Force-Dependent Recruitment from the Myosin Off State Contributes to Length-Dependent Activation

Abstract

Cardiac muscle develops more force when it is activated at longer lengths. The concentration of Ca²⁺ required to develop half-maximal force also decreases. These effects are known as length-dependent activation and are thought to play critical roles in the Frank-Starling relationship and cardiovascular homeostasis. The molecular mechanisms underpinning length-dependent activation remain unclear, but recent experiments suggest that they may include recruitment of myosin heads from the off (sometimes called super-relaxed) state. This manuscript presents a mathematical model of muscle contraction that was developed to investigate this hypothesis. Myosin heads in the model transitioned between an off state (that could not interact with actin), an on state (that could bind to actin), and a single attached state. Simulations were fitted to experimental data using multidimensional parameter optimization. Statistical analysis showed that a model in which the rate of the off-to-on transition increased linearly with force reproduced the length-dependent behavior of chemically permeabilized myocardium better than a model with a constant off-to-on transition rate (F-test, p < 0.001). This result suggests that the thick-filament transitions are modulated by force. Additional calculations showed that the model incorporating a mechanosensitive thick filament could also reproduce twitch responses measured in a trabecula stretched to different lengths. A final set of simulations was then used to test the model. These calculations predicted how reducing passive stiffness would impact the length dependence of the calcium sensitivity of contractile force. The prediction (a 60% reduction in ΔpCa₅₀) mimicked the 58% reduction in ΔpCa₅₀ in myocardium from rats that expressed a giant isoform of titin and had low resting tension. Together, these computational results suggest that force-dependent recruitment of myosin heads from the thick-filament off state contributes to length-dependent activation and the Frank-Starling relationship.

Introduction

The Frank-Starling mechanism helps mammalian hearts to balance cardiac output from the left and right ventricles by increasing stroke volume in response to elevated filling pressure. Changes in calcium handling play an important role, but experimental data from chemically permeabilized cardiac preparations suggest that myofilament-level mechanisms dominate the effect. Specifically, static stretch increases both the Ca²⁺ sensitivity of contraction and the maximal contractile force in permeabilized cells (1). These functional responses have become known as length-dependent activation (2).

Despite decades of research, the molecular basis of length-dependent activation remains unclear. Numerous mechanisms have been tested, but none of the early hypotheses can explain every feature of the underlying data (3). For example, experiments performed by McDonald and Moss in 1995 suggested that length-dependent activation might reflect changes in the radial separation of the thick and thin filaments (4). However, this mechanism became less likely when de Tombe and his colleagues combined mechanics and x-ray diffraction to show that length-dependent activation can occur without an appreciable change in radial spacing (5).

The most recent experimental data suggest that length-dependent activation involves complex force-dependent transitions within the myofilaments. Ait-Mou et al. made an important advance in early 2016 when they used x-ray diffraction techniques to show that passive stretch induces structural rearrangements in both the thick and the thin filaments (6). Intriguingly, the structural changes were greater in myocardium from wild-type (WT) rats than in preparations from mutant rats that expressed a giant splice isoform of cardiac titin and had low resting tension (7).

A second influential work was published by Kampourakis et al. a few months later (8). These authors tested how the thick and thin filaments responded to different levels of Ca²⁺ activation by monitoring fluorescent probes that labeled either myosin regulatory light chain (RLC) or troponin C. The data from Kampourakis et al.’s experiments complemented the results published by Ait-Mou et al. and clearly demonstrated that length-dependent activation is associated with structural changes in the myofilaments.

Although they did not describe precise mechanisms, both groups of authors speculated that mechanosensitive transitions within the thick filament might contribute to length-dependent activation. More specifically, both studies noted that force development was accompanied by thick filaments transitioning from an off state (in which myosin heads are prevented from binding to actin) to an on state (in which actin-myosin binding is permitted) (9, 10, 11). The precise biochemical nature of the off state remains debated (12), but it might correspond to the super-relaxed state of cardiac myosin (13) or the interacting-heads motif known from cryo-electron microscopy (14).

Numerous researchers are now testing these ideas using sophisticated biophysical and biochemical techniques. For example, it is becoming clear that myosin heads can transition between the off and on states with millisecond-level kinetics in active muscle (15) and that the dynamics of the thick filament transitions depend on interactions with myosin binding protein-C (MyBP-C) (9) and on the phosphorylation status of myosin RLC (16). The thick filament also seems to be an exciting new therapeutic target because data currently published in preprint and abstract form suggest that the relative populations of the off and on myosin states can be manipulated with mavacamten (formerly MYK-461) (17) and by peptides targeted to the S2 region of myosin (18).

