Mechanism of contraction rhythm homeostasis for hyperthermal sarcomeric oscillations of neonatal cardiomyocytes

The heart rhythm is maintained by oscillatory changes in [Ca2+]. However, it has been suggested that the rapid drop in blood pressure that occurs with a slow decrease in [Ca2+] preceding early diastolic filling is related to the mechanism of rapid sarcomere lengthening associated with spontaneous tension oscillation at constant intermediate [Ca2+]. Here, we analyzed a new type of oscillation called hyperthermal sarcomeric oscillation. Sarcomeres in rat neonatal cardiomyocytes that were warmed at 38–42 °C oscillated at both slow (~ 1.4 Hz), Ca2+-dependent frequencies and fast (~ 7 Hz), Ca2+-independent frequencies. Our high-precision experimental observations revealed that the fast sarcomeric oscillation had high and low peak-to-peak amplitude at low and high [Ca2+], respectively; nevertheless, the oscillation period remained constant. Our numerical simulations suggest that the regular and fast rthythm is maintained by the unchanged cooperative binding behavior of myosin molecules during slow oscillatory changes in [Ca2+].

half-period of a normal HSO waveform. The frequency of occurrence of this half-period was rare, as indicated by the area ratio of the histogram in Figure 2i.

HSO amplitude
Enlarged view of Figure 5 (a-f) from 0.05 to 0.25 seconds. Panels g-i were reorganized into two panels except for < k-2>. A-E are symbols for explaining the mechanism of HSOs in the text.

Myofibril Model
In the passive elastic model, we assumed only transversely isotropic deformation for each halfsarcomere for simplicity, and we applied the half-sarcomere deformation energy per unit volume in the unloaded condition with the following equation: Here, and are stretches in the longitudinal and transverse directions, respectively.
The first and second terms are the deformation energies associated with the longitudinal and transverse elasticities of the half-sarcomere, respectively. Each of them is determined from the strain solely in the associated direction. The third term is a weak penalty term associated with an inverse SL-LS relationship represented as ( , ) = 0. is a parameter that represents the sarcomere stiffness associated with this penalty. Based on the experimental data for contracted muscle [Ertbjerg2017, Konhilas2002], we adopted a linear function: where = 2 , = 10 kPa , = 20 kPa , and = 1 MPa . These parameters were adjusted to reproduce the experimental data. Regarding the last term, is the penalty function that prevents extraordinary shortening and stretching from the presence of the thick filament and the filamentous protein called titin connected with the Z-disc. is represented as follows: where min = 5GPa, min = 0.87, max = 100MPa, and max = 1.25 [Washio2016].
We also introduced the deformation energy per unit volume in the unloaded condition for the difference in lattice spacing between the adjacent half-sarcomeres, as follows: Under the above deformation energies, the total longitudinal contractile tension in the ith halfsarcomere is given by Here, = 10Pa • s is the longitudinal friction coefficient [Washio2017], and , is the active contractile tension generated by the actomyosin complexes (given later). The total transverse contractile tension in the ith half-sarcomere is given by Here, = 10Pa • s is the transverse friction coefficient. Note that either the third or fourth term is omitted for the half-sarcomeres at both ends. From the longitudinal mechanical equilibriums at the interfaces of the half-sarcomeres and at the right end and from the transversal mechanical equilibrium in each half-sarcomere, the following equations must be fulfilled: where is the total number of half-sarcomeres.
where the sarcomere length 0 = 1.9μm in the unloaded condition. We assumed that the left end (i = 0) was fixed ( 0 ≡ 0) and that the right end ( = ) was connected to a spring with the spring constant . ̅ is the length of the myofibril at which the spring force is zero. For the myofibril model consisting of 40 half-sarcomeres, = 30 kPa/μm and ̅ = 1.07 • • 0 /2 were adopted. These parameters were adjusted to reproduce the experimental data. For the numerical simulation of the single half-sarcomere model, = 30 kPa/μm and ̅ = 2.2 • 0 /2 were adopted.

