The model we employ accounts for the relative distributions of thin filament states (Figure 1). The B, C, and M states of Tm refer to Tm's interactions with actin in these respective positions (Lehman et al., 2000). State C is the equilibrium position (Phillips et al., 1986; Lehman et al., 2000), and states B and M are modeled as competing for Tm in state C. To stabilize the non-equilibrium positions B and M, Tm-bound Tn forms an interaction with actin (Greaser and Gergely, 1973) in position B, and Tm forms a ternary complex with crossbridges and actin in position M (Eaton, 1976; Tobacman and Butters, 2000). The interaction of Tn in state B accounts for the states of Tn that are energetically coupled to the states of Tm. Movement of Tm away from B energetically uncouples Tn from possible interactions with actin (Figure 1).

Coupled and uncoupled states of Tn are designated B and T, respectively (Figure 1). Calcium-dependent states of Tn are designated B_i and T_i, where i represents 1 (Ca²⁺ free), 2 (singly bound), or 3 (doubly bound). Affinity for actin is progressively reduced as i increases. Based on conservation of the mass for two partially overlapping subsystems (Tm and Tn), B + C + M = 1 for Tm and B + T = 1 for Tn.

In the system we describe, M can arise by either rigor or cycling crossbridges, but cooperativity derives solely from cycling crossbridges, as seen in the records of Ca²⁺ binding measurements. With overlapping thin and thick filaments and ATP, M is a state in constant flux (steady-state) rather than at equilibrium; this is readily observed by Ca²⁺-dependent in vitro sliding of regulated actin filaments. Steady-state cooperativity is achieved if crossbridge turn-over generates additional opportunities (second chances) for M formation (Zot et al., 2009, 2012). By analogy, the M state operates like a man whose feet are bound to a ceiling by adhesion: changing positions quickly relative to the rate of deadhesion improves the odds of remaining bound by re-establishing the initial binding conditions. A statistical treatment of a second chance mechanism applied to data from biological systems is available (Zot et al., 2016a). In practice, a second chance mechanism is given in the following rate equation (1)dM/dt=K0′k−0C(1+(α−1)M)n−k−0M where parameters K0′, k₋₀, α, and n are derived elsewhere (Zot et al., 2009, 2012). The parameter α expresses second chance opportunities for reestablishing equilibrium before the decay of M. As steady-state or equilibrium approach, dM/dt → 0. Because Equation (1) acts on the C-M transition (Figure 1), cooperativity does not directly involve Ca²⁺ binding to Tn. Although, Equation (1) performs adequately, any logistic function operating on C can be compatible with our model.

As applied to transient and steady-state striated muscle regulation, the parameters of Equation (1) may have the following interpretations. The equilibrium potential of M as a function of the population of strong binding myosin at any moment of steady-state is expressed by K0′. The forward rate of ensemble M formation is the product K0′k₋₀. Ensemble size, n, expresses the number of Tm subunits acting in concert to form a ternary complex as described above. The orchestrating event could be a lateral stretch imposed on contiguous Tm subunits by an axial force acting on the thin filament (Zot et al., 2009). The parameter α is an expression of the crossbridges poised to replace crossbridges disrupted by internal chemomechanical forces or by active sliding. The value of α may be related to the average number of crossbridges in a target zone (Tregear et al., 2004), as has been described (Zot et al., 2009). We give unit value to K0′ for simplicity and recycle previously discussed values for α, and n (Zot et al., 2009; Table 1) for consistency.

Transitions other than C-M are spontaneous processes governed by simple mass action (Figure 1). Although the model has eight states, only six are independent. If we choose to calculate states C and T₁ by mass conservation (see above), the relative abundances or probabilities of the other six states (Figure 1) as a function of an independent calcium transient are calculated by solving a system of six ordinary differential equations (ODE), i.e., in addition to Equation (1), we have dB1/dt=K1k−1CT1+k−4B2−(k−1+K4k−4[Ca2+])B1;dB2/dt=K3k−3CT2+k−4B3+K4k−4[Ca2+]B1                   −(k−3+k−4+K4k−4[Ca2+])B2;dB3/dt=K5k−5CT3+K4k−4[Ca2+]B2−(k−4+k−5)B3;dT2/dt=K2k−2[Ca2+]T1+k−3B2+k−2T3                   −(k−2+K2k−2[Ca2+]+K3k−3C)T2;dT3/dt=K2k−2[Ca2+]T2+k−5B3−(k−2+K5k−5C)T3.

