Motor Force Homeostasis in Skeletal Muscle Contraction

Abstract

In active biological contractile processes such as skeletal muscle contraction, cellular mitosis, and neuronal growth, an interesting common observation is that multiple motors can perform coordinated and synchronous actions, whereas individual myosin motors appear to randomly attach to and detach from actin filaments. Recent experiment has demonstrated that, during skeletal muscle shortening at a wide range of velocities, individual myosin motors maintain a force of ∼6 pN during a working stroke. To understand how such force-homeostasis can be so precisely regulated in an apparently chaotic system, here we develop a molecular model within a coupled stochastic-elastic theoretical framework. The model reveals that the unique force-stretch relation of myosin motor and the stochastic behavior of actin-myosin binding cause the average number of working motors to increase in linear proportion to the filament load, so that the force on each working motor is regulated at ∼6 pN, in excellent agreement with experiment. This study suggests that it might be a general principle to use catch bonds together with a force-stretch relation similar to that of myosin motors to regulate force homeostasis in many biological processes.

Introduction

The coordinated action of multiple myosin motors has so far been investigated mainly in skeletal muscle contraction (1–5), although it is equally essential for many other biological processes such as cellular mitosis and neuronal growth. Despite a tremendous amount of studies since the 1930s (1), the detailed mechanisms and pathways of efficient energy transduction from ATP hydrolysis into mechanical work in such a molecular system remains an open and fascinating subject of biophysics research. For many years, it was believed that changing actin filament load would result in a proportional change in load on each of the working motors as they slide with respect to the actin filaments (4,5). However, recent experiments revealed that it is the number of myosin motors attached to the actin filaments, rather than the force on individual motors, which changes in proportion to the filament load during skeletal muscle contraction (6). The same experiments have also illustrated that individual myosin motors maintain a force of ∼6 pN while pulling an actin filament through a 6-nm power stroke during muscle contraction in a wide range of velocities (6).

An intriguing question is how such precise regulation of motor force can take place in an apparently chaotic system. To understand this issue, we adapt a recently developed stochastic-elastic model of stress fiber (7) to investigate the mechanism of skeletal muscle contraction. We show that the unique mechanical properties of myosin motor and the stochastic behavior of actin-myosin binding play essential roles in regulating the motor force during muscle contraction. The model also recovers the classical Hill's law (1) between muscle shortening/lengthening velocity and the filament load. The coordinated actions of multiple motors revealed in this article provide insights into a class of self-regulated force homeostasis in a chaotic system and is of importance not only in muscle contraction but also in cellular mitosis, neuronal growth, etc.

Stochastic-elastic model of single myosin motor

According to the Lymn-Taylor scheme (8), an Actin-myosin-ATP working cycle consists of four stages:

• 1.A myosin motor carrying an ATP approaches a binding site on the actin thin filament through Brownian motion.
• 2.The motor binds to the actin filament and the ATP is hydrolyzed with hydrolysis products Pi and ADP.
• 3.Pi is released, followed immediately by the working stroke of myosin and then by the release of ADP.
• 4.At the end of the working stroke, the myosin motor secures an ATP and detaches from the actin filament.

In our model, each myosin motor is assigned two states: working and idle. The working state starts with the power stroke upon Pi release and terminates as the motor detaches from the actin filament. The detachment occurs either due to the limited lifetime of the bond between a motor with ADP and the actin filament, or due to the binding of an ATP after the ADP is released. The idle state covers whenever the myosin motor is detached from the actin filament. It is assumed that Pi is released as soon as the motor attaches to the actin filament. In the case of a relatively long hydrolysis reaction, the extra time taken for the reaction to occur is lumped into the idle state.

During the idle state, a motor is assumed to behave as a simple linear spring with spring constant K_oa. During the working state, the force-stretch relation of a motor is strongly rate-dependent. The typical timescale associated with a myosin motor binding/unbinding with the actin filament is ∼30 ms, which is much larger than the characteristic time for a motor to recover from a sudden displacement shortening (∼0.1 ms) in a transient tension test (9,10). Although the so-called T₁ curve in the transient tension test corresponds to the behavior of myosin motor at a very high rate of deformation, the T₂ curve occurs at the intermediate timescale of 1 ms, which is more relevant to the power stroke. Therefore, we extract the force-stretch curve of a single motor during a normal working stroke from the T₂ curve, which should represent the average behavior of all working motors in series with a linear spring representing the elasticity of the filaments during tension recovery.