This study builds on these important experimental results and presents a new to our knowledge mathematical model of muscle that was formed by coupling a dynamically regulated thick filament to a cooperative Ca²⁺-dependent thin filament. Simulations show that the coupled model can reproduce the effects of length and RLC phosphorylation on myocardial tension-pCa relationships if the rate of the off-to-on myosin transition increases linearly with muscle force. Additional calculations show that the mechanosensitive model can also reproduce twitch contractions measured at four different sarcomere lengths in a living rabbit trabecula. The final phase of the work tests the model by predicting how the Ca²⁺ sensitivity of contraction will be affected if passive tension is reduced. The results of these calculations match experimental data obtained from the mutant rats studied in Ait-Mou et al.’s original experiments (6). Together, these results show that a coupled model incorporating force-dependent thick filament transitions can reproduce many important features of muscle contraction.

Materials and Methods

Overview

The calculations presented in this work were based on the kinetic scheme shown in Fig. 1. Binding sites on the thin filament transitioned between an inactive state termed N_off (which myosin heads could not attach to) and an active state N_on (which was available for myosin binding). Activated binding sites could not switch back to the inactive state if a myosin head was attached. Myosin heads transitioned between an off state (that could not interact with actin), an on state (that could potentially bind to actin), and a single attached force-generating state.

Figure 1.

Kinetic scheme. Sites on the thin filament switch between states that are available (N_on) and unavailable (N_off) for cross-bridges to bind to. Myosin heads transition between an off detached state, an on detached state, and a single attached force-generating state. J terms indicate fluxes between different states. To see this figure in color, go online.

Thin filament transitions

The fraction of binding sites in the active state was defined as N_on. The flux for the N_off to N_on transition (that is, the number of sites per unit time switching from N_off to N_on) was defined as

$$J_{on}=k_{on}\left[C{a}^{2+}\right]\left(N_{overlap}-N_{on}\right)\left(1+k_{coop}\left(\frac{N_{on}}{N_{overlap}}\right)\right),$$ (1)

where k_on is a rate constant, N_overlap is the fraction of binding sites that are in range of myosin heads and depends on the prevailing half-sarcomere length as described in Campbell (19), and k_coop is a constant that defines the strength of thin filament cooperativity.

If k_coop is zero, Eq. 1 reduces to J_on = k_on [Ca²⁺](N_overlap − N_on). This describes a simple interaction in which binding sites activate at a rate equal to a constant times the Ca²⁺ concentration times the number of binding sites in the off state. The (N_overlap − N_on) term ensures that only binding sites that overlap with the thick filaments are available to myosin heads. If k_coop is greater than zero, the k_coop (N_on/N_overlap) term in Eq. 1 produces cooperative activation as binding sites that have switched to the on state increase the rate at which other binding sites activate.

The flux of binding sites through the off transition was defined as

$$J_{off}=k_{off}\left(N_{on}-N_{bound}\right)\left(1+k_{coop}\left(\frac{N_{overlap}-N_{on}}{N_{overlap}}\right)\right),$$ (2)

where k_off is a rate constant and the other terms are as defined previously. The (N₀ − N_bound) term means that only unbound sites can deactivate (that is, myosin heads have to detach before a binding site can revert to the off state). The k_coop((N_overlap − N_on)/N_overlap) term causes binding sites in the off state to induce further deactivation.

In summary, Eqs. 1 and 2 define a thin-filament system that activates in the presence of Ca²⁺. If k_coop is greater than zero, active binding sites induce other sites to activate, whereas inactive sites accelerate deactivation. Equations 1 and 2 thus describe a cooperative system.

Thick-filament transitions

The flux of myosin heads transitioning from the M_OFF state to the M_ON configuration was defined as

$$\begin{array}{c}{J}_1=\left(k_1+H{k}_{MLCP}\right)\left(1+k_{force}{F}_{total}\right)M_{OFF},\\ \text{where}\\ H=0\text{when}\text{myosin}\text{RLCs}\text{are}\text{not}\text{phosphorylated}\\ =1\text{when}\text{myosin}\text{RLCs}\text{are}\text{phosphorylated}\end{array},$$ (3)

where k₁ is a rate constant, H switches between 0 and 1 depending on the phosphorylation status of myosin RLCs, k_MLCP and k_force are constants, F_total is the force in the muscle, and M_OFF is the proportion of myosin heads in the off state.

If k_force is zero in Eq. 3, J₁ = k₁M_OFF or (k₁ + k_MLCP)M_OFF depending on the phosphorylation status of the RLCs on myosin. If k_force is greater than 0, the rate at which myosin heads transition into the on state increases with force.

The flux of myosin heads into the off state was defined as

$$J_2=k_2{M}_{ON},$$ (4)

where k₂ is a rate constant and M_ON is the proportion of myosin heads in the on state.

