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Muscle contraction mechanism is discussed by reforming the model described in an article by Mitsui (

The contraction of muscles takes place by mutual sliding of a thick (myosin) and a thin (actin) filaments. A. Huxley [

Difficulties in the power stroke model are discussed in Sect. 2. Basic ideas of the new model are explained in Sect. 3. Various experimental data are theoretically reproduced in Sect. 4. Additional comments are given in Sect.5. The obtained results are summarized and discussed in Sect. 6. A list of the parameter values used in calculation can be found in the

Generally the first step to construct a molecular model in material physics is to look for a thermodynamic relationship among parameters to appear in the model and put restrictions on the manner to construct models (e. g., cf. [

It is assumed that a myosin head exerts force to an actin filament only when it attached to an actin. The mean force exerted by the myosin head on the actin filament is denoted as _{ATP}. Measured macroscopic quantities during muscle contraction are the tension _{ATP} used for work is given by ε_{ATP}

The tension _{hs} (hs: half sarcomere length) is the total number of myosin heads contained in this layer and _{hs}, then

or

Combining

This is the required thermodynamic relationship

By inserting the values of ε_{ATP}, _{hs} and _{max} (^{3} at the tension

Additional comments on

According to Geeves and Holmes [

The stiffness of a muscle, which varies depending upon the stress in the muscle, is assumed to be a measure of _{0} is 0.35 where _{0} is the isometric tension. Therefore, even in case that the value in _{0} becomes

On the other hand, the value of _{0} can be obtained by _{0} (cf. _{0} (the isometric tension per head). The experimental results of Ishijima _{0} is close to 5.7 pN. We adopt this value in calculation:

Then

X-ray data are favorable to the

It should be noted that the _{0}. The

X-ray diffraction data suggest that

It should be noted that we have

by combining the constant _{0} when we obtain an expression of the quantity as a function of microscopic parameter _{0}. This is very convenient in theoretical treatment.

As mentioned in the preceding section, the ratio of stiffness of the crossbridge at _{0} is 0.35 where _{0} is the isometric tension, and this

According to

On the other hand, much smaller values of _{J} (cf.

In the new model, _{0} and the equatorial X-ray measurements. Originally the large value of

In experiments by Ramsey and Street [

Yagi

Generally, when a molecule A bounds to an assembly of B molecules, structural changes occur in both A and B and rearrange the manner of the binding, resulting in formation of the locally deformed complex of A and B. If molecules A and B are electrically charged and have no center of symmetry (i. e. structurally polar) as in the case of protein molecules, the complex formation will be enhanced by electric and piezoelectric interactions. It seems plausible that sometimes the molecular complex moves in the assembly of B as in the case of a polaron in ionic crystals. Presumably some readers are not familiar with the polaron, and a simplified scheme of the polaron formed by an electron in two-dimensional KCl crystal is illustrated in

In ^{+} ions move closer to e and Cl^{−} ions apart from e, forming the strain field around e. The electron plus the strain field is called a polaron. The polaron moves in the crystal by hopping over a potential barrier, changing the mate ions (c).

The large value of

Rayment

It is assumed that molecular structures change as shown in _{3}. _{eq} in the case of a single myosin head.

Now a contracting muscle is considered. The myosin filament is thought to be moving to the right at a constant velocity _{eq} assuming that the molecular shape is the same as the single myosin head in _{eq} |. The stress which the myosin head exerts on the myosin filament is indicated as _{eq} < 0 (_{eq} > 0 (_{eq} is positive and the mean θ in the isometric contraction is negative as shown in

1973, Mendelson

The crossbridge is the tail plus myosin head. In

The _{eq} is indicated as _{eq} (

By definition, _{J}:

Note that _{J} has the same sign as _{J}. Then,

where κ is the mean stiffness of the crossbridge. Since the head and tail are connected in series as elastic elements, we have

where κ_{H} and κ _{T} are the stiffnesses of the myosin head and tail, respectively. When _{T} seems much larger than κ_{H} due to the α-helical structure of the tail, and κ is nearly equal to κ_{H}. On the other hand, when _{T} should be much smaller than κ_{H} due to the bending flexibility of the tail and κ is nearly equal to κ_{T}. Thus, if κ is expressed by κ_{f} for the forward force (_{b} for the backward force (_{f} is nearly equal to κ_{H} and κ_{b} is nearly equal to κ_{T}, and κ_{f} is expected larger than κ_{b} With these notations,

Then, from

Magnitudes of κ_{f,} and κ_{b} will be determined in Sect. 4. 1.