Fitting the T₂ curve in Fig. 3b of Piazzesi and Lombardi (10) shows that a working motor should obey a bilinear force-stretch relation at the timescale relevant to the power stroke, and can therefore be modeled as a bilinear spring with prestored elastic energy resulting from the ATP hydrolysis, as schematically shown in Fig. 1 a. Although the stiffness of the myosin filament is only 60% higher than that of the actin filament (11,12), for simplicity we treat the myosin filament as rigid while considering the combined spring constant of both actin and myosin filaments. In this sense, the force-stretch relation of the thin filament is taken to be F = k_f (x + ɛ₀), where k_f = 150 pN/nm and ɛ₀ = 3.2 nm, and that of the motors (totaling 84 in number under isomeric load (6)) is taken to be F = k₁(x + x₁), when x < ɛ₁ and F = k₂(x + x₂) when x ≥ ɛ₁.

Figure 3.

The number of working myosin motors versus time at different filament loads. In the beginning of the simulation, all motors are assumed to be attached to the thin filament. The simulation shows that the number of working motors rapidly approaches a steady-state value with a fluctuation of 10 motors: (purple) 85 × 6 pN; (green) 65 × 6 pN; (red) 45 × 6 pN; and (blue) 25 × 6 pN.

Figure 1.

Molecular model of skeletal muscle contraction. (a) Schematics of force-stretch curve of a single myosin motor over an intermediate timescale of ∼1 ms. During the idle state (Segment oa), the motor behaves as a linear spring with spring constant K_oa. Upon Pi release, the motor switches to the working state (Segment bcd), behaving as a bilinear spring with two different spring constants, K_bc and K_cd, carrying prestored elastic energy drawn from ATP hydrolysis. In the working state, the motor can either detach from the thin filament as a catch bond, or it can release the ADP and then detach upon capturing an ATP. Upon ADP release, for example at point e, the motor immediately behaves as a linear spring with spring constant K_oa. (b) T₂ curve in a typical tension transient test. (Crosses) Experimental data from Fig. 3b in Piazzesi and Lombardi (10). (Solid lines) Fitting curves F = 1.12 × 84(x + 10.2) and F = 0.256 × 84(x + 23.4), respectively. (c) An ensemble of myosin motors drives an elastic actin thin filament toward the M-band on the left at velocity V, whereas the filament is subjected to an external load P to the right. The actin filament is modeled as an elastic beam, whereas the myosin filament is considered rigid. (Right to left, numbered) Bonds between myosin motors and the thin filament.

The composite force-stretch relation of the filament-motor system also has a bilinear relationship. Fitting to the T₂ curve (10), the result is F = 1.12 × 84 (x + 10.2) when x < ɛ₂ and F = 0.0256 × 84 (x + 23.4) when x ≥ ɛ₂, as shown in Fig. 1 b. It follows that k₁ = 3 × 84 pN/nm, k₂ = 0.3 × 84 pN/nm, x₁ = 7 nm, x₂ = 20.2 nm, ɛ₁ = −5.6 nm, and ɛ₂ = −6.3 nm, leading to K_bc = 0.3 pN/nm, K_cd = 3 pN/nm, x_c = −5.6 nm, and x_d = −7 nm in Fig. 1 a. The estimated stiffness K_cd = 3 pN/nm is close to the experimentally reported compliance value of 0.3 nm/pN for an individual myosin motor (10). Note that the reported value of myosin spring constant varies in the literature and can appear to be substantially smaller than 3 pN/nm (13) due to the rate-dependent elasticity of myosin. The bilinear spring reflects a smaller spring constant K_bc associated with the relatively slower initiation phase of the working stroke starting at b.

