Analysis of spontaneous oscillations for a three-state power-stroke model

A Multiple Step Active Stiffness Integration Scheme to Couple a Stochastic Cross-Bridge Model and Continuum Mechanics for Uses in Both Basic Research and Clinical Applications of Heart Simulation

In a multiscale simulation of a beating heart, the very large difference in the time scales between rapid stochastic conformational changes of contractile proteins and deterministic macroscopic outcomes, such as the ventricular pressure and volume, have hampered the implementation of an efficient coupling algorithm for the two scales. Furthermore, the consideration of dynamic changes of muscle stiffness caused by the cross-bridge activity of motor proteins have not been well established in continuum mechanics. To overcome these issues, we propose a multiple time step scheme called the multiple step active stiffness integration scheme (MusAsi) for the coupling of Monte Carlo (MC) multiple steps and an implicit finite element (FE) time integration step. The method focuses on the active tension stiffness matrix, where the active tension derivatives concerning the current displacements in the FE model are correctly integrated into the total stiffness matrix to avoid instability. A sensitivity analysis of the number of samples used in the MC model and the combination of time step sizes confirmed the accuracy and robustness of MusAsi, and we concluded that the combination of a 1.25 ms FE time step and 0.005 ms MC multiple steps using a few hundred motor proteins in each finite element was appropriate in the tradeoff between accuracy and computational time. Furthermore, for a biventricular FE model consisting of 45,000 tetrahedral elements, one heartbeat could be computed within 1.5 h using 320 cores of a conventional parallel computer system. These results support the practicality of MusAsi for uses in both the basic research of the relationship between molecular mechanisms and cardiac outputs, and clinical applications of perioperative prediction.

A reverse stroke characterizes the force generation of cardiac myofilaments, leading to an understanding of heart function

Changes in the molecular properties of cardiac myosin strongly affect the interactions of myosin with actin that result in cardiac contraction and relaxation. However, it remains unclear how myosin molecules work together in cardiac myofilaments and which properties of the individual myosin molecules impact force production to drive cardiac contractility. Here, we measured the force production of cardiac myofilaments using optical tweezers. The measurements revealed that stepwise force generation was associated with a higher frequency of backward steps at lower loads and higher stall forces than those of fast skeletal myofilaments. To understand these unique collective behaviors of cardiac myosin, the dynamic responses of single cardiac and fast skeletal myosin molecules, interacting with actin filaments, were evaluated under load. The cardiac myosin molecules switched among three distinct conformational positions, ranging from pre- to post-power stroke positions, in 1 mM ADP and 0 to 10 mM phosphate solution. In contrast to cardiac myosin, fast skeletal myosin stayed primarily in the post-power stroke position, suggesting that cardiac myosin executes the reverse stroke more frequently than fast skeletal myosin. To elucidate how the reverse stroke affects the force production of myofilaments and possibly heart function, a simulation model was developed that combines the results from the single-molecule and myofilament experiments. The results of this model suggest that the reversal of the cardiac myosin power stroke may be key to characterizing the force output of cardiac myosin ensembles and possibly to facilitating heart contractions.

Effects of high-pressure treatment on the structure and function of myofibrils

The effects of high pressure (40-70 MPa) on the structure and function of myofibrils were investigated by high pressure microscopy. When this pressure was applied to myofibrils immersed in relaxing solution, the sarcomere length remained almost unchanged, and the A band became shorter and wider. The higher the applied pressure, the faster the change. However, shortening and widening of the A band were not observed when pressure was applied to myofibrils immersed in a solution obtained by omitting ATP from the relaxing solution. However, even under these conditions, structural loss, such as loss of the Z-line structure, occurred. In order to evaluate the consequences of this pressure-treated myofibril, the oscillatory movement of sarcomere (sarcomeric oscillation) was evoked and observed. It was possible to induce sarcomeric oscillation even in pressure-treated myofibrils whose structure was destroyed. The pressurization reduced the total power of the sarcomeric oscillation, but did not change the average frequency. The average frequency did not change even when a pressure of about 40 MPa was applied during sarcomeric oscillation. The average frequency returned to the original when the pressure was returned to the original value after applying stronger pressure to prevent the sarcomere oscillation from being observed. This result suggests that the decrease in the number of myosin molecules forming the crossbridge does not affect the average frequency of sarcomeric oscillation. This fact will help build a mechanical hypothesis for sarcomeric oscillation. The pressurization treatment is a unique method for controlling the structure of myofibrils as described above.