Together, Eqs. 3 and 4 describe a situation in which the equilibrium between heads in the off and on relaxed states depends on both the proteomic and the mechanical status of the muscle.

Myosin heads bound to actin with a flux J₃ defined as

$$J_3\left(x\right)=k_3{e}^{\frac{-k_{cb}{x}^2}{2k_{B}T}}{M}_{ON}\left(N_{on}-N_{bound}\right),$$ (5)

where k₃ is a rate constant, k_cb is the stiffness of the cross-bridge link, k_B is Boltzmann’s constant (1.38 × 10⁻²³ J K⁻¹), and T is the temperature. Equation 5 mimics a second-order reaction so that the rate at which myosin heads attach to the thin filament increases with the product of the proportion of myosin heads that are able to attach (M_ON) and the fraction of available binding sites (N_on − N_bound). The Gaussian form of the myosin strain dependence reflects the probability of the cross-bridge spring being extended to x by random Brownian motion when the myosin head binds to actin (20).

Similarly, the flux through the myosin detachment step was defined as

$$J_4\left(x\right)=\left(k_{\mathrm{4,0}}+k_{\mathrm{4,1}}{\left(x-x_{ps}\right)}^4\right)M_{FG}\left(x\right),$$ (6)

where k_4,0 is a rate constant, k_4,1 is a parameter that sets the strain dependence of the cross-bridge detachment rate, x_ps is the power stroke of an attached cross-bridge, and M_FG(x) is the proportion of cross-bridges attached to actin with spring lengths between x and x + δx.

Calculations and software

The kinetic scheme shown in Fig. 1 was simulated by discretizing the relevant differential equations to yield

$$\begin{array}{l}\frac{d{N}_{off}}{dt}=-J_{on}+J_{off}\hfill \\ \frac{d{N}_{on}}{dt}=J_{on}-J_{off}\hfill \\ \frac{d{M}_{OFF}}{dt}=-J_1+J_2\hfill \\ \frac{d{M}_{ON}}{dt}=\left(J_1+\sum _{i=1}^n{J}_{4,x_i}\right)-\left(J_2+\sum _{i=1}^n{J}_{3,x_i}\right)\hfill \\ \frac{d{M}_{FG,i}}{dt}=J_{3,x_i}-J_{4,x_i}\text{where}\text{i}=1\text{…n}\hfill \end{array},$$ (7)

Cross-bridge populations were evaluated with 0.5 nm resolution over the range −10 nm ≤ x ≤ 10 nm. n was thus equal to 41, giving a complete set of 45 equations. These were integrated numerically using routines from the GNU Scientific Library (21) that implemented an embedded Runge-Kutta 2,3 method with adaptive step-size control. Calculations were initiated with all binding sites in the N_off configuration and all myosin heads in the M_OFF state.

Dynamic interfilamentary movement was incorporated into simulations when required by using polynomial interpolation to displace the distribution that described the number of heads bound with each spring length (22). Filament compliance effects (23, 24) were mimicked by assuming that a half-sarcomere length change of Δx displaced each actin-myosin link by 1/2.Δx (19, 25).

Muscle force was calculated as

$$\begin{array}{c}{F}_{total}=F_{active}+F_{passive},\\ \text{where}\\ F_{active}=N_0{k}_{cb}\sum _{i=1}^n{M}_{FG,i}\left(x_i+x_{ps}\right)\end{array}$$ (8)

and N₀ is the number of myosin heads in a hypothetical cardiac half-sarcomere with a cross-sectional area of 1 m². N₀ was set to 6.9 × 10¹⁶ m⁻² throughout this work based on the assumptions that 1) myofibrils occupy ∼60% of the cross-sectional area of myocardium, 2) half-thick filaments contain 283 myosin heads, and 3) half-thick filaments have a spatial density of 4.07 × 10¹⁴ m⁻² within myofibrils (22, 26).

For the simulations of permeabilized myocardium, passive force was assumed to increase linearly with sarcomere length such that F_passive = k_p(L − L₀), where k_p is the stiffness of the passive elastic element, L is the sarcomere length, and L₀ is the sarcomere length at which passive force is zero. This relationship was chosen for these simulations because experimental data were only available at two lengths.

Experimental data at four lengths were available for the measurements performed using an intact trabecula, and a nonlinear relationship was found to fit these passive tension data better. Specifically, passive force was defined as F_passive = σ(exp(L/L_c) − 1), where σ is a constant and L_c sets the curvature of the relationship. The model of the twitching trabecula also included a series elastic spring of stiffness k_s. As described by Chung et al. (27), this improved the fit to the experimental data during relaxation.

