† Corresponding author. E-mail:
Project supported by Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region, China (Grant Nos. NJZY16493 and NJZC17458).
The spontaneous oscillatory contraction (SPOC) of myofibrils is the essential property inherent to the contractile system of muscle. Muscle contraction results from cyclic interactions between actin filament and myosin II which is a dimeric motor protein with two heads. Taking the two heads of myosin II as an indivisible element and considering the effects of cooperative behavior between the two heads on rate constants in the mechanochemical cycle, the present work proposes the tenstate mechanochemical cycle model for myosin II dimer. The simulations of this model show that the proportion of myosin II in different states periodically changes with time, which results in the sustained oscillations of contractive tension, and serves as the primary factor for SPOC. The good fit of this model to experimental results suggests that the cooperative interaction between the two heads of myosin II dimer may be one of the underlying mechanisms for muscle contraction.
Skinned skeletal and cardiac muscle fibers can exhibit the dynamic behavior of spontaneous oscillatory contraction (SPOC), that is, the sarcomere length changes periodically with the saw-tooth waveform.[1–6] The SPOC is considered to be the third state of muscle, which occurs at the activation level that is intermediate between full activation and relaxation.[7] This phenomenon is observed for both skeletal and cardiac muscle in the physiological conditions realized at around pCa∼6 (termed Ca-SPOC),[4] or at high concentration of MgADP (ADP: adenosine diphosphate) with MgATP in the absence of Ca2+ (termed ADP-SPOC).[5] Besides, even at full activation level with Ca2+, SPOC also occurs by the addition of inorganic phosphate (Pi).[6] In summary, SPOC is a universal phenomenon that occurs under a wide range of conditions.
Some theoretical models for SPOC have been proposed.[8–15] Julicher and Prost[8] presented a physical mechanism which could lead to the oscillatory motion of molecular motors cooperating in large groups when the system was elastically coupled to its environment. Smith and Stephenson[9,10] presented a concrete model based on the experimentally obtained physiological properties of muscle. This model considered the structural and functional factors inherent to the striated muscle, attributing the essential cause of auto-oscillation to the elastic force provided by titin. In recent years, Sato et al.[11–13] put forward several similar theories for SPOC. They assumed that the probability of cross-bridge formation was depended on the lattice spacing between the thin and thick filament, while the lattice spacing decreased with the increase of the sarcomere length. Based on the two-state ratchet mode, Li and Wang[14] studied the dynamics of collective myosin II motors. The model results showed that spontaneous oscillations could be generated if the system was elastically coupled to its environment. All the above models help us understand SPOC in different ways, but these theories involve little about the mechanochemical cycle of myosin, failing to exam the dependence of SPOC on the chemical conditions in the experiments. Guo and Luo[15] introduced a chemical kinetic model to describe the muscle contraction and SPOC, considering the dependence of chemical rate constants on the sliding velocity of myofilament and the activation level of muscle. Nevertheless, this model attributed the agent of SPOC to the external factors.
Muscle contraction results from cyclic interactions between the contractile proteins myosin and actin, driven by the turnover of adenosine triphosphate (ATP). Knowing the mechanochemical cycle process of myosin is critical for understanding the mechanism of muscle contraction. Myosin II is a dimeric protein with two heads. The mechanochemical cycle for single myosin head has been intensely studied, but the role of the second head in the myosin II and whether the inter-head action will affect the rates in the ATPase cycle are still uncertain.[16] Most models for muscle contraction assumed that the two heads of myosin II acted independently from each other, thinking little about the effects of the interhead action on the mechanochemical cycle. In recent years, however, the single molecular experiments showed that both the inter-molecular force and the external load on molecular motors could greatly affect the rates of the mechanochemical cycle.[17–20] In addition, some experimental results suggested different forms of cooperative behavior between the two heads of myosin II.[21–25] Thus, myosin II dimer may show some different mechanochemical characteristics from a single myosin head.