_{J} = 0 and _{eq} and _{J} becomes positive as shown in _{J} increases so that the transition from well 2 to 3 becomes possible. _{J} becomes so large that the catalytic domain is going to translate to the next site. _{eq} < 0 and the head will pull the myosin filament forward till θ reaches θ_{eq.} After θ increases over θ_{eq}, the state will become as shown in

Based upon Eyring's theory of the rate process [

where _{J} increases for

where _{J} is given by κ_{b}

Hence _{c} (c: critical). Hereafter, we suppose that all transitions occur simultaneously when _{c}. This approximate treatment will make various calculations simple.

_{c} defined in the preceding section. The head in _{c} − _{c} to _{c} − _{c}. Thus _{c} − _{c} with the filament sliding in the process from _{c} is large for fast sliding as shown in

A question arose of what determines the isometric tension, where the sliding velocity is zero and the above cycle stops. In this connection, the double hyperbolic force-velocity relation found by Edman [_{0} becomes larger than about 0.68 (cf. _{c0} (_{c} at _{0}) but this explanation does not seem very realistic. Instead, another mechanism is proposed here that the isometric tension is related with a tolerance limit for “pull-up” detachment of the head from the potential well 3 toward well 2 in _{0} increases beyond 0.68. In _{c} = _{c}* which corresponds to _{0} = 0.68, increasing with decreasing _{c}. Then, _{c} − _{c}* − _{c} < _{c}* as indicated by the arrow “Pull-up transition”. The _{c} − _{c}* − _{c} < _{c} as the region _{c} < _{c}* is widened, implying that reduction of _{0}) – 0.68} (cf.

The potential barrier _{J} as _{0} − _{J} (Eq, 3-5-2), where _{J} is the force exerted on the head by the myosin filament. This force is originally produced by other myosin heads attached to the myosin filament. In this sense, myosin molecules belonging to the same myosin filament help each other to make their step motion easy by decreasing others’_{0} in Sect. 4.3 (cf.

Lymn and Taylor [_{ATP} stored in the head are used in forming the complex MA_{3} (_{i} corresponds to ith actin molecule. The force _{J} lowers the potential barrier _{i–1} and bind to A_{i}. Associated with this step, myosin head produces force by spending the energy indicated as

It is assumed that the probability

This idea is similar to the model proposed by Huxley in 1957 [_{ATP} (about 21_{ATP} is used in each cycle of force production. The magnitudes of the energy fraction will be discussed in Sect. 4.3.

Huxley and Simmons [

Isotonic velocity transients were studied by Podolsky [

As already mentioned n Sect. 2.2, the stiffness of a muscle varies depending upon stress, and is regarded as a measure of

The X-ray diffraction studies on the extensibility of myosin and actin filaments by Huxley

The stress _{c}. In this section, we discuss the muscle stiffness on the same simplified scheme. Since

Thus we obtain

From _{c} is given by

Let us denote _{c} at _{0} as _{c0} and _{c} at _{c}(0), then from

The stiffness is κ_{f} in the region _{c} − _{b} in the region 0 < _{c}, and

_{0} = _{0} (3-1-2). By the same reason we have

where _{0}) and _{0}) are, respectively, _{0}. Then, from

The ratio _{0}) was experimentally determined by Ford

The muscle stiffness _{0}) is 0.35 as seen in

Then, from

Since _{0} = 5.7 pN (_{f}, κ_{b}, _{c0}, _{c}(0)) which are related by the three _{c}(0) is _{c} for the fastest contraction, _{c}(0) is relatively close to _{f}, κ_{b,}_{c0}, and _{c} by _{0}) by _{c}(0). After several calculations, good agreement with experimental data was obtained with

Calculated result is shown by the curve in _{c}(0) in

Now we can calculate _{c} as a function of _{0} by _{c} vs. _{0} relation will be used for various calculations.

The probability _{0} − _{0} as discussed concerning _{c0}, where _{co} becomes _{c} at _{0}. It follows from the above equations that _{c} as discussed in Sect. 3. 5. Then the mean time _{c} needed to complete the transition is approximately given by the inverse of

where

From the other viewpoint, the transition is approximately completed while _{c0} to _{c} with the velocity _{c} can be expressed by

From

where

The velocity _{0} when _{c} are given in

Edman [

Agreement with the experimental data is good for _{0} < 0.7, but there is a distinct deviation from the calculation curve in larger _{0} region. In Sect. 3.6, the pull-up mechanism was considered as a possible origin of the deviation.

