Locally and globally coupled oscillators in muscle

A thermodynamically consistent monte carlo cross-bridge model with a trapping mechanism reveals the role of stretch activation in heart pumping

Changes in intracellular calcium concentrations regulate heart beats. However, the decline in the left ventricular pressure during early diastole is much sharper than that of the Ca²⁺ transient, resulting in a rapid supply of blood to the left ventricle during the diastole. At the tissue level, cardiac muscles have a distinct characteristic, known as stretch activation, similar to the function of insect flight muscles. Stretch activation, which is a delayed increase in force following a rapid muscle length increase, has been thought to be related to autonomous control in these muscles. In this numerical simulation study, we introduced a molecular mechanism of stretch activation and investigated the role of this mechanism in the pumping function of the heart, using the previously developed coupling multiple-step active stiffness integration scheme for a Monte Carlo (MC) cross-bridge model and a bi-ventricular finite element model. In the MC cross-bridge model, we introduced a mechanism for trapping the myosin molecule in its post-power stroke state. We then determined the rate constants of transitions for trapping and escaping in a thermodynamically consistent manner. Based on our numerical analysis, we draw the following conclusions regarding the stretch activation mechanism: (i) the delayed force becomes larger than the original isometric force because the population of trapped myosin molecules and their average force increase after stretching; (ii) the delayed force has a duration of more than a few seconds owing to a fairly small rate constant of escape from the trapped state. For the role of stretch activation in heart pumping, we draw the following conclusions: (iii) for the regions in which the contraction force decreases earlier than the neighboring region in the end-systole phase, the trapped myosin molecules prevent further lengthening of the myocytes, which then prevents further shortening of neighboring myocytes; (iv) as a result, the contraction forces are sustained longer, resulting in a larger blood ejection, and their degeneration is synchronized.

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.

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.

Making waves: A proposed new role for myosin-binding protein C in regulating oscillatory contractions in vertebrate striated muscle

Myosin-binding protein C (MyBP-C) is a critical regulator of muscle performance that was first identified through its strong binding interactions with myosin, the force-generating protein of muscle. Almost simultaneously with its discovery, MyBP-C was soon found to bind to actin, the physiological catalyst for myosin’s activity. However, the two observations posed an apparent paradox, in part because interactions of MyBP-C with myosin were on the thick filament, whereas MyBP-C interactions with actin were on the thin filament. Despite the intervening decades since these initial discoveries, it is only recently that the dual binding modes of MyBP-C are becoming reconciled in models that place MyBP-C at a central position between actin and myosin, where MyBP-C alternately stabilizes a newly discovered super-relaxed state (SRX) of myosin on thick filaments in resting muscle and then prolongs the “on” state of actin on thin filaments in active muscle. Recognition of these dual, alternating functions of MyBP-C reveals how it is central to the regulation of both muscle contraction and relaxation. The purpose of this Viewpoint is to briefly summarize the roles of MyBP-C in binding to myosin and actin and then to highlight a possible new role for MyBP-C in inducing and damping oscillatory waves of contraction and relaxation. Because the contractile waves bear similarity to cycles of contraction and relaxation in insect flight muscles, which evolved for fast, energetically efficient contraction, the ability of MyBP-C to damp so-called spontaneous oscillatory contractions (SPOCs) has broad implications for previously unrecognized regulatory mechanisms in vertebrate striated muscle. While the molecular mechanisms by which MyBP-C can function as a wave maker or a wave breaker are just beginning to be explored, it is likely that MyBP-C dual interactions with both myosin and actin will continue to be important for understanding the new functions of this enigmatic protein.