In this paper, we take into account the effect of cooperativity between the two heads on the rates of the mechanochemical cycle, and propose a mechanochemical model for myosin II. This model gives all possible states of myosin II dimer and the transition between these states. According to mass action law, the proportions of enzyme species (the myosin II dimmers in different states) are calculated. To our surprise, if we assume that the attached myosin heads will help the partner heads rapidly attach to the actin filament,[25] the proportion of myosin II dimers in different states will periodically change with time, which leads to the sustained oscillations of contractive tension and serves as the major contribution to SPOC. As a check on the model, the equation of force balance for SPOC under isotonic conditions is built, by which the time-dependent change in the sarcomere length is calculated. The simulation results are consistent with the experimental observations. This theoretical work may throw some light on further understanding of muscle contraction. Furthermore, since the step manner and cooperating mechanisms of myosin II are different from those of kinesin, myosin V, and cytoplasmic dynein, the present mechanochemical model supplies a new quantitative analysis method to study the mechanism of cooperative behavior in systems of molecular motors.
In order to produce contractive tension, myosin II couples a chemical cycle of ATP hydrolysis to a mechanical actin–myosin interaction. Even at the simplest level, a minimal chemical cycle involves ATP binding, hydrolysis, and subsequent release of Pi and ADP.[26,27] Many models for the mechanochemical cycle of single myosin head have been proposed.[28–31] To study the cooperative interaction between the two myosin heads and the effect of product concentration on SPOC in a simple way, this paper adopts the four-state mechanochemical cycle model for single myosin head, as shown in Fig.
In Fig.
The mechanochemical cycle models of single myosin head neglect the cooperative behavior between the two heads. But different forms of cooperative behavior have been proposed to interpret the experimental results. One type of the cooperative behavior is positive cooperativity, that is, the binding of one head promotes the attachment of the second head. This positive cooperativity helps to explain the apparently faster attachment rate of cross-bridges during muscle stretch and shortening.[25] Based on these facts, this paper takes the two heads as an indivisible whole and proposes a mechanochemical model for myosin II dimer, as shown in Fig.
In Fig.
In this model, the cooperative behavior between the two heads is characterized by the effect of inter-head interaction on the rates in the chemical cycle, and it is expressed quantitatively by the ratio of
According to the mass action law, the set of chemical dynamics equations describing the graph in Fig.
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The adenine nucleotides are transported through skeletal muscle fibers during muscle activation, and the myofilament lattice significantly reduces the diffusion of adenine nucleotides.[33–35] As a result, from the outside of myofilament to the center of myofilament, the ATP concentration varies from place to place. Due to the reaction–diffusion effect, the concentration of ATP inside myofilament may change with time. Thus, the average concentration of ATP in the local neighborhood of myosin active sites, y, is a time-dependent variable. Since the influence of myofilament lattice on the diffusion process is very complex, it is difficult to establish the reaction–diffusion equation accurately. The focus of this paper is to study the effect of cooperation between two myosin heads on the mechanochemical cycle. Therefore, the detailed analysis of the diffusion process is not conducted, but only a relatively simple mathematical expression is given below.
