase T1. Then we have

$$\Delta y_{\text{cT}1}=\Delta L_{\text{hs}}/C_{\text{CBT}}=\{r_{\text{CB}}/(r_{\text{CB}}+r_{\text{M}})\}\hspace{0.17em}\Delta L_{\text{hs}}$$

(26)

Thus,

$$C_{\text{CBT}}=(r_{\text{CB}}+r_{\text{M}})/r_{\text{CB}}$$

(27)

As assumed above, the load change period is long enough and elastic changes of the crossbridge, myosin filament and actin filament contribute to Phase V1. Thus, Δy_cV1 is given by

$$\Delta y_{\text{cV}1}=\Delta L_{\text{hs}}/C_{\text{CBV}}=\{r_{\text{CB}}/(r_{\text{CB}}+r_{\text{M}}+r_{\text{A}})\}\hspace{0.17em}\Delta L_{\text{hs}}=r_{\text{CB}}\Delta L_{\text{hs}}$$

(28)

Hence,

$$C_{\text{CBV}}=1/{\text{r}}_{\text{CB}}$$

(29)

By using the values of C_CBT (Equation 22) and C_CBV (Equation 23), Equations 25, 27 and 29 give

$$r_{\text{CB}}=0.22$$

(30a)

$$r_{\text{M}}=0.26$$

(30b)

$$r_{\text{A}}=0.52$$

(30c)

The extensibility ratios were investigated by X-ray diffraction by Huxley et al. [20] and Wakabayashi et al. [21]. The values reported by Wakabayashi et al. [21] are r_CB = 0.31, r_M = 0.27 and r_A = 0.42. Considering approximate nature of the theory and experimental errors, our values in Equation 30 are in reasonable agreement with the X-ray values.

In this section, it is assumed that elastic change of the actin filament does not take place during the fast length change in Phase T1. Then the elastic change of the actin filament should contribute to the next step, the tension recovery in Phase T2. This problem is discussed in next section.

5. Isometric Tension Transient

As noted in Section 4, Huxley [15] divided the transient responses to the sudden reduction of length into four phases. Time courses of these Phases are as follows. Phase 1 is instantaneous drop of tension. Phase 2 is rapid early tension recovery in next 1∼2 ms. Phase 3 is extreme reduction or even reversal of rate of tension recovery during next 5∼20 ms. Phase 4 is the gradual recovery of tension, with asymptotic approach to the isometric tension.

As in Section 4, we use the terms Phase Tn (n = 1, 2, 3, 4) in the tension transient to avoid confusion. Phases Tn are more related with molecular process rather than time sequence. Phase T1 is simultaneous drop of tension caused by elastic shortening of the crossbridge and myosin filament. Phase T2 is due to elastic shortening of the actin filament. Phase T3 is related with the U*₁₂ transition, and divided into T3a and T3b. Phase T3a is the first part of Phase T3. Both Phases T2 and T3a contribute to the rapid early tension recovery (Phase 2 of Huxley). Phase T3b is the second part of T3 where the extreme reduction or reversal of rate of tension recovery occurs (Phase 3 of Huxley). Phase T4 is the gradual recovery of tension, with asymptotic approach to isometric tension (Phase 4 of Huxley). (There were misprints in Section 4.4 of the article [2], and the last three sentences of the section should be neglected.)

Figures 11 and 12 illustrate how the rectangular ρ distribution changes during these Phases. Figure 11 is for the case of Δy_cT1 < y_cdev and Figure 12 is for the case of Δy_cT1 > y_cdev, where Δy_cT1 is y_c just after Phase T1. (cf. Figure 10(b)).

As mentioned above, it is assumed that the elastic changes of the crossbridge and myosin filament occur in Phase T1 and then the elastic change of actin filament occurs in Phase T2. Figure 11(a) represents the states just after Phase T1. Figure 11(b) shows the state just after Phase T2. There is a time lag for the U*₁₂ transition to occur, and its effect is neglected in Phases T1 and T2, so that the width of the violet area is kept the same as the isometric value x₀.