Simulations were fitted to experimental data by using simplex minimization algorithms (28) to optimize the mean normalized deviation, defined as

$$\text{Mean}\text{normalized}\text{deviation}=\frac{1}{n_{conditions}}\sum _{i=1}^{{\mathsf{n}}_{\mathsf{conditions}}}\left(\frac{\sqrt{\sum _{j=1}^{n_{points}}{\left(y_{simulation,i,j}-y_{experiment,i,j}\right)}^2}}{\frac{1}{n_{points}}\sum _{j=1}^{n_{points}}{y}_{experiment,i,j}}\right),$$ (9)

where n_points is the number of data points for each of the n_conditions experimental conditions. For example, in Fig. 3, n_conditions is 4 and n_points ranges from 8 to 11.

Figure 3.

Simulations of tension-pCa curves measured at short/long lengths and before/after treatment with myosin light chain kinase. Experimental data are replotted from Kampourakis et al. (8). Model parameters are shown in Table S1. To see this figure in color, go online.

Confidence limits for each parameter were determined as previously described (22) by adjusting each parameter until the mean normalized deviation exceeded the best-fit value by 5% (29). Two additional statistical techniques that were used to analyze the k_force parameter are described in Results.

All of the simulations were performed using version 2.5 of the MyoSim software package (22). The program and source code are available for free download at http://www.myosim.org. The calculations ran quickly—∼10 times faster than real time on a standard PC. For example, the total computation time for the seven traces shown in Fig. S1 (each trace shows 12 s calculated with 1 ms resolution) was ∼8 s.

Values for pCa₅₀ and the Hill coefficient n_H were calculated by fitting a function of the form

$$y=A+B\frac{{\left[C{a}^{2+}\right]}^{n_H}}{{\left[C{a}^{2+}\right]}^{n_H}+{\left[C{a}_{50}^{2+}\right]}^{n_H}},$$ (10)

where A and B are constants and pCa = −log₁₀[Ca²⁺], to the relevant experimental data.

Experimental data

The experimental data shown in Figs. 2, 3, and 4 were published by Kampourakis et al. (8). The passive tension data were not included in the original publication and were supplied to the current authors by Dr. Malcolm Irving, the corresponding author for Kampourakis et al.’s manuscript.

Figure 2.

Simulations of thin-filament activation reproduce experimental data. Experimental data are replotted from Fig. 5 D of Kampourakis et al. (8) (open circles, “After Blebb” condition). k_off was held constant at 100 s⁻¹. Values for k_on and k_coop were deduced by fitting the simulations to the experimental data and were equal to 2.08 ± 0.03 × 10⁷ M⁻¹ s⁻¹ and 5.70 ± 0.52, respectively. As described in Materials and Methods, the confidence limits for the two parameters were determined by adjusting each parameter until the mean normalized deviation exceeded the best-fit value by 5% (22, 29).

Figure 4.

Ablating mechanosensitivity of thick-filament transitions eliminates length-dependent activation. The figure shows the best fit to the experimental data published by Kampourakis et al. (8) that was attained when k_force was fixed at zero. Model parameters are shown in Table S2. To see this figure in color, go online.

Figs. 5 and 6 show force records and Ca²⁺ transients measured from a single rabbit trabecula at 37°C. The Ca²⁺ transients were measured from the fluorescence of iontophoretically loaded bis-fura-2. These data are representative of many similar experiments performed by one of the authors and were collected using techniques that have been described previously (30).

Figure 5.

Simulations driven by measured Ca²⁺ transients reproduce force records measured from a rabbit trabecula at four different sarcomere lengths. Parameter values are shown in Table S3. To see this figure in color, go online.

Figure 6.

Myofilament-based mechanisms have a bigger impact on the length dependence of twitch contractions than Ca²⁺ transients. Simulations at four sarcomere lengths driven by the Ca²⁺ transient measured at the shortest length are shown. Parameter values are shown in Table S3. To see this figure in color, go online.

Results

Overview of modeling approach for chemically permeabilized myocardium

The first goal of this work was to test whether the coupled model could reproduce the length-dependent contractile behavior of chemically permeabilized myocardium measured by Kampourakis et al. (8). Because the full model for permeabilized muscle had 14 parameters, it was important for model credibility (31) to minimize the number of values that needed to be adjusted at each stage of the fitting process. This was accomplished by 1) using values derived from the published literature whenever possible and 2) adopting a sequential approach in which values defining the behavior of the thin-filament system were deduced first and then held fixed during subsequent calculations.