Nanoscopic changes in the lattice structure of striated muscle sarcomeres involved in the mechanism of spontaneous oscillatory contraction (SPOC)

Muscles perform a wide range of motile functions in animals. Among various types are skeletal and cardiac muscles, which exhibit a steady auto-oscillation of force and length when they are activated at an intermediate level of contraction. This phenomenon, termed spontaneous oscillatory contraction or SPOC, occurs devoid of cell membranes and at fixed concentrations of chemical substances, and is thus the property of the contractile system per se. We have previously developed a theoretical model of SPOC and proposed that the oscillation emerges from a dynamic force balance along both the longitudinal and lateral axes of sarcomeres, the contractile units of the striated muscle. Here, we experimentally tested this hypothesis by developing an imaging-based analysis that facilitates detection of the structural changes of single sarcomeres at unprecedented spatial resolution. We found that the sarcomere width oscillates anti-phase with the sarcomere length in SPOC. We also found that the oscillatory dynamics can be altered by osmotic compression of the myofilament lattice structure of sarcomeres, but they are unchanged by a proteolytic digestion of titin/connectin—the spring-like protein that provides passive elasticity to sarcomeres. Our data thus reveal the three-dimensional mechanical dynamics of oscillating sarcomeres and suggest a structural requirement of steady auto-oscillation.

Self-excited oscillation of the director field in cholesteric liquid crystalline droplets under a temperature gradient

In this study, we demonstrate a self-excited oscillation induced in cholesteric liquid crystalline droplets under a temperature gradient. At equilibrium, a winding Maltese cross pattern with a point defect was observed via polarised microscopy in the droplets dispersed in an isotropic solvent. When the temperature gradient was applied, the pattern was deformed owing to the Marangoni convection induced by the gradient. Here, when both the droplet size and temperature gradient were sufficiently large, the periodic movement of the defect together with the pattern deformation was observed, which demonstrated the self-excited oscillation of the director field. To describe this phenomenon, we theoretically analysed the flow and director fields by using Onsager's variational principle. This principle enabled the simplified description of the phenomenon; consequently, the time evolution of the director field could be expressed by the phenomenological equations for the two parameters characterising the field. These equations represented the van der Pol equation, which well expressed the mechanism of the self-excited oscillation.

A chemical kinetic model for Ca2+ induced spontaneous oscillatory contraction of myocardium

The Ca²⁺ induced Spontaneous Oscillatory Contraction (Ca-SPOC) of cardiac myofibrils oscillate with a period similar to resting heartbeat of several animal species, and its auto-oscillatory properties set the basic rhythm of cardiac contraction. To explain the dynamics of Ca-SPOC, the present paper constructs a novel chemical kinetical model based upon the cooperative behavior between the two heads of myosin II dimer, also considering the reaction-diffusion effect of ATP inside myocardial fibers. The simulation results show that the concentration of ATP inside myocardial fibers oscillates over time under some special conditions, together with the proportions of myosin II dimers in different states periodically changing with time, which contributes to produce the sustained oscillations of contractive tension. These results indicate that the SPOC of muscles may be partly due to chemical oscillation involved in the actomyosin ATPase cycle, which has been ignored by the previous theoretical studies.

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.

Radial stiffness characteristics of the overlap regions of sarcomeres in isolated skeletal myofibrils in pre-force generating state

We have studied the stiffness of myofilament lattice in sarcomeres in the pre-force generating state, which was realized by a relaxing reagent, BDM (butane dione monoxime). First, the radial stiffness for the overlap regions of sarcomeres of isolated single myofibrils was estimated from the resulting decreases in diameter by osmotic pressure applied with the addition of Dextran. Then, the radial stiffness was also estimated from force-distance curve measurements with AFM technology. The radial stiffness for the overlap regions thus obtained was composed of a soft and a rigid component. The soft component visco-elastically changed in a characteristic fashion depending on the physiological conditions of myofibrils, suggesting that it comes from cross-bridge structures. BDM treatments significantly affected the soft radial component of contracting myofibrils depending on the approach velocity of cantilever: It was nearly equal to that in the contracting state at high approach velocity, whereas as low as that in the relaxing state at low approach velocity. However, comparable BDM treatments greatly suppressed the force production and the axial stiffness in contracting glycerinated muscle fibers and also the sliding velocity of actin filaments in the in vitro motility assay. Considering that BDM shifts the cross-bridge population from force generating to pre-force generating states in contracting muscle, the obtained results strongly suggest that cross-bridges in the pre-force generating state are visco-elastically attached to the thin filaments in such a binding manner that the axial stiffness is low but the radial stiffness significantly high similar to that in force generating state.