As noted referring to Figure 5, the mean time (τ_step) needed for the U* transition is relatively large (e.g., larger than 10 ms for T/T₀ > 0.5). It is assumed that the potential barrier U*_12f (f: forward) is lower than U* as shown in Figure 3 (b) and thus the U*_12f transition starts before the U* transition. Phase T3a is regarded as the state that the U*_12f transition is present but the U* transition does not occur yet, while the U*_12f and U* transitions coexist in Phase T3b. In Phase 4, only the U* transition exists. Figure 11(c) and (d), respectively, show the states at the end of Phases T3a and T3b. The violet area in (c) is smaller than in (b) because a portion of the crossbridges in the violet area change their binding partners from A1 to A2 through U*_12f transition. Thus the red area increases and the stress is stored and the tension increases in Phase T3a. The U* transition starts at (c) and the stress stored during Phase T3a is released by the shift of the edge y_c from y_cT3a in (c) to y_cT3b in (d). The shift of the edge from y_cT3a (c) to y_cT3b (d) causes increase of the blue area and decrease of the violet area. These two effects tend to cancel each other and reduce change of the red area in the time course from (c) to (d), i.e., in Phase T3b. The reduced change of the red area may cause the “extreme reduction of rate of tension recovery” mentioned at the beginning of this section. In Phase 4, the U* transition and the filament sliding continue and the state approaches to the tetanus state (e).

Figure 12 shows changes of the ρ distribution in the isometric tension transient for the case of Δy_cT1 > y_cdev. Figure 12(a) and (b) are similar to Figure 11(a) and (b). Since Δy_cT1 > y_cdev, the U*_12f transition does not leave the violet area and the red area significantly increases in (c). Then the U* transition starts and the ρ distribution shifts to the right as shown in (d). The red area significantly decreases from (c) to (d), and tension decreases and thus the “reversal of rate of tension recovery” cited at the beginning of this section is expected. The filament sliding continues in Phase 4, and the ρ distribution asymptotically approaches the isometric tetanus state (e).

Figure 13 illustrates changes in Phases T1∼T4 in another way by the T/T_0obs vs. ΔL_hs relation. The solid black arrows “Phase Tn” indicate an example of change of relative tension in Phase Tn for the case of Δy < y_cdev and the dashed black arrows for the case of Δy > y_cdev. The edge Δy_cT1 in Figures 11(a) or 12(a) is resulted from the elastic changes of the crossbridge and myosin filament, and is given as a function of ΔL_hs by combining Equations 18 and 22:

$$\Delta y_{\text{cT}1}=-\Delta L_{\text{hs}}/2.2$$

(31)

The tension variation T_T1/T_0obs in Phase T1 can be calculated as a function of ΔL_hs by using Equations 20 and 31. Calculation results are shown by the green curve in Figure 13. The edge y_cT2 in Figures 11(b) or 12(b) are the same quantity as Δy_cV1 in Figure 10(c), since they result from the elastic changes of the crossbridge, myosin filament and actin filament. Then, from Equations 19 and 23, we have

$$y_{\text{cT}2}=-\Delta L_{\text{hs}}/4.6$$

(32)

Δy_cT2 = Δy_cV1 means T_T2/T_0obs = T_V1/T_0obs. By using Equation 32 and replacing T_V1 by T_T2 and Δy_cV1 by Δy_cT2 in Equation 21,T_T2/T_0obs can be calculated as a function of ΔL_hs. The calculation result is given by the brown curve in Figure 13. In the steady filament sliding, there is the definite relation between the tension T and the parameter y_c as shown in Figure 8, where the red curve shows the relation for the Hill-type filament sliding and the blue curve shows the relation when the filament sliding deviates from the Hill-type. The parameter y_c in Figure 8 is related with Δy_c in Figures 11 and 12 by the relation y_c = y_c0obs + Δy_c since Δy_c is the change of y_c from the value at the isometric tension (y_c0obs). In the steady filament sliding, all elastic elongations of the crossbridge, the myosin filament and the actin filament contribute to the muscle elongation and thus Δy_c (=y_c − y_c0obs) is equal to −ΔL_hs/4.6 as in Equation 32. The relative tension T/T₀ in Figure 8 can be converted to T/T_0obs by using the ratio T_0obs/T₀ = 0.88 (Equation 2). Based upon these considerations, the red and blue curves in Figure 8 are reproduced with the same colors in Figure 13.