Activation of thin filaments

Kampourakis et al. measured the Ca²⁺-dependence of thin filament activation in rat myocardium by abolishing active force with blebbistatin and monitoring how Ca²⁺ modulated the orientation of bifunctional rhodamine probes positioned on troponin C molecules (8). Their experimental data showed that the <P₂> order parameter of the troponin C probe decreased from ∼0.30 to ∼0.22 as the free Ca²⁺ concentration was raised from pCa 6.2 to pCa 4.2 (Fig. 5 D of (8)). In the current work, these <P₂> values were assumed to correspond to minimal and full activation of the thin filament, respectively. It was also assumed that the blebbistatin treatment had completely prevented cross-bridge binding so that the experimental data represented the intrinsic behavior of thin filaments in chemically permeabilized tissue.

These experiments were simulated by fixing k_off at 100 s⁻¹ (based on data published by Siddiqui et al. (32)) and by setting k₃ equal to zero (which mimics the effect of blebbistatin by eliminating cross-bridge binding). k_on and k_coop were then adjusted in an attempt to match the predicted values of N_on to Kampourakis et al.’s data. The results of this fitting procedure are shown in Fig. 2. The mean normalized deviation (Eq. 9) for the fit was 0.019, which suggests that Eqs. 1 and 2 may provide a useful and simple way of modeling thin-filament activation.

Force-pCa relationships

A second set of simulations was then performed to test whether the model could reproduce the effects of sarcomere length and myosin light-chain phosphorylation on contractile force. The number of free parameters in this minimization was minimized by 1) constraining k_on, k_off, and k_coop to the values used for the simulations in Fig. 2; 2) setting k_cb to 0.001 N m⁻¹ (based on data published by Pinzauti et al. (33)); 3) setting k₂ to 200 s⁻¹ (based on data published by Fusi et al. (15)); and 4) fixing k_4,1 at 10 nm⁻⁴ so that the detachment rate of myosin heads exhibited moderate strain dependency. Eight parameters (Table S1) were then adjusted to produce the best attainable fit to the experimental data. k₁, k_MLCP, and k_force (all Eq. 3) modulated transitions from the super-relaxed state. k₃, k_4,0, and x_ps (Eqs. 5, 6, and 8) defined the kinetics and force-generating properties of bound cross-bridges. Finally, k_p and L₀ (both Eq. 8) set the muscle’s passive elastic properties.

The results of the optimization procedure are shown in Fig. 3. The mean normalized deviation was 0.008, indicating that the model produced a good fit to the experimental data.

Mechanosensitivity of thick filaments

Throughout this work, thick filaments were considered to exhibit “mechanosensitivity” if force accelerated the rate at which myosin heads transitioned out of the off state. Two different statistical approaches were used to test the hypothesis that thick filaments were indeed mechanosensitive (i.e., k_force > 0, Eq. 3). The first approach utilized the extra sum-of-squares F-test (34, 35) and compared the fit shown in Fig. 3 to the best fit attained using a simpler model in which the thick filaments were not mechanosensitive (i.e., k_force was fixed at zero) (Fig. 4).

When the two fits were compared, the sum of squares for the mechanosensitive model (k_force > 0, Fig. 3) was 6% of that calculated for the nonmechanosensitive simulations (k_force = 0, Fig. 4). The mechanosensitive model had an extra degree of freedom (k_force) so it would be expected to produce a lower sum of squares. However, the F-ratio for the model comparison was 457, which corresponds to a p value of ∼10⁻¹⁸. This implies that the mechanosensitive model (k_force > 0) fits the experimental data significantly better than the nonmechanosensitive model.

The second statistical approach established confidence bounds for the k_force parameter. These were calculated using an iterative process in which k_force was initially reduced by a small amount to a new value, k_test. The model was then refitted to the experimental data, holding k_force equal to k_test but allowing all other parameters to vary. This produced a new sum of squares that could be compared to the sum of squares for the global fit shown in Fig. 3 using an F-test. The process was then repeated using successively smaller values of k_test until the p value for the F-test was 0.05. The value of k_test at this point is the lower 95% confidence bound for k_force. The upper 95% confidence bound was then calculated by repeating the entire procedure using successively larger values of k_test. This approach yielded 8.8 × 10⁻⁵ N⁻¹ m² < k_force < 2.1 × 10⁻⁴ N⁻¹ m². Because the lower bound was positive, it was statistically appropriate to reject the null hypothesis that k_force equaled zero.

Both approaches therefore provided statistical justification for concluding that thick filaments are mechanosensitive and that the rate at which myosin heads transition out of the off state increases with force.

Additional simulations of chemically permeabilized myocardium

Results from many additional calculations are included in the Supporting Material. Figs. S1–S4 illustrate how the simulations evolve with time. Each figure shows a different combination of sarcomere length and kinase treatment and includes 1) activations in solutions with different pCa values and 2) length perturbations that mimic those used to measure k_tr, the rate of tension recovery (36).