Energy liberation rate in the crossbridge mechanism is denoted as _{max}, _{max}/2, and 0. It is assumed that 30% of the heat production at _{0}, which is 4.4 kW/m^{3}, comes from a non-crossbridge source and this amount is subtracted from the experimental values, with the results

These values are shown by filled circles in

_{1}. Generally there is an energy loss in working machine, which is called the maintenance heat for muscle and denoted as _{0}:

(_{1} consists of two parts. A fraction of ε_{ATP} is spent for the force production by each head attached to actin during crossing over _{1,1}. For such force production, the complexes (MA_{3}) have to be formed. A fraction ofε_{ATP} is used firstly to create (MA_{3}) in the process from _{1,2}. Then

Let us also denote the portion of ε_{ATP} that the head spends to produce force during each step motion as _{1,1} is given by

where (

where _{g}

where

As for (_{1,2}, the frequency of the attachment-detachment of all the heads in unit volume is given by (_{1}/ε_{ATP}. Let the energy spent by each head for the attachment-detachment be _{1}, then

Thus

That is,

Thus

where

The maintenance heat _{0} is assumed to be a constant as was done by Hill [^{3} given in

_{g}_{0} with the value of

Calculated values of

Now we can calculate _{c}) = _{0} − _{c} (_{c} in _{0} in

Hill [_{0} + 1.18_{0}. It should be noted that _{0}−_{c}) in

The portion of ε_{ATP} used for each force production step, _{g}_{c}) (_{g}_{g}_{ATP}) (_{g}_{g}_{g}_{c}). With _{g}_{g}_{c}) was calculated and the result is shown by dashed curve in

From

Huxley and Simmons [

The fastest response, phase 1, seems to be related with elastic length changes in the crossbridge. The muscle length is controlled from outside in the isometric tension transient and the initial and final states are the stationary state. It is assumed that, in the intermediary, the barrier

To discuss phase 2, potential distribution other than the vicinity of wells 1 and 2 can be treated as infinity since there is practically no translation over _{af}. Now we consider heads in a muscle, where the head is under the influence of the elastic force _{J}. The nature of _{J} is already known through _{J} is discussed by using an elastic potential _{el}. Thus the total potential energy

The stepwise length changes cause a step-change in tension and thus a step-change in _{el}. Then the ratio of head populations in wells 1 and 2 will alter so that _{el} changes toward the value at the stationary isometric state. The changes in population and tension are calculated as functions of time based upon the statistical mechanics in [_{hs}. Here, the notations used by Ford _{0} is the isometric tension. _{0} are the same as _{0}, respectively, in the preceding sections. The experimental data on _{0} by Ford

Values of _{1}/_{0}, _{2}/_{0} and _{2}/_{0} were obtained for various length changes Δ_{hs} [_{1} is for _{2} is for _{2}’ is for _{1}/_{0} and _{2}/_{0} by solid curves and _{2}’/_{0} by dotted curve in _{1}/_{0} and by squares for _{2}/_{0}. The solid curves well reproduce the experimental data. The calculation results are similar to those reported by Ford et al [

As mentioned in Sect.3.10, isotonic velocity transients were studied by Podolsky [

Discussion on the isotonic velocity transient in Sect.V of [_{c} increases from _{c0} to a new value _{c}’. Then the heads in _{c0} < _{c}’ will cross over _{el} (cf. _{el} will change the ratio of head populations in wells 1 and 2, producing a balance between the outside and inside stresses. The balance is realized around e in _{J} and reduces

Some readers of [

Yanagida _{ATP} _{hs}_{hs},

Giving their notations in parenthesis, Yanagida _{F}), ATP activity of sarcomere (^{s}_{ATP}) and the average number of myosin molecules in half a thick filamenit overlapping with thin filament during sliding (_{m}).

Harada _{F}), the number of ATP molecules spent per unit length of an actin filament per unit time (d_{i}/d^{8} as the most probable number in calculation [

It is assumed that 41% of myosin heads are attached to actin in contracting muscles (

There are two actin filaments per myosin filament in frog skeletal muscle. An actin filament has two actin molecules per 5.46 nm. A myosin filament has three myosin molecules per 14.3 nm. Consequently, the relevant densities per nm are 2×2/5.46 = 0.73/nm for actin molecule and 3×2/14.3 = 0.42/nm for myosin head. If 41% of the heads are attached to actin, the density of the attached myosin head is 0.42×0.41 = 0.17/nm. Therefore, available actin molecules per attached myosin head is 0.73/0.17 = 4.3. If the head uses 3 out of the 4.3 actin molecules to form the complex MA_{3}, the remaining 1.3 actin molecules will provide a space which makes movement of MA_{3} possible. Hence, the figure 4.3 means almost the full use of actin molecules for the filament sliding. As already mentioned in Sect. 2. 2, by X-ray diffraction study, Matsubara

Concerning the cooperativity of myosin heads discussed in Sect.3.7, the observation by Nothnagel and Webb [

Myosin heads move as fast as 60μm/s in the algae, _{c})/

The new model described in the present paper is characterized by the constant _{3} (Sect. 3. 3), the nonlinear elastic property of the crossbridge (_{J}-dependence of of

Calculations based on the model well explain experimental data on muscle stiffness (

In the power stroke model, an actin filament is treated as a relatively passive element like a ladder for a myosin head. In general, however, protein molecules have their proper active functions in biological systems. There was a question of why an actin molecule plays such relatively passive role in muscle. In the new model, actin molecules play more active roles in mutual cooperation with myosin.