Active Contractile Tension
All the parameter values adopted in the actomyosin complex model described here are listed in Table S1. To determine the active contractile tension act , we applied a stochastic crossbridge cycling model [Washio2019] consisting of the five states (detachment, weak binding, prestroke, poststroke, and rigor states) depicted in Fig. 4. First, we describe the transitions between the strong binding states (the prestroke, poststroke, and rigor states), which are related to active tension. In this model, the rate constants of the power stroke ( + , = 3,4) and the stroke reversal ( − , = 3,4) are determined as a function of the rod strain such that they fulfill the relationship given by the following statistical equilibrium: where and denote the Boltzmann constant and the temperature, respectively, and and +1 are the free energies of the actomyosin complex before and after the power stroke, respectively. The stroke distance . is the rod strain before the power stroke. is the strain energy in the myosin rod. In our numerical model, we put one-dimensional filament pairs in each half-sarcomere to determine the active tension at time generated by the bound myosin molecules, as follows: (S11) Here, is the most recent time at which the myosin molecule was attached, is the initial strain at the attachment, is the total power stroke distance after the attachment, and ̇ is the half-sarcomere shortening velocity given by Thus, half-sarcomere shortening ( ̇> 0) implies a decrease in rod strain resulting in facilitation of the power stroke transition (increase in + ( )/ − ( + ) ), while half-sarcomere lengthening ( ̇< 0) implies an increase in rod strain, resulting in facilitation of stroke reversal (decrease in + ( )/ − ( + )).
( + /2) is introduced to take the energy barrier between the two states before and after the power stroke into account [Washio2017]. Note that the contribution of the free energy in the myosin head at the barrier between the states before and after the power stroke is included in the constant ℎ . The power stroke transitions are thought to accompany the release of Pi and ADP from the nucleotide binding pocket in the myosin head at the first and second power strokes, respectively [Wulf2016]. If that is case, the reversal stroke transitions must accompany the re-binding of these molecules. Therefore, it is reasonable to suppose that the transition rates are limited by the rates of these chemical reactions. We assumed the upper bounds due to these limitations from the release and the re-binding, which are given by ̅ + and ̅ − , respectively. With these upper limits, the temporary rate constants given above are modified as follows: The elastic force of a myosin rod is nonlinear with respect to the strain, as described by Kaya et al. [Kaya2010]. We assumed that a myosin rod behaves as a linear spring for positive stretches, . (S18) In our model, we assumed that attachment (the transition from the weak binding state to the prestroke state in Fig. 4c) is allowed only in the single overlapping region of the thin and thick filaments. We also assumed that myosin molecules are arranged on a thick filament at regular intervals except in the bare zone (B-zone). Therefore, the myosin head (# ) is situated in the single overlapping region if and only if the following condition is fulfilled: Here, the middle term is the distance from the center of the sarcomere; , and are the lengths of the thick filament, the B-zone and the thin filament, respectively; and = ( • 0 /2) is the half-sarcomere length for stretch . The parameters for the sarcomere geometry were determined for a cardiac sarcomere [Kolb2016, Lodish, Rice2008, Rodriguez1993].
The thin filament is divided into segments called troponin/tropomyosin (T/T) units (Fig. 1b). plays an important role in the force-pCa relationship [Washio2018]. We assumed that one thin filament in the three-dimensional arrangement corresponds to two thin filaments in our halfsarcomere model. This is because we assumed that cooperative behavior exists along the tropomyosin and tropomyosin molecules wrapped around the thin filament in a double spiral fashion, and only one of the spirals is considered in our half-sarcomere model. The constants np0 , np1 , pn0 , and pn1 are determined from , basic , and , as follows: (S23) Here, > 1 controls the degree of crossbridge inhibition for T/T units in states other than Caon, and controls the ratio of binding states of the myosin heads. The greater the value of , the larger the ratio of binding states for a given [Ca 2+ ]. In this study, we assumed that elevated temperature facilitates the transition from the detachment state to the weak binding state by relaxing the binding inhibition of tropomyosin. This effect was introduced by multiplying the temperature-dependent coefficient to define .
= 0 (S24) = 1 was adopted for the case before heating, while = 2 was adopted for the case after heating. To reproduce the sarcomere length ( = 0 ) dependence in the active contraction tension, the following function ( ) was multiplied to define np0 and np1 :
The rate constants of attachment ( +2 ) and detachment ( −2 ) are given based on the assumed free energies 2 and 3 , respectively, in the weak binding state and the prestroke state, as follows: The initial rod strain at the attachment is given stochastically from the Boltzmann distribution determined by the rod strain energy ( ) [Washio2016]. The detachment rate constant from the rigor state to the detachment state is assumed to be strain-dependent for the negative strain, as follows: =6pN was applied in this study.
We also took the forced detachment due to the extreme strain of the myosin rod into account with the rate constant:  Table 1.
Here, we assumed that the forced detachment from the prestroke state is directed to the weak binding state, while the forced detachment from the poststroke or rigor state is directed to the detachment state. For detachment from the prestroke state and the rigor state, the rate constant in Equation (S28) was added to the original rate constant.