In Figure 13, the Phases are indicated by the solid black arrows “Phase Tn” in the case of Δy < y_cdev (the case of Figure 11), and by the dashed black arrows in the case of Δy > y_cdev (the case of Figure 12). The rapid tension change in Phase T1 occurs along the green curve as indicated by the solid or dashed arrows “Phase T1”. Phase T2 is a result of the elastic change of the actin filament and thus the arrows “Phase T2” start from the green curve and end on the brown curve. The main part of Phase T3a corresponds to the rapid tension increase from Figure 11(b) to (c) or from Figure 12(b) to (c). It is difficult to determine where Phase T3a turns into Phase T3b in Figure 13. Trial calculations, however, show that fairly good agreement can be obtained by assuming that Phase T3a ends at the blue curve or the red curve. Thus, in the case of Δy < y_cdev, the solid arrow “Phase T3a” is depicted with its tip on the blue curve. As discussed above, it is plausible that the red area in Figure 11(d) is nearly equal to that in (c). Taking this point into account, the solid arrow “Phase T3b” is depicted almost parallel to the abscissa, in accordance with the observation of the “extreme reduction of rate of tension recovery” noted at the beginning of this section. In the case of Δy > y_cdev, as discussed above, it is plausible that the red area significantly decreases from Figure 12(c) to (d). Accordingly, the dashed arrow “Phase T3b” is depicted downward, in accordance with the observation of the “reversal of rate of tension recovery ” noted at the beginning of this section. After Phase T3b, the tension T approaches T_0obs by the filament sliding as shown by the arrows “Phase T4”. The above argument on the solid and dashed arrows “Phase T3b” suggests that the reversal of rate of tension recovery occurs when |ΔL_hs| is large. Supporting this conclusion, Figure 1 of the paper by Julian and Sollins [19] shows that the reversal of the rate occurs when |ΔL_hs| is large.

Ford et al. [16] reported experimental data on variation of T/T_0obs during 0–9 ms, as cited by green circles in Figure 14. The range of ΔL_hs was +1.5∼−6.0 nm in their experiment. Since our model is not applicable for positive ΔL_hs, the case of +1.5 nm is omitted in Figure 14. Now we shall try to reproduce these results by calculation.

In Figure 14, values of T/T_0obs quickly increase within about 1 ms and then gradually approach to the value at 9 ms. The mean time duration needed for one U* transition is longer than 10 ms for T/T₀ > 0.5 in Figure 5. Accordingly, effect of U* transition is not considered. The experimental length changes | ΔL_hs | are smaller than 7.0 nm as seen in Figure 14. Hence, the following discussion is done referring to the solid arrows which are in this | ΔL_hs | range. The tension changes in Phases T1, T2 and T3a are denoted as T₁, T₂ and T_3a, respectively. Also symbols T₁₂ and T_123a are defined as follows:

$$T_{12}=T_1+T_2$$

(33)

$$T_{123\text{a}}=T_{12}+T_{3\text{a}}$$

(34)

According to the scheme in Figure 13, the change of T₁ is given by the green curve, and T₂ changes from the green curve to the brown curve. In Figure 14, the origin of time t is set equal to 0 when the change of Phase T1 finishes. The parameters C_ini2 and C _fin2 are defined as the initial and final values of T₂/T_0obs, which are given by the green and brown curves, respectively in Figure 13. If T₂ is approximately expressed by a single decay constant τ_T2, T₁₂/T_0obs is given by,

$$T_{12}/T_{0\text{obs}}=C_{\text{fin}2}-(C_{\text{fin}2}-C_{\text{ini}2})\text{exp}(-t/{\tau }_{\text{T}2})$$

(35)