This system of ODE is solved for each free Ca²⁺ concentration of a given transient. The free Ca²⁺ transient of a muscle fiber (Matsuo et al., 2010) is reproduced by a linear rise from the origin to the peak (time to peak is 0.005 s), followed by an exponential decay (rate constant is 100 s⁻¹; Figure 2). We use the same Ca²⁺ transient for all calculations as a control. Hence, the model varies only the contribution of crossbridges in fitting data from overlap and non-overlap preparations. A Matlab program is provided to reproduce calculations presented here (see Supplementary Materials).

Standard conditions refer to a set of constants governing steady-state potentials (upper case, “K”), α, and n that we hold constant (Table 1). Thermodynamic principles dictate that K₁/K₃ = K₃/K₅ = K₂/K₄, which allows K0′, K₁, K₂, and K₄ to be selected independently. The values used here for these four parameters, α, and n (Table 1) are the same shown elsewhere to fit steady-state activation of skinned fibers of fast muscle (Zot et al., 2009) and cardiac muscle (Zot et al., 2016b) by Ca²⁺. Component rate constants (lower case, “k”) are varied to fit transient data. Slow and fast dissociation rates of Ca²⁺ from Tn (k₋₂ and k₋₄; Table 1), are taken from Rosenfeld and Taylor (1987).

Steady-state or equilibrium calculations are produced for standard conditions (Table 1) by nullifying the system of ODE (cf. Matlab program in Zot et al., 2016b). Calculated Ca²⁺-dependent activation curves on absolute and normalized scales (inset, Figure 2) reproduce fits of steady-state tension and ATPase data of diverse native and mutant protein preparations of fast skeletal and cardiac muscles (Zot et al., 2009, 2016b). To verify that the model reproduces established Ca²⁺ binding properties of Tn, Ca²⁺-bound Tn (B₂ + B₃ + T₂ + T₃) is calculated as a function of constant concentrations of Ca²⁺. With cycling crossbridges, the mathematical solution is a cooperative function of Ca²⁺ (inset, Figure 2), which reproduces the distinctive binding-activation relationship observed previously with fluorescently labeled Tn in reconstituted myofibrils (Zot et al., 1986; Zot, A. S. and Potter, J. D., 1987; Brandt and Poggesi, 2014). To model equilibrium achieved by non-overlap or rigor, K0′ is set either to zero or to a relatively large value (10⁶), respectively. Both solutions predict simple mass action (non-cooperative) calcium binding (inset, Figure 2), and both simulations reproduce published simple mass action Ca²⁺ binding curves for regulated actin and regulated actin with rigor myosin binding, respectively (Rosenfeld and Taylor, 1987). Hence, the model is able to reproduce cooperative and non-cooperative Ca²⁺ binding measurements of both steady-state and equilibrium preparations, respectively.

The transient of calcium-bound troponin, which is modeled as the sum of B₂, B₃, T₂, and T₃, is compared with the transient change measured by fluo-3 fluorescence (Matsuo et al., 2010) in overlap muscle preparations. Rather than choosing parameters to fit the fluo-3 data, we fit tension data (Figure 3) from the same preparation with calculated value of M (Equation 1) by adjusting the rate constants (Table 2) that comprise standard conditions (Table 1). Decreasing K0′k₋₀ shifts the calculated tension transient rightward, and k₋₀ is decreased in tandem to maintain constant K0′. The same constraint is used throughout to maintain standard conditions. Although adjusting either K0′k₋₀ or K₁k₋₁ changes the tension transient equivalently, using K0′k₋₀ for lateral adjustments and K₁k₋₁ for vertical adjustments yields the best shape of the tension transient relative to the data. Given the optimum fit of the tension data, the predicted Ca²⁺-bound Tn transient is seen to fit most of the fluo-3 fluorescent data (Figure 3). If we accept a slightly faster time to peak tension, the model predicts slower decay of Ca²⁺-bound Tn, which may better capture the entire trend of fluo-3 data (see Supplementary Materials).

The dissociation of Ca²⁺ from the Ca²⁺ regulatory sites of Tn is not uniform over time. Owing to a faster off-rate, Ca²⁺-bound B states (B₂ + B₃) release Ca²⁺ faster than Ca²⁺-bound T states (T₂ + T₃; Figure 3). Hence, a significant fraction of Tn has released Ca²⁺ before tension reaches a peak. A protracted dissociation of residual Ca²⁺ comes mainly from T states, which represent a pool of Tn molecules held away from interaction with actin in the B position owing to crossbridges (crossbridge-dependent Ca²⁺-bound Tn).