Embedding dual function into molecular motors through collective motion

Protein motors, such as kinesins and dyneins, bind to a microtubule and travel along it in a specific direction. Previously, it was thought that the directionality for a given motor was constant in the absence of an external force. However, the directionality of the kinesin-5 Cin8 was recently found to change as the number of motors that bind to the same microtubule is increased. Here, we introduce a simple mechanical model of a microtubule-sliding assay in which multiple motors interact with the filament. We show that, due to the collective phenomenon, the directionality of the motor changes (e.g., from minus- to plus- end directionality), depending on the number of motors. This is induced by a large diffusive component in the directional walk and by the subsequent frustrated motor configuration, in which multiple motors pull the filament in opposite directions, similar to a game of tug-of-war. A possible role of the dual-directional motors for the mitotic spindle formation is also discussed. Our framework provides a general mechanism to embed two conflicting tasks into a single molecular machine, which works context-dependently.

Model simulation of the SPOC wave in a bundle of striated myofibrils

SPOC (spontaneous oscillatory contraction) is a phenomenon observed in striated muscle under intermediate activation conditions. Recently, we constructed a theoretical model of SPOC for a sarcomere, a unit sarcomere model, which explains the behavior of SPOC at each sarcomere level. We also constructed a single myofibril model, which visco-elastically connects the unit model in series, and explains the behaviors of SPOC at the myofibril level. In the present study, to understand the SPOC properties in a bundle of myofibrils, we extended the single myofibril model to a two-dimensional (2D) model and a three-dimensional (3D) model, in which myofibrils were elastically connected side-by-side through cross-linkers between the Z-lines and M-lines. These 2D and 3D myofibril models could reproduce various patterns of SPOC waves experimentally observed in a 2D sheet and a 3D bundle of myofibrils only by choosing different values of elastic constants of the cross-linkers and the external spring. The results of these 2D and 3D myofibril models provide insight into the SPOC properties of the higher-ordered assembly of myofibrils.

Transient and periodic spatiotemporal structures in a reaction-diffusion-mechanics system

We study transient spatiotemporal structures induced by a weak space-time localized stimulus in an excitable contractile fiber within a two-component globally coupled reaction-diffusion model. The model which we develop allows us to analyze various regimes of excitation spreading and determine origin of the induced structures for various contraction types (defined by the fiber fixation) and global coupling strengths. One of the most notable effects we observed is the after-excitation effect. It leads to emergence of multiple excitation pulses excited by a single external stimulus and can result in long-lasting transient activity and appearance of new oscillatory attractor regimes, including the ones with multiple phase clusters.

Various oscillation patterns in phase models with locally attractive and globally repulsive couplings

We investigate a phase model that includes both locally attractive and globally repulsive coupling in one dimension. This model exhibits nontrivial spatiotemporal patterns that have not been observed in systems that contain only local or global coupling. Depending on the relative strengths of the local and global coupling and on the form of global coupling, the system can show a spatially uniform state (in-phase synchronization), a monotonically increasing state (traveling wave), and three types of oscillations of relative phase difference. One of the oscillations of relative phase difference has the characteristic of being locally unstable but globally attractive. That is, any small perturbation to the periodic orbit in phase space destroys its periodic motion, but after a long time the system returns to the original periodic orbit. This behavior is closely related to the emergence of saddle two-cluster states for global coupling only, which are connected to each other by attractive heteroclinic orbits. The mechanism of occurrence of this type of oscillation is discussed.