Fig. S5 shows that k_tr-force plots simulated by the model reproduce the characteristic J-shaped relationship, with the minimal rate of force development occurring at intermediate levels of Ca²⁺ activation (37, 38, 39). This is an important result because no k_tr data were utilized in the optimization procedures; instead of being a computational fit, the J-shaped k_tr-force plots are an independent prediction of the model. Moreover, the simulations predicted that the maximal value of k_tr was reduced at long sarcomere lengths in agreement with some published results (40, 41). Interestingly, the sarcomere-length dependence of k_tr was markedly reduced in simulations that did not exhibit thick-filament mechanosensitivity (Fig. S6). This implies that destabilizing the myosin off state will decrease the length dependence of kinetics, a hypothesis that can potentially be tested in future experiments.

Figs. S7–S17 show sensitivity analyses in which model parameters were sequentially adjusted to 0.1, 0.5, 1.0, 2.0, or 10.0 times the corresponding value shown in Table S1. These plots show how each parameter influences the predicted contractile behavior and reveal clear trends. For example, all modifications that increase force (e.g., increases in any of k₃, x_ps, or k_p or decreases in either of k_4,0 or L₀) increase the pCa₅₀ for contractile force.

Simulations of living myocardium

The next set of simulations were driven by smoothed versions of the Ca²⁺ transients measured in experiments performed using a living rabbit trabecula. The goal of this phase of the work was to determine whether the mechanosensitive model could reproduce the length dependence of twitch contractions.

No data quantifying the Ca²⁺ dependence of thin-filament activation in living myocardium were available, so it was not possible to duplicate the sequential fitting procedures used for the chemically permeabilized data. The optimization process thus involved judicious adjustment of 11 model parameters (Table S3). The model framework was identical to that described for the simulations of permeabilized tissue except that 1) the linear passive tension relationship was replaced by a nonlinear function (defined by σ and L_C), and 2) a series elastic element (stiffness k_series) was included to improve the fit during relaxation (27).

The results of the optimization process are shown in Fig. 5. The mean normalized deviation (Eq. 9) was only 0.013, which implies that the model can reproduce dynamic twitch contractions with high fidelity.

Simulations driven by the same Ca²⁺ transient at each of four lengths are shown in Fig. 6. The peak twitch force predicted for the longest sarcomere length under these conditions was 25.8 kN m⁻². This value is 61% of the corresponding twitch force predicted for the same sarcomere length using the experimentally appropriate Ca²⁺ transient (Fig. 5, orange trace). The difference in the Ca²⁺ transients measured at the long and short lengths therefore contributes 39% of the change in peak twitch force. These calculations imply that myofilament-based mechanisms have a slightly larger impact on twitch forces than length-dependent changes in Ca²⁺ transients.

Model validation

The last phase of this work tested the model by using it to predict experimental data that had not been included in the parameter fitting procedures. An opportunity for independent validation was offered by data obtained by Patel et al. using chemically permeabilized myocardium isolated from mutant rats that expressed a giant isoform of titin (41). Note that Ait-Mou et al. were studying the same strain of rats when they discovered that passive tension could induce structural changes in the myofilaments (6).

Patel et al. measured reduced passive tension in the mutant myocardium as well as significant changes in length-dependent activation. The model could thus be tested by decreasing passive stiffness in the simulations to match Patel et al.’s data, calculating force-pCa curves while leaving all other parameters unchanged, and then comparing the predictions to the published experimental results. Fig. 7 summarizes the test.

Figure 7.

Simulations predicting the effects of altered passive stiffness mimic experimental data measured from rat myocardium expressing different titin isoforms. (A) Simulations of WT and mutant myocardium are shown. k_p and L₀ for the WT myocardium were set to 118 N m⁻² nm⁻¹ and 984 nm, respectively. The corresponding values for the mutant myocardium were 14.0 N m⁻² nm⁻¹ and 912 nm. These values were obtained by fitting a linear function to the data reported by Patel et al. (Table 1 of (41)). All other parameter values are as listed in Table S1. (B) A comparison of experimental data (Table 1 of (41)) and predicted data for pCa₅₀, n_H, and relative Ca²⁺-activated force is shown. To see this figure in color, go online.