The constant τ_T2 is related with the elastic change of the actin filament and is independent of ΔL_hs. The solid arrow “Phase T3b” is drawn between the brown curve and the blue curve in Figure 13, so that the magnitude of tension of T/T_0obs in T3b is given by the difference between these curves. It is uncertain when the U*₁₂ transition and thus Phase T3a start. To make calculation simple, it is assumed that it starts at the same moment as Phase T2, i.e., at t = 0. Also the change of T_3a is approximately expressed with a single decay constant τ_T3a:

$$T_{3\text{a}}/T_{0\text{obs}}=(C_{\text{fin}3\text{a}}-C_{\text{ini}3\text{a}})\{1-\text{exp}(-t/{\tau }_{\text{T}3\text{a}})\}$$

(36)

where (C_ini3a − C_fin3a) represents the magnitude of the tension variation. C_ini3a and C_fin3a are given by the brown and blue curves, respectively in Figure 13.

Now the problem is how the parameter τ_T3a changes with ΔL_hs. In the discussion on the force-velocity relation in [2], the average time t_c for a myosin head at y_c to cross over the potential barrier U*(y_c) is expressed as t_c = (1/A) exp (U*(y_c)/kT) (Equations 4–2–1 in [2]), and U*(y_c) is expressed by U*(y_c) = U*₀ − by_c (Equations 4–2–2 in [2]). These formulae give t_c = B exp (−cy_c) where B and c = b/kT are constants. In Figure 11(b) the boundary between the violet and red areas is indicated as −L + y_cT2 + x₀. The relative relation between this boundary and A1 is similar to that between y_c and A2. Then, in analogy to the above relation, with constants B’ and c’, the relation _T3a = B’ exp (−cy_c’) is expected as an approximate expression, where y_c’ = −L + y_cT2 + x₀ (Figure 11(b)). Then, as L and x₀ are constants, τ_T3a in Equation 36 is approximately given by τ_T3a = Bexp(−c’y_cT2), where c’ is constant. Since y_cT2 = −ΔL_hs/4.6 (Equation 32), this relation can be rewritten as,

$${\tau }_{\text{T}3\text{a}}=A_{3\text{a}}\hspace{0.17em}\text{exp}(c_{3\text{a}}\Delta L_{\text{hs}})$$

(37)

where A_3a and c_3a are constants. This relation implies that τ_T3a becomes small and the tension variation becomes fast when | ΔL_hs | increases.

Trial calculations were done for T_123a = T₁₂ + T_3a (Equation 34) to explain the experimental observations by changing parameters τ_T2, A_3a and c_3a in Equations 35, 36 and 37. Fairly good agreement with the experimental data is obtained as shown in Figure 14 by using the following parameter values:

$${\tau }_{\text{T}2}=0.7\hspace{0.17em}\text{ms}$$

(38a)

$$A_{3\text{a}}=3\hspace{0.17em}\text{ms}$$

(38b)

$$c_{3\text{a}}=0.5\hspace{0.17em}(1/\text{nm})$$

(38c)

In Section 4, it is assumed that the elastic change of the actin filament occurs 0.4–1 ms after the length changes of the crossbridge and myosin filament. The time constant τ_T2 = 0.7 ms is in consistency with this assumption.

6. Isotonic Velocity Transient

Isotonic velocity transients were studied by Civan and Podolsky [17], Huxley et al. [15] and Sugi and Tsuchiya [22,23]. A muscle was stimulated and initially held at a constant length. It was then released suddenly and allowed to shorten under a constant load. In this section, discussion will refer to the experimental data presented in Figure 3 of the article by Civan and Podolsky [17]. Responses of muscle to the sudden load change are classified into four Phases by Huxley [15]. Analogously, we use terminology “Phase Vn” as mentioned in Section 4. While Phase number of Huxley is related with time sequence of the length changes, Phase Vn is more related with molecular processes rather than the time sequence. Phase V1 is the length change which simultaneously occurs with load change. Phase V2 is rapid early shortening. Phase V3 is extreme reduction or reversal of shortening speed. Phase V4 is responsible to the fact that the filament sliding velocity temporarily becomes larger than the steady value. Phases V2, V3 and V4 overlap each other in their time courses.