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⁺].

Active Force Generation in Cardiac Muscle Cells: Mathematical Modeling and Numerical Simulation of the Actin-Myosin Interaction

Cardiac in silico numerical simulations are based on mathematical models describing the physical processes involved in the heart function. In this review paper, we critically survey biophysically-detailed mathematical models describing the subcellular mechanisms behind the generation of active force, that is the process by which the chemical energy of ATP (adenosine triphosphate) is transformed into mechanical work, thus making the muscle tissue contract. While presenting these models, that feature different levels of biophysical detail, we analyze the trade-off between the accuracy in the description of the subcellular mechanisms and the number of parameters that need to be estimated from experiments. Then, we focus on a generalized version of the classic Huxley model, built on the basis of models available in the literature, that is able to reproduce the main experimental characterizations associated to the time scales typical of a heartbeat-such as the force-velocity relationship and the tissue stiffness in response to small steps-featuring only four independent parameters. Finally, we show how those parameters can be calibrated starting from macroscopic measurements available from experiments.

Effect of myofibril passive elastic properties on the mechanical communication between motor proteins on adjacent sarcomeres

Rapid sarcomere lengthening waves propagate along a single muscle myofibril during spontaneous oscillatory contraction (SPOC). In asynchronous insect flight muscles, SPOC is thought to be almost completely synchronized over the entire myofibril. This phenomenon does not require Ca²⁺ regulation of the dynamics of the motor proteins, and cannot be explained simply by the longitudinal mechanical equilibrium among sarcomeres in the myofibril. In the present study, we rationalize these phenomena by considering the lateral mechanical equilibrium, in which two tensions originating from the inverse relationship between sarcomere length and lattice spacing, along with the lattice alignment, play important roles in the mechanical communication between motor proteins on adjacent filaments via the Z-disc. The proposed model is capable of explaining various SPOC phenomena based on the stochastic power-stroke mechanism of motor proteins, which responds to temporal changes in longitudinal mechanical load.

Coupling Langevin Dynamics With Continuum Mechanics: Exposing the Role of Sarcomere Stretch Activation Mechanisms to Cardiac Function

High-performance computing approaches that combine molecular-scale and macroscale continuum mechanics have long been anticipated in various fields. Such approaches may enrich our understanding of the links between microscale molecular mechanisms and macroscopic properties in the continuum. However, there have been few successful examples to date owing to various difficulties associated with overcoming the large spatial (from 1 nm to 10 cm) and temporal (from 1 ns to 1 ms) gaps between the two scales. In this paper, we propose an efficient parallel scheme to couple a microscopic model using Langevin dynamics for a protein motor with a finite element continuum model of a beating heart. The proposed scheme allows us to use a macroscale time step that is an order of magnitude longer than the microscale time step of the Langevin model, without loss of stability or accuracy. This reduces the overhead required by the imbalanced loads of the microscale computations and the communication required when switching between scales. An example of the Langevin dynamics model that demonstrates the usefulness of the coupling approach is the molecular mechanism of the actomyosin system, in which the stretch-activation phenomenon can be successfully reproduced. This microscopic Langevin model is coupled with a macroscopic finite element ventricle model. In the numerical simulations, the Langevin dynamics model reveals that a single sarcomere can undergo spontaneous oscillation (15 Hz) accompanied by quick lengthening due to cooperative movements of the myosin molecules pulling on the common Z-line. Also, the coupled simulations using the ventricle model show that the stretch-activation mechanism contributes to the synchronization of the quick lengthening of the sarcomeres at the end of the systolic phase. By comparing the simulation results given by the molecular model with and without the stretch-activation mechanism, we see that this synchronization contributes to maintaining the systolic blood pressure by providing sufficient blood volume without slowing the diastolic process.