The diffusion effect of ATP is mainly reflected in the radial direction of muscle fibers. According to Fickʼs law, the diffusion of ATP follows the equation
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn4.gif)
Assuming that the volume of myofilaments is V, the side area is A, and the average concentration of ATP in myofilament is y, then the amount of ATP diffused into the muscle fiber per unit time is
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If the shape of the myofilament is a cylinder with a length of l and a radius of r, then
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There is still no experimental data on the concentration gradient of ATP diffusion in the muscle fibers. For the sake of simplicity, we assume that the concentration gradient is equal everywhere inside the muscle fibers, position-independent, but only as a function of the average ATP concentration y. So
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn5.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn6.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn7.gif)
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In the experiments of SPOC, the concentration of ATP outside the myofilaments is constant. Here, suppose its value is a mM. The ATP concentration in the center of myofilaments is less than the average concentration of ATP due to the reaction–diffusion effect. For simplicity, let us set
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Considering that the values of
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According to Eq. (
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn10.gif)
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The SPOC is often studied under isotonic and isometric conditions. Yasuda et al. made a deep research into SPOC and got the time-dependent change in the sarcomere length (SL) under the isotonic condition.[1] To check the proposed mechanochemical model of myosin dimer, the present work establishes the kinetic equation describing the force balance along the myofilament, as shown by
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn11.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn12.gif)
In Eq. (
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn13.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn14.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn15.gif)
![](cpb_27_11_118702/cpb_27_11_118702_ieqn16.gif)
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![](cpb_27_11_118702/cpb_27_11_118702_ieqn17.gif)
The second term in Eq. (
The last two terms in Eq. (
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To analyze the cooperative effects of the two heads of myosin II dimer, the proportion of myosin II dimer in different states is numerically calculated for given specified rate constants. Table
![]() | Table 1. Model parameters. . |
The constants k1 and k2 are related to the level of muscle activation, which depends on the concentration of ADP and Ca2+. The value of k1 is taken as 0.1 s−1, while k2 changes with various muscle active conditions.[39,40] The actual value k2 employed in our model is 0.5 s−1.
The rate k3 is the first-order rate constant of ATP binding to myosin, and K3 is the second-order rate constant of ATP binding to myosin. So,
The constant k4 is the binding rate of single myosin head occupied by ATP to thin filament, and k8 is the association rate of single myosin head while another head has been attached to thin filament, as shown in Fig.
The rate constant k5 is determined by the inverse process of ATP hydrolysis, and the rate of 1 s−1–10 s−1 for k5 is adopted by this work.[40,42–44] The constants k6 and k7 are related to the reversible steps of ADP release. In the experimental research by Yasuda et al., the concentration of ADP is 4 mM. As the second-order rate constant of ADP binding to myosin (K2) is about
The diffusion coefficient of small molecules in solution is around 10−8 m2/s–10−9 m2/s, and the radius of myofilament used in experiments is in the ranges of 10−5 m–10−6 m. Referring to the related literature,[1,34,35] the value of
Substituting the above parameter values into Eqs. (
The numerical simulation results show that the proportion of myosin II dimers in different states, xi, periodically changes with time. The muscle contraction force is proportional to the number of power strokes, which is mainly decided by x5 and x6 in the given activation level. Thus, the force cyclically changes with time, and this is the basic reason for SPOC. It is also found that all the values of xi tend to be constants if the rate of k4 is equal to k8. This brings about the conclusion that one of the necessary conditions for SPOC is
The simulation results also show that the concentration of the ATP around the local area of active site changes periodically with time, as shown in Fig.
The time-dependent changes in sarcomere length are calculated by Eq. (
Substituting the above related parameter values into Eq. (
In the process of muscle contraction, the lattice spacing between the thin and thick filaments varies with the sarcomere length. So the cross-bridge formation probability coupling to the muscle structure is not a constant. If the coupling effect is not taken into account, the value of q will hold as a constant
In addition, the simulation results show that the term Fk in Eq. (
As a further check on our model, we test the dependence of SPOC on the concentration of ADP ([ADP]), that is, the change in SL–time relationship with the parameters of k1 and k7. The parameter k7, the result of multiplying [ADP] and K2 (the second-order rate constant of ADP binding to myosin), characterizes the influence of [ADP] on the transition processes,
The simulation results show that the wave form, amplitude, and period of SPOC are influenced by [ADP]. Both the amplitude and the period decrease with the increase of k7, accompanied by the change of oscillation from the saw-tooth waveform to the triangular type. A similar tendency is also observed when k1 is increased, because the increase in the ADP concentration may also change the activation level of muscle contraction.
The force equation (
In our model, the level of muscle activation, the cooperative effect between the two heads of myosin, and the concentrations of ATP and ADP are important conditions for spontaneous oscillation of muscle contraction.