The structure of a protein molecule is polar, i. e., the center of symmetry is lacking in its structure. Generally mechanical properties of the polar systems are treated as four-variable systems i. e., with strain, stress, electric field and polarization. Two of the four (e. g., stress and electric field) are adopted as independent variables and the others are dependent variables (cf, e.g., [

We would like to thank Prof. Reuven Tirosh of Bar Llan University for suggesting submission of the present article to this journal.

3.77×10^{−21} J at 0°C

period of actin strand projection onto the filament axis: 5.46 nm

number of myosin heads in 1 m ^{3}, calculated with _{hs} and ^{23} m^{−3}

_{hs}

number of myosin heads in a volume with a base of 1 m^{2} and a thickness of half the sarcomere length : 1.76×10^{17} m^{−2} [

_{0}

isometric tension : 4.1×10^{5} N/m^{2} [

_{0}

mean force produced by one myosin head in isometric contraction: 5.7×10^{−12} N/head [

ratio of the number of myosin heads simultaneously in the attached state to the number of all the heads: 0.41 (

sarcomere length: 2.10 μm [

_{max}

shortening velocity under no load near 1.8 °C: 2.25(

_{max}

velocity of filament sliding under no load in muscle at 1.8°C: 2.36 μm/s [

_{c}(0)

critical

_{c0}

critical

_{ATP}

energy liberated by the hydrolysis of one ATP molecule: 8.0×10^{−20} J/molecule [

_{f}

stiffness of crossbridge when the myosin head exertes positive force on a myosin filament: 2.80×10^{−3} N/m = 2.80 pN/nm (

_{b}

stiffness of crossbridge when the myosin head exerts negative force on a myosin filament: 0.26×10^{−3} N/m = 0.26 pN/nm (

The formation of a polaron in two-dimensional models of ionic crystal, following Figure 19 of [

Diagram showing sequential changes in the potential of force acting on ATP-activated single myosin head M (subfragment-1) in binding to an actin filament. The numbers 1, 2, 3 are assigned to the potential wells at the binding site on actin molecules on the same strand. (a) Periodic potential distribution when M is sitting at a position apart from the actin filament. (b) Just after M attaches to actin 1. (c) Molecules deform and potential distribution changes. (d) Equilibrium potential distribution. M is statistically distributed in wells 1 and 2.

Formation of a complex of myosin head and actin molecules (MA_{3}). (a) Just after attachment of myosin head to actin molecule corresponding to _{3} corresponding to _{eq} is the bending angle of neck domain at equilibrium.

Deformation of a myosin head while the catalytic domain stays at the same actin molecule. The small square on the tail is to show symbolically the bending flexibility of the tail. (a) Myosin head is at the equilibrium angle, θ_{eq}. (b) Myosin head is pulling myosin filament forward. (c) Myosin head is pushing myosin filament backward.

Various quantities in the domain of _{eq} (

Step motion and deformation of a myosin head in shortening muscle. (a) Rightward tilt of neck domain just before catalytic domain moves to right. (b) Leftward tilt of neck domain just after catalytic domain moves to the new site. (c) Rightward tilt of neck domain when catalytic domain is ready to next movement to right.

Cycle of variation of _{0} > 0.68.

Energy flow and chemical reaction associated with force production. Chemical reaction is a series of detachment from actin A_{i−1} and attachment to A_{i}. The fraction, _{ATP} given to the head by the ATP hydrolysis.

Relative stiffness _{0}) of muscle as a function of _{0}. Open circles: Experimental data cited from

_{c} as a function of _{0}, calculate by

Force-velocity relation. Black circle are experimental data by Edman [

Energy liberation rate _{0}. Filled circles: experimental data by Homsher

Calculated _{c})/_{g}_{c})/_{0}. _{c}) = _{0} – _{c} (_{g} is 1.69 (_{c}) (cf, _{c}) and _{g}_{c}).

_{0} relationship at about 0°C calculated using _{ATP}.

_{0} as a function of time _{hs} in the isometric tension transient. _{0}: the isometric tension _{0}. Circles: experimental data cited from Figure 23 in the article by Ford

_{1}/_{0} and _{2}/_{0} as functions of Δ_{hs}. Circles and squares: experimental data cited from _{0} is the isometric tension _{0}, _{1} is for _{2} is for _{2}’ is for

Illustration for the isotonic velocity transient, drawn following _{0} = (_{0}−_{0}, which is suddenly altered at