Figure 15 shows changes of the ρ distribution in these four Phases. Figure 15 (a) is the ρ distribution just after Phase V1. Since y_c = 0 at the isometric tension (cf. Figure 10 (a)), y_c in Figure 15 (a) is equal to the variation from the isometric tension, Δy_cV1 (cf. Figure 10 (c)). We shall discuss the case of Δy_cV1 > y_cdev, since most experiments on the velocity transient were done for Δy_c > y_cdev.

In Figure 15, the vertical red arrows show time courses of Phases V2, V3 and V4. They overlap each other indicating the overlap of their time courses. The width of the violet area x₀ in Figure 15(a) is the same as in the isometric tension since the U*₁₂ transition does not occur yet. Then the U*₁₂ transition starts, i.e., Phase V2 starts. As mentioned above, there is a time delay in occurrence of the U* transition and thus of the filament sliding. Figure 15 (b1) shows the state in midway of Phase V2, where the U* transition starts. In (b1), a portion of the violet area has turned into red due to the U*_12f transition and internal stress increases, which causes shrinkage of the ρ distribution, and thus of the muscle. At (b2), the U*_12f transition is over and all the violet area is turned into the red area as Δy_cV1 > y_cdev. The process from (a) to (b2) is related with the U*₁₂ transition and is called Phase V2, where the red area increases and the internal stress increases. The width of ρ distribution changes from L in (a) to L-ΔL in (b2). Accordingly, a fast shortening of the muscle is expected.

During Phase V2 the U* transition starts at (b1). There is a mutual interaction between the U* transition and the internal stress. The U* transition causes the filament sliding, releases the internal stress and tends to make y_c larger. The internal stress tends to expand the ρ distribution, increases y_c and accelerates the U* transition. The former process is called Phase V3. The latter process is called Phase V4. The mutual interaction almost disappears leaving a large value of y_c at (c). Then the frequency of the U* transition and y_c change toward their steady values from (c) to (d), where the ρ distribution at the steady filament sliding is shown. Phases V3 and V4 occur in parallel starting with the U* transition at (b1) and end near (d).

Figure 16 illustrates changes in Phase V1∼V4 in another way. This figure is drawn considering the case of small T/T_0obs. (An example of the experimental data for small T/T_0obs can be seen in Figure 18 (c) given later where ΔT/T_0obs = 0.87, i.e., T/T_0obs = 0.13.) The red, blue and brown curves are the same as the curves of the same colors in Figure 13. The black arrows indicate an example of changes of ΔL_hs in the four Phases. As discussed above, there is the overlap between the Phases, and the arrow “Phase Vn” indicates the Phase which mainly contributes to the process.

The length change in Phase V1 occurs along the brown curve, since the elastic changes of the crossbridge, the myosin filament and the actin filament contribute to this change. The arrow “Phase V2” corresponds to the rapid decrease of muscle length from Figure 15 (a) to (b2). The length of the arrow “Phase V2” is tentatively depicted as 6 nm (the same order of magnitude of the rapid drop of 5 nm in Figure 18 (c) given later). As discussed above, the time courses of Phases 3 and 4 overlap each other. The direction of arrow “Phase V3” is reversed, representing the muscle elongation from Figure 15 (b2) to (c). Through “Phase 4”, the system approaches to the red curve corresponding to the change from (c) to (d) in Figure 15. The green arrow “L_hsv” symbolically represents the steady filament sliding corresponding to the state in Figure 15 (d).

Now let us numerically reproduce the experimental data presented in Figure 3 of the paper by Civan and Podolsky [17]. The length change per half sarcomere at the steady filament sliding is denoted as L_hsv which corresponds to the dashed line in Figure 3 of [17]. Discussion will be done referring to L_hsv expressed by

$$L_{\text{hsv}}=-vt$$

(39)

Here v is the sliding velocity determined by the experimental data in [17].

The length change caused by the U*₁₂ transition in Phase V2 is denoted as L_hs2. The speed of L_hs2 depends upon the frequency of U*₁₂ transition and will be large at the beginning and gradually decay. Its time course is approximately expressed with decay constant τ_V2 by

$$L_{\text{hs}2}=-B_{\text{V}2}(1-\text{exp}(-t/{\tau }_{\text{V}2}))$$

(40)

where B_V2 is a constant.

