Fig 1.
Each regulatory unit (RU) along the actin-tropomyosin thin filament contains troponin C, which binds calcium, and troponin I, which can bind either to actin or troponin C. These reactions, in addition to the binding of crossbridges, define each of the states. In the schematic, state names with + have calcium bound, state names which include an ‘X’ have crossbridge(s) bound, and B, U, S refer to the labels ‘blocked’, ‘unblocked’, ‘stable unblocked’ at the bottom of the schematic. ‘Blocked’ refers to troponin I bound to actin, which blocks myosin binding. ‘Unblocked’ refers to troponin I being not bound to actin, and ‘stable unblocked’ refers to troponin I being held in place by troponin C. Each state allows for crossbridge binding, although this is very improbable in the ‘blocked’ states, such that states (‘BX’ and ‘BX+’) rarely occur. Note that all transitions between neighbouring states exist, in addition to transitions between the top and bottom rows. The green arrows indicate the main pathway during activation, with Ca2+ binding to TnC, and TnI moving from actin to TnC⋅Ca2+ to allow crossbridge binding. Red arrows indicate the main deactivation pathway, TnI detaching from TnC⋅Ca2+, followd by Ca2+ detaching from TnC and TnI binding to actin to block crossbridge binding.
Table 1.
Index of model parameters.
Fig 2.
Equivalent tropomyosin states.
The top two states are considered equivalent, and part of the same class of states, as they both have three adjacent unblocked RU’s and one isolated unblocked RU. The bottom state is part of a different class, even though it also has four unblocked RU’s, as it has two stretches with two adjacent unblocked RU’s each.
Fig 3.
Influence of TnI affinity for actin and TnC on muscle cooperativity.
Panel A shows cooperativity plotted as a function of the dissociation constant of TnI for actin (KDA) and the dissociation constant of TnI for TnC⋅Ca2+ (KDI). There is a relatively large triangular region in parameter space in which cooperativity is high, with a slight tendency for higher cooperativity at very low KDA, KDI reflecting more extreme competitive binding of TnI. Panel B shows calcium sensitivity, which follows a smooth gradient The yellow ‘X’ indicates our choice of parameters, and the red contours indicate the regions within which nH ≥ 5. At high KDA, affinity for actin is insufficient to block tropomyosin effectively, leading to a permanent high level of activation (indicated by the blue text and contour line for minimum force greater than 1% of maximum force at the top of the plot). When the affinity of TnI for actin is much lower than for TnC, muscle activation is decreased, (indicated by the green text and contour line for maximum force less than 95% of overall maximum force in the bottom right of the plot).
Fig 4.
Comparison of different modelling approaches on a short filament.
Shown are the different modelling approaches on a filament with n = 7 RU’s and 18 crossbridges. Results for the ‘brute-force solution’, and the representative state approximation overlap and are indicated by a single line. The Monte Carlo approximation performs well, with only a ∼ 1% difference at higher force levels. Comparison with the independent crossbridge approximation shows the importance of including XB-RU cooperativity, which increases calcium sensitivity, as well as XB-XB cooperativity, which increases maximum force development as shown by the difference in the number of crossbridges bound per half-sarcomere at high calcium. Note that due to the lower number of 7 RU’s, cooperativity is significantly lower than in realistic models with 26 RU’s presented in other results, and the duty ratio is moderately reduced to approximately 2.7/18 = 15%.
Fig 5.
Panel A shows the Force-calcium curve relationship of the model alongside the RU unblocking as a function of calcium. This shows that the Force-calcium curve is significantly steeper than RU activation, as indicated by its higher Hill coefficient. Panel B explains this effect by looking at the expected number of crossbridges on tropomyosin chains with a fixed number of RU’s unblocked, where the probability of tropomyosin sub-states is according to Boltzmann’s law. This shows that there is significant inhibition of crossbridge binding at low numbers of RU’s, compared to models where the ‘linear response’ of crossbridge binding being proportional to the number of unblocked RU’s is assumed.
Fig 6.
Force-pCa curves for the model and effects of changes in crossbridge affinity.
The effects of crossbridge inhibition by substances such as blebbistatin and sodium vanadate were simulated by changing the dissociation constant of myosin (KDM), resulting in significant changes in calcium sensitivity. Shown in panel (A) are the default, highly cooperative, force-calcium relationship of the model (nH = 5.1), along with the following virtual experiments: (1) In red: a factor 3 decrease in crossbridge affinity. This reproduces data from experiments with the cross-bridge inhibitor blebbistatin [16], showing a decrease in calcium sensitivity and a mild decrease in cooperativity (nH = 4.2). (2) In blue: a 33% increase in crossbridge affinity. This reproduces data from experiments with the cross-bridge augmenter 2-deoxy-ATP (dATP) [55], showing an increase in calcium sensitivity and a small increase in cooperativity (nH = 5.3). Panel (B) shows the KDM-dependence of force at zero Ca2+ and at pCa 4.5 [45], showing the model produces maximal force at both calcium levels for a sufficiently high myosin affinity, and a sigmoidal relationship between KDM and force. The dashed line indicates the value of KDM used in the model, which intersects the pCa 4.5 curve at approximately 0.25, the duty ratio of myosin used in the model. Thus, the maximal force generated in panel (B) for KDM → 0 is approximately 4× higher than the ‘default model’ curve in panel (A).
Table 2.
Model predictions for shifts in pCa50 after changes in crossbridge affinity.
Table 3.
List of model parameters.
Table 4.
Parametrization of the model for different species.
Fig 7.
Isometric tension in mouse, rat, and human.
Shown in panel A are the tension transients described in Table 4. The last 500 ms of the human tension transients are not shown, as force is very low throughout. These results were driven by fixed calcium transients shown in panel B, based on recent data in mouse [66] and rat (see S1 Text [67, 68]) measured at 37°C and 6 Hz, and data from Coppini et al. [69] (see S1 Text [70]). Two sets of results are shown, corresponding to a constant crossbridge unbinding rate (q = 1 in Eq 22) and a crossbridge unbinding rate that is variable, modified by the average difference in free energy for the tropomyosin deformation (q = 0.5 in Eq 22)
Fig 8.
Tension redevelopment rates in mouse, rat and human.
Panel A shows results for q = 1 (constant crossbridge unbinding rate), and Panel B plot results for q = 0.5 (variable crossbridge unbinding rate). Each plot shows both normal kinetics, and with RU (un)blocking due to TnI-actin binding disabled for mouse, rat, and human parametrizations of the model. The difference in ktr when disabling RU (un)blocking reveals the influence of transient blocking of RU’s on tension redevelopment kinetics. This transient blocking causes a significant difference between the minimum ktr and the ktr at high [Ca2+], the ratio of which is indicated with the results. Note the shift in the minimum ktr is not significant here as it is a direct result of the difference in peak calcium shown in Fig 7A and subsequent parametrization.