The release of ADP is assumed to be thermally activated with a rate of k_ADP = k⁰_ADP exp(−ΔE/k_BT), where ΔE is the change in elastic energy of the motor concomitant with the work stroke and k⁰_ADP has lumped contributions from the rest of the free energy barrier. Similar expression has been used by Duke (13). The backward rate of this reaction is neglected.

The bond between a myosin motor with ADP and the actin filament manifests a catch-bond behavior (14). Following Pereverzev et al. (15), the detachment rate of myosin can be described as

$$k_{off}^{ADP}=\alpha \left(50e^{-|F_i/1.5|}+e^{|F_i/6|}\right),$$ (1)

where α = 8.3/s. This gives a maximal lifetime of ∼30 ms under a stretching force of 6 pN (14). Here we have fitted a formula for catch-bond breaking rate proposed in Pereverzev et al. (15) with experimental data reported in Guo and Guilford (14). Note that our simulation results only depend on how the detachment rate varies with the force and, as such, should not be affected by the question whether the fitting parameters are unique or which model of catch bonds is used, as long as the experimental data can be represented. Upon ADP release, the motor is assumed to immediately acquire an ATP and then detaches from the actin filament at a much higher rate, K^ATP_off (14). The attachment site of motor to the thin filament should be biased so that the elastic energy drawn from ATP hydrolysis would not be severely reduced (4). This is achieved by selecting a binding site which allows reduction of no more than 30% of spacing between neighboring sites in the subsequent power stroke. Considering an idle motor as a linear spring with zero rest length, its binding rate to a site on the actin filament at distance u can be expressed as (16–20)

$$k_{on}=\sqrt{\frac{\beta }{\pi }}\frac{2\gamma \text{exp}\left(-\beta U^2\right)}{erf\left(\sqrt{\beta }U\right)+erf\left(0.3\sqrt{\beta }\right)},$$ (2)

where U = u/l₀ is the normalized distance, with l₀ being the spacing between neighboring sites; β = K_oal₀²/(2 k_BT); and γ is a prefactor.

Stochastic-elastic model of actin filament sliding

Here we adapt a coupled stochastic-elastic numerical framework (7,18–20) to incorporate the stochastic processes of myosin motor binding/unbinding and the continuum elasticity of actin filament contraction. This framework takes advantage of the fact that the elastic response of actin filament occurs at a much smaller timescale than the stochastic events of motor binding/unbinding. Therefore, we proceed with a two-level hierarchical scheme each associated with a different characteristic timescale. At the upper timescale of 30 ms, we employ a Monte Carlo scheme to simulate the binding/unbinding of myosin motors. At the lower timescale (<1 ms), the system is considered elastic. At each step of Monte Carlo simulation, we first solve for the forces and displacements of the filament-motor complex based on the elastic equations. The results from the elastic calculations are then used to determine the reaction rates of motor binding/unbinding and to update the system configuration. Our model is based on the structure unit of sarcomere consisting of a thick filament made of a bundle of myosin stalks surrounded by a hexagonal arrangement of thin actin filaments.

The myosin motors along the thick filament are active force generators. The thin filaments are anchored to the Z-disk (21), whereas the thick filaments are anchored to the M-line (22). As shown in Fig. 1 c, the thin filament is modeled as an elastic beam with tension stiffness EA (E = Young's modulus and A = cross section) attached to a substrate via a cluster of myosin motors, and is pulled by an external force P at its right end, corresponding to the Z-disk (22). The substrate representing the thick filament is considered rigid. The myosin motors undergo stochastic processes of attaching to or detaching from the actin filament according to a set of forward and backward reactions rates similar to those formulated by Bell (23). In our simulation scheme, the thin filament is discretized into N-1 segments, with N nodes coincident with N motor biting sites. Initially, all motors are placed in the working state, and the i^th motor on the substrate is connected to the i^th biting site on the thin filament. The deformation in the thin filament is assumed to be tension-dominated. The N biting sites/nodes are equally spaced at distance l₀, as shown in Fig. 1 c. The total elastic energy in the system (filament and motors) can be expressed as

$$\Pi /l_0=\frac{1}{2}EA\sum _{i=1}^{N-1}{\left(U_i-U_{i+1}\right)}^2+\sum _{i=1}^N\varpi \left[U_i+\left(m_i-i\right)\right]-P{U}_1,$$ (3)