As discussed above, there is the mutual interaction between the U* transition and the internal stress. (a) The U* transition causes the filament sliding which releases the internal stress. (b) The internal stress tends to expand the ρ distribution, pushes y_c forward and accelerates the U* transition. The interaction (a) is discussed first. The interaction (b) will be discussed later in relation with Phase 4.

The interaction (a) releases the internal stress and thus elongates the muscle. As shown in Figure 15, there is an overlap between Phase V2 and V3, and thus there is an overlap between length change L_hs2 and the length change due to the interaction (a). The combined length change is denoted as L_hs23, which is expressed by multiplying L_hs2 by exp(−t/τ_V3):

$$L_{\text{hs}23}=-B_{\text{V}2}\{1-\text{exp}(-t/{\tau }_{\text{V}2})\}\hspace{0.17em}\text{exp}(-t/{\tau }_{\text{V}3})$$

(41)

An example of L_hs23 is shown by the dotted black curve in Figure 17 to demonstrate its characteristics, together with L_hsv (the blue line).

The decay times τ_V2 and τ_V3 should be functions of T/T_0obs or v. The magnitude of τ_V2 is proportional to the time duration of occurrence of the U*₁₂ transition. Analogously to the manner to derive Equation 37, the relation τ_T3a = Bexp (−cy_c’) is expected as an approximate expression, where y_c’ is the y at the boundary between the violet area and red area in Figure 15(a), where y_c’ = −L + Δy_cV1+x₀. The tension T becomes smaller as Δ y_cV1 becomes larger. As a simple approximation, ΔT = T_0obs − T is set proportional to Δ y_cV1. Then the decay time τ_V2 is given approximately by

$${\tau }_{\text{V}2}=a_{\text{V}2}\hspace{0.17em}\text{exp}(c_{\text{V}2}T/T_{0\text{obs}})$$

(42)

where a_V2 and c_V2 are constants. The time constant τ_V3 in Phase V3 is related with the U* transition, and its magnitude seems to be an order of τ_step = L/v (Equation 5). Hence τ_V3 is set as

$${\tau }_{\text{V}3}=b_{\text{V}3}(L/v)$$

(43)

where b_V3 is a constant.

To see the characteristics of combination of L_hsv and L_hs23, L_hs23v is defined by

$$L_{\text{hs}23\text{v}}=L_{\text{hsv}}+L_{\text{hs}23}$$

(44)

An example of L_hs23v is illustrated by the dashed violet curve in Figure 17.

Now let us consider about the part (b) of the interaction that the internal stress tends to expand the ρ distribution, pushes y_c forward and accelerates the U* transition. This effect causes muscle elongation. In Figure 15, this effect is illustrated by depicting y_cVc in (c) larger than y_cVd in (d). In Phase 4, y_cVc changes into y_cVd and the filament sliding approaches to the steady value. As shown by the overlap of the arrows of Phases V3 and V4 in Figure 15, these effects overlap with each other. The length change in Phase V4 is denoted as L_hs4. In analogy to the expression of L_hs23 (Equation (41)), L_hs4 is approximately expressed by

$$L_{\text{hs}4}=B_{\text{V}4}{\{1-\text{exp}(-t/{\tau }_{\text{V}4})\}}^2\hspace{0.17em}\text{exp}(-t/{\tau }_{\text{V}5})$$

(45)