The predictions mimicked many features of the experimental data. The most striking result was the prediction for pCa₅₀ (Fig. 7 B, top panel). The experiments had shown that the length-dependent shift in pCa₅₀ was reduced by 58% in the mutant myocardium. The corresponding model prediction was 60%. Patel et al. had also shown that the Hill coefficient was reduced at long length, with the reduction being slightly smaller in the mutant myocardium (Fig. 7 B, middle panel). The model predicted both of these effects, although it overestimated the magnitude of the differences. Note that the differences in baseline values for pCa₅₀ and n_H probably reflect systematic differences in the experimental data measured by Kampourakis et al. (8) and Greaser et al. (41). Finally, the model predicted Patel et al.’s finding that maximal force was less sensitive to sarcomere length in the mutant myocardium (Fig. 7 B, bottom panel). Simulated force was not lower for these mutant samples, but that is the only feature of the experimental data that the calculations did not predict in at least a qualitative manner. Taken together, the accuracy of these predictions validates the coupled mechanosensitive model as a useful resource for studies of sarcomere-level function.

Discussion

Force-dependent recruitment of myosin heads

The modeling studies presented in this work provide key insights into the mechanisms that modulate cardiac contractility. The breakthrough publications from Ait-Mou et al. (6) and Kampourakis et al. (8) had demonstrated that when myocardial force rises, thick filaments undergo structural changes that are consistent with recruitment of myosin heads from the off state (9, 11). What remained unclear was whether these structural effects were sufficient to explain length-dependent activation. Our simulations answer this question and demonstrate that force-dependent recruitment of myosin heads from the off state is sufficient to explain the length-dependent behavior of both permeabilized myocardium and electrically stimulated trabeculae.

Two statistical approaches were used to test the significance of thick-filament mechanosensitivity. Both showed that a model exhibiting force-dependent recruitment of myosin heads produced a better fit to the experimental data than a model with a constant off-to-on transition rate. Our work thus adds corroborating evidence to the emerging theory that dynamic recruitment from an inhibited myosin state is the dominant mechanism underlying length-dependent activation and ultimately the Frank-Starling relationship (42).

Our model was based on simple assumptions. Myosin heads transitioned between an off relaxed state (that could not interact with actin), an on state (that could bind to actin), and a single attached configuration (Fig. 1). The rate of the k₁ transition between the super-relaxed and disordered-relaxed states was assumed to increase linearly with force and also with the relative phosphorylation of myosin light chains (Eq. 3). These choices were inspired by the emerging literature focusing on the super-relaxed state and interacting-heads motif (12, 15, 43, 44). The thin filament was modeled as a simple cooperative system (Eqs. 1 and 2) with the parameters constrained by experimental data (Fig. 2).

Coupled together, these basic assumptions proved sufficient to explain the complex data set reported by Kampourakis et al. (Fig. 3). Modifications to the model provided additional insights. For example, setting k_force to zero prevented force-dependent recruitment and eliminated length-dependent changes in Ca²⁺ sensitivity (Fig. 4) and k_tr (Fig. S6). This computational test reinforces the pivotal role of mechanosensitive transitions within the thick filament.

Additional calculations demonstrated that the model could also reproduce dynamic twitch responses measured at different sarcomere lengths (Fig. 5). This is significant because sarcomere length affects not only peak twitch stress but also the shape and duration of the twitch response. The model was thus able to reproduce multiple facets of length-dependent contractile behavior simultaneously. Further calculations drove models stretched to different lengths with identical Ca²⁺ transients (Fig. 6). These results suggested that myofilament-based mechanisms account for ∼60% of the length-dependence of peak twitch force, with length-dependent changes in Ca²⁺ transients responsible for the remaining ∼40%.

The final calculations presented in the main text (Fig. 7) predicted how reducing passive stiffness would impact force-pCa relationships. The predictions matched most features of the experimental results published by Patel et al. (41) in at least a qualitative sense and were remarkably precise for the pCa₅₀ parameter. Patel et al. had shown a 58% reduction in the length-dependent shift in pCa₅₀ in myocardium from mutant rats that expressed a giant isoform of titin. The prediction was 60%. These calculations are important because they predicted experimental data that had not been utilized in the development of the model. The accuracy of the results validates the model.

Potential role of MyBP-C

Length-dependent activation is driven in our model by dynamic transitions within the thick filament. We recognize, however, that the structure of the thin filament is also sensitive to stretch. In particular, we are intrigued by data published by Zhang et al. (45) that show that the pCa₅₀ for thin-filament activation in rat myocardium increases by 0.05 ∼ 0.10 log units when sarcomere length is increased from 1.9 to 2.3 μm and active force is abolished with blebbistatin. The molecular mechanisms underlying this effect remain unclear, but Zhang et al. suggested that it might involve mechanical signaling through MyBP-C. Fig. S18 shows a model that was developed to investigate this hypothesis.