where U_i is the displacement of the i^th node of the thin filament normalized by l₀, ϖ is the elastic energy of the myosin motor, and m_i is the sequence number of motors anchored on the substrate which is currently connected to the i^th node of the thin filament. The equilibrium condition δΠ = 0 yields N equations to determine N nodal displacements U_i. Subsequently, the force on each working motor, F_i, is determined. After the force and displacement on each node of the thin filament are determined, the time and position for the next event (which can be the attachment of a motor, the catch-bond breaking, the ADP release from a motor, or the detachment of a motor with ATP) are selected according to Gillespie's algorithm (24,25). In determining the time, a series of independent random numbers ξ_μ uniformly distributed over the interval [0, 1] are generated for each node. Let a_μ denote the plausible reaction rate at each motor. The time for the next reaction is chosen to be the smallest among a series of values dT_μ calculated according to

$$dT=\min \left(d{T}_{\mu }\right)=\min \left(-\frac{\ln {\xi }_{\mu }}{a_{\mu }}\right).$$ (4)

The location for the next event is registered as the reaction site where dT is chosen. Subsequently, the bond state at the chosen reaction site is adjusted. Without specification, the default parameters used in the simulations are N = 300, α = 8.3/s, β = 11, γ = 6.0/s, K_cd = 3.0 pN/nm, K_bc = 0.3 pN/nm, k^ATP_off = 2.0/ms, k⁰_ADP = 1.0/s, l₀ = 5.5 nm, EA/l₀ = 3000 pN/nm, x_c= −5.6 nm, and x_d = −7 nm.

Simulation results

The simulation results in Fig. 2, a and b, indicate that the detachment of myosin motors from the actin filament is essentially random, although a pattern resembling frictional slippage behavior (26) appears when the filament load is relatively high, as in Fig. 2 b. At the beginning of simulation, all myosin motors are assumed to be in the working state. The number of attached motors is then plotted against time in Fig. 3, indicating that the number of working motors decreases rapidly in the beginning and then approaches a steady-state number, with a fluctuation ∼10, in close agreement with experimental observations (6). Fig. 4 a shows that the average number of working motors increases in linear proportion to the filament load and the average force of each working motor is thus regulated at ∼6 pN, again in agreement with experiment (6). Driven by the motors, the thin filament approaches a steady-state sliding velocity, which decreases as the filament load increases (as shown in Fig. 5), consistent with Hill's law (1). The effect of changing reaction rates on the muscle performance is also investigated. As seen from Fig. 4 b, the average number of working myosin motors continues to be regulated by the filament load, but the muscle sliding velocity varies significantly at different reaction rates in Fig. 6.

Figure 2.

Site squence of detachment of myosin motors from a thin filament. (a) At low filament loads, the myosin motors randomly detach from (and attach to) the thin filament. (b) At high filament loads, the sequence of myosin detachment resembles a wave of frictional slippage along the interface.

Figure 4.

Average number of working myosin motors versus filament load. (a) The number of working motors increases in linear proportion to the filament load with force per motor at ∼6 pN. (Dashed line) ${\overline{N}}_{on}$ = 0.98P/6 + 9.5. (b) The effect of reaction rates: (diamonds) γ = 4.0/s, k⁰_ADP = 0.5/s; (squares) γ = 8.0/s, k⁰_ADP = 2.0/s; (triangles) γ = 6.0/s, α = 12.5/s; and (crosses) γ = 6.0/s, α = 4.2/s.

Figure 5.

Simulated Hill's law of contractile velocity of thin filament versus the applied load. (Triangles) Simulation results. (Solid line) Fitted curve for the contractile part according to the original Hill's equation (P + 150) (V + 920) = Const (1). (Dashed line) Fitted curve with equation P = 570(1–1.2V/(V + 650)) for the contractile part and P = 630(1 + 15/π arctan(−12V/1800)) for the lengthening part (35). The units for P and V are pN and nm/s, respectively.

Figure 6.