The characteristics of L_hs4 expressed by this equation are illustrated by the dashed green curve in Figure 17. The internal stress is zero at t = 0 and increases by the term {1 − exp(−t/τ_V4)}². The term exp(−t/τ_V5) corresponds to the decrease of the muscle length due to the filament sliding. Concerning the magnitude of B_V4, it is plausible that the internal stress pushes y_c forward more effectively when the internal stress rapidly increases, i.e., when the frequency of the U*₁₂ transition rapidly increases. The frequency of the U*₁₂ transition is given by the reciprocal of the duration τ_V2 given by Equation 42. Thus B_v4 is expressed by

$$B_{\text{V}4}=G_{\text{V}4}/{\tau }_{\text{V}2}$$

(46)

where G_v4 is a constant. The time constants τ_V4 and τ_V5 will be mainly related with the U* transitions, and in analogy to Equation 43 they are expressed as

$${\tau }_{\text{V}4}={\text{b}}_{\text{V}4}(L/v)$$

(47)

$${\tau }_{\text{V}5}={\text{b}}_{\text{V}5}(L/v)$$

(48)

where b₄ and b₅ are constant.

Since the sum of L_hsv, L_hs23 and L_hs4 is the total length change, it is denoted as ΔL_hs’

$$\Delta L_{\text{hs}}’=L_{\text{hs}23}+L_{\text{hs}4}+L_{\text{hsv}}$$

(49)

Note that ΔL_hs’ is the length change from the moment when Phase 1 finishes, while the abscissa ΔL_hs in Figure 16 includes the length change in Phase 1.

Time course of ΔL_hs’ was calculated with various trial values of the parameters looking for good agreement with the experimental data in the cases of (T_0obs − T)/T_0obs = 0.22, 0.44 and 0.87. Calculation results with the following parameter values are shown in Figure 18.

$$B_{\text{V}2}=12\hspace{0.17em}\text{nm}$$

(50a)

$$a_{\text{V}2}=0.14\hspace{0.17em}\text{ms}$$

(50b)

$$c_{\text{V}2}=7.6$$

(50c)

$$G_{\text{V}4}=15.0\hspace{0.17em}\text{pm}/\text{s}$$

(50d)

$$b_{\text{V}3}=0.20$$

(50e)

$$b_{\text{V}4}=0.8$$

(50f)

$$b_{\text{V}5}=0.3$$

(50g)

Characteristic features of the series of experimental data are fairly well reproduced by the calculation. The drastic change of the curve shape for different ΔT/T_0obs is mainly due to the T dependence of the frequency of the occurrence of the U*₁₂ transition, i.e., the relation τ_V2 = a_V2exp(c_V2T/T_0obs) (Equation 42).

7. Deviation from Hill-Type Force-Velocity Relation

Let us consider the deviation from Hill-type force-velocity relation in connection of the U*₁₂ transition. The deviation is determined by change of crossbridge population in well A2 and the U* potential barrier height. The maximum ratio of crossbridge population in A1 per that in A2 is given by x₀/(L−x₀) in Figure 7 (b3). This ratio is not very large and the effect of population change is neglected in the following approximate formulation.

The relative tensions T^H/T₀ and T*/T₀ are given as functions of y_c^H and y_c* in Figure 8. The same relationship can be expressed by giving y_c^H and y_c* as functions of T/T₀. From Equation 8, we have

$$y_{\text{c}}^{\text{H}}=L[{\kappa }_{\text{f}}-\{{\kappa }_{\text{f}}^2-({\kappa }_{\text{f}}-{\kappa }_{\text{b}}){\{{\kappa }_{\text{f}}-2p_0{T}^{\text{H}}/(T_{0}L)\}}^{1/2}\}]/({\kappa }_{\text{f}}-{\kappa }_{\text{b}})$$

(51)

From Equation 12, we have

$$y_c^{*}={[\{b-(T^{*}/T_0)\}/a]}^{1/2}$$

(52)

These relations and y_c^H − y_c* are shown as functions of T/T₀ = T^H/T₀ = T*/T₀ in the range between T_dev/T₀ and T_0obs/T₀ in Figure 19 (a).