The level of muscle activation is determined by the concentration of Ca2+ or ADP in solution and is expressed mainly by the rate constant k1. When the rate of k1 is relatively larger or smaller, the muscle is in a contracted or relaxed state. Conversely, only when the value of k1 is moderate, the sustained oscillations of contractive tension may be generated. Therefore, muscle activation at a moderate level is the basic condition for SPOC.
The cooperative effect between the two heads of myosin dimer is reflected by the ratio of
The concentration of ATP and ADP in the solution is the external condition at which the muscle produces spontaneous vibration. The concentration of ATP determines the value of the parameter a, and thus the value of the rate k3 can be calculated; the concentration of ADP determines the value of k7 while affecting the value of k1. The rest of the rate constants k2, k4, k5, k6, and the value of parameter c, are constant for the given experimental conditions. Some experimental observations show that the concentration ratio of ATP to ADP ([ATP]/[ADP]) has a great influence on SPOC. By the numerical simulations, we find that increasing the concentration of ADP (i.e., increasing the rate of k7) at a given ATP concentration ([ATP] = a mM) gradually changes the period and the amplitude of SOPC until the vibration disappears. This conclusion is consistent with the observed phenomenon in the experiments.
The reaction–diffusion effect of ATP inside muscle fibers is also a necessary condition for SPOC. Without considering the reaction–diffusion effect, the average value of ATP concentration near the active sites of myosin, y, should be assumed as a constant, then the steady-state solutions of the differential equations are also constant and do not change with time. In this case, the muscle will be in a contracted or relaxed state. However, the reaction–diffusion equation (
The conditions for SPOC to occur are very extensive. It means that SPOC is the characteristic of the dynamic system determined by the muscle structure and the mechanochemical cycle of myosin. Based on the idea that the cooperative behavior between the two heads of myosin II dimer may affect the rate in the mechanochemical cycle, this paper proposes a kinetic model for myosin II dimer to describe the spontaneous oscillatory contraction of myofibrils. The chemical kinetic analysis for the model indicates that the coordination of two heads leads to the cyclic change in the proportions of myosin dimer in different states. As a result, the contractive force also periodically changes with time, which is the intrinsic cause of SPOC. This kind of remarkable nonlinear behavior derived from the mechanochemical model for myosin II dimer suggests that the inter-head cooperative behavior may play a critical role in the muscle contraction. Although some mechanochemical models have been proposed to explain the cooperative behavior between the two heads of processive motors, such as myosin V, kinesin, and dynein, the cooperative mechanisms of these motors are obviously different from that of myosin II, which works in a collective and collaborative manner.[54] Thus, the mechanochemical model established by this paper may supply a new quantitative analysis method to study the cooperative characteristics of molecular motor systems.
The present work also investigates the effect of muscle structure on SPOC. The simulation results show that the muscle structure could significantly affect the waveform and amplitude characteristics of SPOC. During muscle contraction, the lattice spacing between the thin and thick filaments varies with the sarcomere length, corresponding to the change in the cross-bridge formation probability. So, the coupling relation between the muscle structure and the cross-bridge formation is an important factor for SPOC. Furthermore, the passive force originated from the elastic element in the myofibril helps to keep the amplitude of SPOC in a reasonable range, although the passive force is not a necessary condition for SPOC.
Considering that the myofilament lattice can significantly reduce the diffusion of adenine nucleotides, this paper establishes the reaction–diffusion equation about ATP. It is unexpected that the concentration of ATP inside myofilament periodically changes with time. Since the SPOC phenomena occur under a fixed chemical condition, early theoretical studies have considered that SPOC is not driven by the oscillation of chemical parameters, such as the concentration of ATP, Pi, or Ca2+, but owes to other factors that contractile apparatus inherently possesses. These theoretical studies may ignore the diffusion of adenine nucleotides from the solution to myofibril. Further research is needed to tell whether the oscillation of chemical parameters plays an important role in the occurrence of SPOC.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] | |
[46] | |
[47] | |
[48] | |
[49] | |
[50] | |
[51] | |
[52] | |
[53] | |
[54] |