The effect of reaction rates on the velocity-force relation: (diamond) γ = 4.0/s, k⁰_ADP = 0.5/s; (square) γ = 8.0/s, k⁰_ADP = 2.0/s; and (triangle) γ = 6.0/s, α = 12.5/s.

Discussion

It was reported that the number of working motors is modulated by the filament load with force per motor maintained at ∼6 pN over a 6-nm power stroke (6). This suggests relatively small variations in the motor force during most of the power stroke. This behavior is correlated with the existence of a relatively small spring constant when the power stroke initiates, as shown in the force-stretch relation in Fig. 1 b. The small variation in motor force during a 6-nm stroke can be important in the modulation of the number of working motors by the filament load. In fact, when the motor is represented as a soft linear spring with spring constant 0.5 pN/nm, similar modulation of the number of working motors by filament load still exists (Fig. 7 a) and the predicted muscle contraction velocity still follows Hill's law (Fig. 7 b). However, if the spring constant is increased to 0.7 pN/nm, the number of working motors is no longer linearly proportional to the filament load (Fig. 7 a). Note that, in our model, the low spring constant of bc segment aims to capture the force-stretch behavior of myosin motors over intermediate timescales (∼1 ms) and therefore should not be regarded as the instantaneous elastic stiffness of myosin.

Figure 7.

Predicted muscle performance when the myosin motor is modeled as a soft linear spring with spring constant K_ss and a prestretch of D₀. (a) The number of working motors versus the filament load: (triangle) K_ss = 0.5 pN/nm, D₀ = 14 nm; and (circle) K_ss = 0.7 pN/nm, D₀ = 10 nm. (b) Muscle contraction velocity when K_ss = 0.5 pN/nm, D₀ = 10 nm. Other parameters are γ = 5.0/s, k⁰_ADP = 0.3/s.

Upon skeletal muscle shortening, the force of a single motor decreases slightly from 6 pN over the entire power stroke, during which both the rate of ADP release and that of catch-bond breaking are relatively low. However, as shown in Fig. 8, the ADP release rate continues to rise as the motor movement approaches 6 nm, at which point the motor rapidly detaches from the thin filament by capturing an ATP. On the other hand, at the filament level, the mechanical balance between the filament load and forces on working motors takes place much faster than the rates of ADP release or catch-bond breaking. It is this difference in dynamic responses at motor and filament levels that leads to the rather precise modulation of the number of working motors by the filament load upon muscle contraction. That the motor force varies gently over motor movement from 0 to 6 nm is a necessary condition for the force homeostasis to occur. On the other hand, without the particular stochastic features of the myosin motors, the number of working motors would not be modulated by the filament load and the elasticity alone could not lead to the observed force homeostasis. It can be inferred from Fig. 8 that the rate of ADP release is very small upon muscle lengthening. Therefore, in this case, the motors mostly detach by catch-bond breaking and the simulations show that the motor force can still be regulated by the filament load up to a certain value.

Figure 8.

ADP release rate (solid line) and the catch-bond breaking rate (dashed line) of a working motor.

The adhesive apparatus of bacteria Escherichia coli or leukocytes also display a force-stretch relation with a long and relatively flat region (27), similar to that of the myosin motors. Interestingly, the bonds formed between receptors at terminals of these adhesive apparatus and their associated ligands also manifest catch behavior (27). Our work thus suggests that the total number of working bonds upon adhesion of E. coli or leukocytes to a surface can be modulated by the applied load. The force in each of these working bonds may be regulated around the optimal value for the catch bond, and the velocity of E. coli or leukocytes would then decrease with the increase of the applied load, which was indeed observed (27), although we caution about possible other interpretations. In addition, Talin, which displays a similar force-stretch relation (28), might have cooperated with the catch behavior of integrin (29) to regulate tensional homeostasis in focal adhesion. Thus, our work suggests that it might be a general principle in biology to use catch bonds together with such a force-stretch curve to regulate the bond forces into homeostasis.