The relationship between calculated Ca²⁺-bound Tn and fluo-3 data differs dramatically in muscle fibers stretched beyond overlap. To simulate no overlap, we nullify K0′k₋₀ in Equation (1), but otherwise preserve the rates established for the overlap condition (Table 1). Absent crossbridges, the decays of calculated Ca²⁺-bound troponin, whether expressed as a total or separated into components B and T, are much faster than the decay of fluo-3 fluorescence (Figure 4). No combinations of rate constants will allow the model to fit the tension data of the overlap preparation and the fluo-3 data of overlap and non-overlap preparations. By contrast, the decay of fluo-3 fluorescence is constant for overlap and non-overlap conditions (Figures 3, 5). Therefore, there is not a causal relationship between the calculated Ca²⁺-bound state of troponin and fluo-3 fluorescence.

Matsuo et al. (2010) show that in a non-overlap sarcomere preparation, the fluo-3 fluorescence decay is slower than the decay of meridional 1/38.5 nm⁻¹ reflection intensity, which corresponds to the troponin repeat in the thin filament (Figure 5). We find that the calculated time courses of Ca²⁺-bound Tn and the C position of Tm correlate with the transient of meridional 1/38.5 nm⁻¹ reflection intensity (Figure 5). A systematic sensitivity test of all rate constants shows that the decay of Ca²⁺-bound Tn is limited only by k₋₄ of the model (Figure 5, Table 2). This is the faster of two dissociation rates determined for Ca²⁺ from regulated actin (Rosenfeld and Taylor, 1987). Increasing k₋₄ by a factor of 1.33, which is within the range of measured values (Rosenfeld and Taylor, 1987), and maintaining the same steady-state conditions (inset, Figure 2) bring the decay rate of Ca²⁺-bound Tn closer in alignment with the decay of meridional 1/38.5 nm⁻¹ reflection intensity (see Supplementary Materials). Furthermore, the decays of Ca²⁺-bound Tn and state C of Tm align more closely by increasing K₁k₋₁ (Figure 5, Table 2; see Supplementary Materials). Hence, the model predicts that the decay rate of meridional 1/38.5 nm⁻¹ reflection intensity in the non-overlap preparation gives an in vivo measure of the rate of Ca²⁺ dissociation from Tn. A characteristic intensity increase in response to Ca²⁺ represents Tn-dependent structural changes of the thin filament (Yagi, 2003). Modeling suggests that Ca²⁺-bound Tn regulates completely the structural changes related to the movement of Tm to position C in the non-overlap preparation.

In the overlap preparation, the structural changes related to meridional 1/38.5 nm⁻¹ reflection intensity are more complex (Matsuo et al., 2010), showing intensity changes with positive and negative slopes over the time course of a twitch (Figure 6). The early rise in reflection intensity correlates with a brief period at the beginning of the Ca²⁺ pulse in which the model predicts a rise in Ca²⁺-bound Tn and state C of Tm before the transition to state M of Tm begins. The large negative change in reflection intensity has roughly the same time course as the calculated fraction of Tm in the M position (Figure 6). The minimum reflection intensity comes at a time after the Ca²⁺ transient has decayed and the calculated M state has peaked.

The calculated decay rate of Ca²⁺-bound Tn can be made more rapid than the measured decay of fluo-3 fluorescence by increasing rate constants, K₁ k₋₁, at fixed K₁ (Figure 6, Table 2). However, both absolute rates and the competition between states M and B for Tm in state C (Figure 1) must be made more extreme to hold constant the calculated tension transient (Figure 3). Thus, a balance of competing factors explains the correlation between the calculated Ca²⁺-bound state of Tn and the fluo-3 fluorescence transient in the overlap preparation (Figure 6).

Crossbridge-dependent Ca²⁺-bound Tn is the difference between Ca²⁺-bound Tn calculated for overlap and non-overlap conditions, holding the transient of free Ca²⁺ constant (Figure 7). The residue is a pool of crossbridge-dependent Ca²⁺-bound Tn, which represents about 30% of the total area under the curve. The rise in the pool of crossbridge-dependent Ca²⁺-bound Tn begins near the end of the free Ca²⁺ transient and continues during the decay of the tension transient. Peaking at ~60 ms, the time course of rising crossbridge-dependent Ca²⁺-bound Tn correlates most closely with the time course of the declining phase in meridional 1/38.5 nm⁻¹ reflection intensity, which reaches a minimum at ~70 ms.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.