The filament sliding velocity of Hill-type is denoted as v^H and the velocity in the presence of the U*₁₂ transition as v*. The period for the crossbridge to move over L is given as τ_step = L/v^H in Figure 5, which should have a similar nature as the time duration of U* transition, t_c = (1/A)exp(U*/kT) (Equation 4–2–1 in [2]), where A is a constant and U* is the potential barrier for the U* transition. Thus v^H = L/τ_step is approximately proportional to 1/exp(U*/kT). Here U* is the potential barrier height when there is no U*₁₂ transition. The difference y_c^H − y_c* shown in Figure 19 (a) can be regarded as a measure of the magnitude of structural change in MA₃. The potential barrier U* should be affected by this structural change, and the change of the U* height is approximately set as b_V(y_c^H − y_c*) where b_V is a constant proportional to 1/kT. Then exp(U*/kT) changes into exp((U*/kT) + (b_V(y_c^H − y_c*)) in T_dev < T < T_0obs. Then, if we put v^H = C/exp(U*/kT) and v* = C/exp((U*/kT) + b_v(y_c^H − y_c*)) with a constant C, we have

$$v^{*}=v^{\text{H}}/\text{exp}(b_{\text{V}}(y_{\text{c}}^{\text{H}}-y_{\text{c}}^{*}))$$

(53)

Since y_c^H = y_c* = y_cdev (cf. Figure 8), Equation 53 gives v* = v^H at y_cdev, as required. The blue curve in Figure 19(b) shows results of calculation by Equation 53 with the parameter value

$$b_{\text{V}}=7.9\hspace{0.17em}(1/\text{nm})$$

(54)

Agreement with the experimental data is fairly good. Accordingly, the deviation from Hill-type force-velocity relation is mainly due to the change of the potential barrier height U* caused by occurrence of U*₁₂ transition.

8. Summary and Discussion

In our previous paper [2] (the first part of this Remarks series), difficulty of the power stroke model is pointed out and a new model is proposed to avoid the difficulty. In the model, it is proposed that about 41% of the myosin heads are bound to actin filament and each bound head forms complex MA₃ with three actin molecules. The complex MA₃ translates along the actin filament changing its partner actin molecules in cooperation with U* transition. This model well explains the properties in the steady filament sliding such as the tension-dependence of the muscle stiffness, the Hill-type force velocity relation and the tension-dependence of energy liberation rate. In the present paper, the isometric tension transient and isometric velocity transient are studied based upon the model. Statistical ensemble of crossbridges is considered and the binding probability density ρ is introduced. On rectangular ρ approximation, the edge of the rectangle, y_c determines dynamic properties of muscle in cooperation with U* and U*₁₂ transitions. The internal structure of the MA₃ complex becomes temporally unstable by the sudden length change or by the sudden load change. The complex muscle behaviors observed in these transients are related with the process that the disturbed internal structure returns to its stationary state.

Results reported in the present paper are summarized as follows.

• The tension variations in the first Phases in the isometric tension transient (Ford et al. [16]) and the isotonic velocity transient (Civan and Podolsky [17]) are well explained as shown in Figure 9.
• Ratios of extensibilities of crossbridge, myosin filament and actin filament are estimated as 0.22, 0.26 and 0.52 (Equation 30), in reasonable agreement with the approximate values (0.31, 0.27, 0.42) determined by X-ray diffraction by Wakabayashi et al. [21].
• The experimental data on the isometric tension transient reported by Ford et al. [16] are fairly well explained as shown in Figure 14.
• The characteristic features of muscle in the isotonic velocity transient observed by Civan and Podolsky [8] are fairly well explained as shown in Figure 18.
• The deviation from the Hill-type force-velocity relation observed by Edman [12] is reproduced as shown in Figure 19(b).

It should be noted that the above-mentioned agreements between experimental data and calculation results are obtained by using the muscle stiffnesses κ_f and κ_b determined in [2], whose numerical values are given in Appendix.

The obtained results suggest that the ideas of the ensemble of crossbridges and the rectangular ρ approximation are useful tools in theoretical studies of muscle contraction.

In Figure 8, tangent of the T*/T₀ vs. y_c* relation (the blue curve) is 0 at y_c* = 0. Accordingly, small fluctuation of tension around T_0obs can produce significant variation of the muscle length. In this connection, the spontaneous oscillatory contraction (SPOC) of muscle (cf. the review by Ishiwata and Yasuda [24]) seems interesting. As mentioned in [24], it is a possibility that SPOC has some relation with the activities of cardiac muscles (cf. [25]).