We note that the model presented in this article bears some similarity to (as well as a number of important differences from) a previous milestone work on molecular model of muscle contraction by Duke (13). In Duke's model, the thin filament was assumed to be rigid, the motor was treated as a soft linear spring, and the chemical reaction rates were modified according to the work done in moving the arm. As a result of these assumptions, it was inferred that the chemical energy of the motor was insufficient to power the stroke from the zero-strain state, which then led to the assertion that a large numbers of motors need to be in a waiting-state before a synchronized power stroke can occur. In this article, we have included the effect of elasticity of the thin filament while allowing the motors to bind to discrete attachment sites on the filament.

More importantly, in predicting the muscle performance at timescales associated with filament contraction, the force-stretch curve of a single motor is deduced from the experimentally measured T₂ curve from the standard transient tension test, and the catch-bond behavior of motors is included to model the muscle lengthening velocity above the isotonic filament load. Neither of these features was in Duke's model (13). Our model shows that the average number of working motors increase in linear proportion to the filament load with force per motor self-regulated at ∼6 pN, in agreement with recent experiments (6). It is worth pointing out that the average movement of a motor during one work stroke may be <6 nm in our simulation, as inferred from the ADP release rate in Fig. 8, which becomes significant above 3 nm. We suggest here that the dependence of the ADP release rate on the concomitant elastic energy change in motor might have been tuned to enable the motor to carry out an average movement of 6 nm within one work stroke.

The force-generating process in the skeletal muscle of frog is known to strongly depend on the temperature (30–32). As the temperature is increased from 2 to 17°C, the motor force increases by 70% and the rate of quick force recovery increase ∼3 times (30). Although the amount of filament sliding caused by a small reduction in filament load from the maximum isometric force varies little, it is significantly reduced under very low filament load (31) as the temperature rises from 2 to 17°C. Motivated by these observations, we have also tempted to investigate the effect of temperature by modifying the parameters for both elasticity and reaction rates of motors. At a high temperature, we let K_cd = 4.2 pN/nm, K_bc = 0.45 pN/nm, x_c = −3.2 nm, and x_d = −5 nm (31), which is exacted from a virtual T₂ curve of similar trend as those in Fig. 4 of Piazzesi et al. (30).

The temperature is expected to increase the reaction rates and have a strong influence on the filament sliding velocity, as seen from Fig. 6, although more quantitative information is not available. We let k⁰_ADP = 3/s and γ = 18/s at the high temperature. According to the simulation results in Fig. 9 a, the average number of working motors is still well regulated by the filament load, although the motor force has increased to ∼9 pN at the high temperature. As seen from Fig. 9 b, the filament sliding velocities at the high temperature are much higher than those at the low temperature under the same filament load taken from Fig. 5, which is consistent with the experimental observation (30). Fig. 9 b also plots the experimental data from Fig. 3A in Piazzesi et al. (6), which shows agreement with our simulation results at the low temperature.

Figure 9.

The effect of temperature is qualitatively investigated through modifying the parameters for both the elasticity and reaction rates of motors in the simulation. At a high temperature, we let k⁰_ADP = 3.0/s, γ = 18.0/s, K_cd = 4.2 pN/nm, K_bc = 0.45 pN/nm, x_c = 3.2 nm, and x_d = −5 nm. (a) The average number of working motors at the high temperature versus the applied load. (b) The shortening velocity of thin filament versus the applied load. (Triangles) Simulation results at the high temperature and (circles) simulation results at the low temperature taken from Fig. 5, which agrees with the experimental data from Fig. 3A in Piazzesi et al. (6) (crosses).

We also point out some limitations of our model. For example, a myosin motor has two heads competing for the same binding site and the spacing between neighboring motors is different from that between neighboring binding sites on the actin filament. The myosin filament has been treated as a rigid substrate. We find that changing the stiffness of actin filament by a few times has a trivial effect on the modulated number of working motors in the simulation, which suggests that the results may not be significantly different when the elasticity of the myosin filament is rigorously considered in our simulations. The spacing between neighboring motors can be much more important than the stiffness of myosin filament.

Despite these deficiencies, the predictions of our model show very broad agreement with experimental results and can serve as a platform to develop more sophisticated models with additional features (33,34) to understand motor-driven contraction mechanisms in skeletal muscle, as well as in other biological processes driven by multiple motors, such as neuronal growth and cellular mitosis, where elasticity of the involved structures can be important.