The concepts underlying the supplementary model are as follows. Increased passive tension destabilizes a complex formed by MyBP-C and the myosin off state (12, 16). This releases MyBP-C molecules so that their N-terminus ends can compete with myosin for binding sites on actin (46, 47). Actin-MyBP-C links mimic non-force-generating cross-bridges and thus augment bound myosin heads as they activate the thin filament through a cooperative mechanism (48). In situations in which myosin binding is inhibited, the actin-MyBP-C links drive the changes in thin-filament structure.

Fig. S19 shows that these assumptions are sufficient to explain the length-dependent shift in the pCa₅₀ for thin-filament activation observed by Zhang et al. (45). However, the supplementary model has several key weaknesses. First, the kinetics of the postulated actin-MyBP-C interactions are not grounded by corresponding experimental data. Second, the structure of the current MyoSim source code makes it difficult to couple myosin and MyBP-C in noninhibited muscle. Third, the model does not allow for the fact that MyBP-C molecules are localized to the central region of each sarcomere. All of these weaknesses can be overcome, but we concluded that simulations focusing on the specific role of MyBP-C are best left for future work using spatially explicit models (49, 50, 51).

Force-dependent recruitment and thin-filament cooperativity are synergistic

Further analysis of the twitch simulations demonstrated that force-dependent recruitment of myosin and thin-filament cooperativity are synergistic; both mechanisms are required to reproduce the length dependence of twitch forces (Fig. 5). The best fit attained when force-dependent recruitment was eliminated (Fig. S20) did not match the time course of contraction at the longest length. Similarly, simulations without thin-filament cooperativity (Fig. S21) could not relax quickly enough at long lengths because the thin filament remained activated as Ca²⁺ declined.

These results show that the potency of force-dependent recruitment depends on thin-filament cooperativity. When a myosin head attaches to actin, it increases the availability of binding sites in adjacent regions of the thin filament (reviewed recently in (48)). Myosin heads bind to these new sites and generate force, which in turn recruits additional myosin heads from the super-relaxed state. The end result is a positive feedback mechanism that supports successive rounds of cooperative activation, myosin force generation, and further myosin recruitment. Eliminating thin-filament cooperativity reduces the positive feedback and renders force-dependent recruitment less potent.

These hypotheses are supported by experimental data. Tachampa et al. (52) observed diminished length-dependent activation after exchanging mutant troponin I into cardiac fibers. In light of our simulations, it now seems reasonable to suggest that the troponin I T144P mutation repressed length-dependent activation by diminishing cooperative interactions along the thin filament. Indeed, our model now helps to explain why a wide range of molecular perturbations to thick and thin filaments can modulate length-dependent contraction.

Passive and active force

This manuscript only investigated the simplest mechanism for force-dependent recruitment of myosin. All forces within the sarcomere, whether passive or active in origin, were assumed to impact the thick filament and modify the number of available heads.

For some portions of the data, recruitment in response to passive force was critical. For example, in Fig. 3, the increased passive tension at the longer sarcomere length raised the proportion of myosin heads in the off state at low levels of activation irrespective of the relative phosphorylation of myosin light chain (compare magenta triangles to green squares and blue triangles to red stars). Similarly, the increased passive tension at the longest sarcomere length raised the proportion of on heads before the Ca²⁺ transient in simulations of the twitching trabecula (Fig. 5).

However, recruitment due to myosin-based active force also played an important role modifying myofilament-length dependence and Ca²⁺ sensitivity in simulations of permeabilized myocardium. For example, the Hill coefficients dropped by an average of 2.6 when k_force was set to zero (Figs. 3 and 4). Similarly, eliminating force-dependent recruitment reduced the rate of tension development in simulations of the twitching trabecula (Fig. S20).

We note that the first report of the myosin super-relaxed state in cardiac muscle by Hooijman et al. (13) suggested that the population of super-relaxed heads remained unchanged at pCa 5.7. In our simulations (Fig. 3), heads have been pulled out of the M_OFF state at this Ca²⁺ concentration at a sarcomere length of 2.3 μm but remain relatively unperturbed at 1.9 μm. Hooijman et al. did not monitor force or sarcomere length during their measurements, so it is possible that force remained too low to recruit heads from the M_OFF state in their specific experiments.

Conclusion

We show that a mechanism whereby myofilament force triggers the release of myosin heads from the off state is sufficient to produce realistic length-dependent activation, provided that it occurs in the context of cooperative thin filaments. The quantitative framework provided by this new to our knowledge model leads naturally to new hypotheses, including the idea that any molecular perturbation that affects thin-filament cooperativity could also affect length-dependent activation.

Author Contributions

K.S.C. planned the project, developed the model, created the computer software, ran the simulations, and wrote the manuscript. P.M.L.J. performed experiments, interpreted the data, and edited the manuscript. S.G.C. planned the project, developed the model, and wrote the manuscript.

Editor: David Thomas.