Spontaneous oscillatory contraction (SPOC) in cardiomyocytes

SPOC (spontaneous oscillatory contraction) is a characteristic state of the contractile system of striated (skeletal and cardiac) muscle that exists between the states of relaxation and contraction. For example, Ca-SPOCs occur at physiological Ca2+ levels (pCa ∼6.0), whereas ADP-SPOC occurs in the virtual absence of Ca2+ (pCa ≥ 8; relaxing conditions in the presence of MgATP), but in the presence of inorganic phosphate (Pi) and a high concentration of MgADP. The concentration of Mg-ADP necessary for SPOC is nearly equal to or greater than the MgATP concentration for cardiac muscle and is several times higher for skeletal muscle. Thus, the cellular conditions for SPOC are broader in cardiac muscle than in skeletal muscle. During these SPOCs, each sarcomere in a myofibril undergoes length oscillation that has a saw-tooth waveform consisting of a rapid lengthening and a slow shortening phase. The lengthening phase of one half of a sarcomere is transmitted to the adjacent half of the sarcomere successively, forming a propagating wave (termed a SPOC wave). The SPOC waves are synchronized across the cardiomyocytes resulting in a visible wave of successive contractions and relaxations termed the SPOC wave. Experimentally, the SPOC period (and therefore the velocity of SPOC wave) is observed in demembranated cardiomyocytes and can be prepared from a wide range of animal hearts. These periods correlate well with the resting heartbeats of a wide range of mammals (rat, rabbit, dog, pig and cow). Preliminary experiments showed that the SPOC properties of human cardiomyocytes are similar to the heartbeat of a large dog or a pig. This correlation suggests that SPOCs may play a fundamental role in the heart. Here, we briefly summarize a range of SPOC parameters obtained experimentally, and relate them to a theoretical model to explain those characteristics. Finally, we discuss the possible significance of these SPOC properties in each and every heartbeat.

High-frequency sarcomeric auto-oscillations induced by heating in living neonatal cardiomyocytes of the rat

In the present study, we investigated the effects of infra-red laser irradiation on sarcomere dynamics in living neonatal cardiomyocytes of the rat. A rapid increase in temperature to >~38 °C induced [Ca(2+)]i-independent high-frequency (~5-10 Hz) sarcomeric auto-oscillations (Hyperthermal Sarcomeric Oscillations; HSOs). In myocytes with the intact sarcoplasmic reticular functions, HSOs coexisted with [Ca(2+)]i-dependent spontaneous beating in the same sarcomeres, with markedly varying frequencies (~10 and ~1 Hz for the former and latter, respectively). HSOs likewise occurred following blockade of the sarcoplasmic reticular functions, with the amplitude becoming larger and the frequency lower in a time-dependent manner. The present findings suggest that in the mammalian heart, sarcomeres spontaneously oscillate at higher frequencies than the sinus rhythm at temperatures slightly above the physiologically relevant levels.

Spontaneous oscillations of elastic contractile materials with turnover

Single and collective cellular oscillations driven by the actomyosin cytoskeleton have been observed in numerous biological systems. Here, we propose that these oscillations can be accounted for by a generic oscillator model of a material turning over and contracting against an elastic element. As an example, we show that during dorsal closure of the Drosophila embryo, experimentally observed changes in actomyosin concentration and oscillatory cell shape changes can, indeed, be captured by the dynamic equations studied here. We also investigate the collective dynamics of an ensemble of such contractile elements and show that the relative contribution of viscous and friction losses yields different regimes of collective oscillations. Taking into account the diffusion of force-producing molecules between contractile elements, our theoretical framework predicts the appearance of traveling waves, resembling the propagation of actomyosin waves observed during morphogenesis.

Sarcomere length nanometry in rat neonatal cardiomyocytes expressed with α-actinin-AcGFP in Z discs

Nanometry is widely used in biological sciences to analyze the movement of molecules or molecular assemblies in cells and in vivo. In cardiac muscle, a change in sarcomere length (SL) by a mere ∼100 nm causes a substantial change in contractility, indicating the need for the simultaneous measurement of SL and intracellular Ca(2+) concentration ([Ca(2+)]i) in cardiomyocytes at high spatial and temporal resolution. To accurately analyze the motion of individual sarcomeres with nanometer precision during excitation-contraction coupling, we applied nanometry techniques to primary-cultured rat neonatal cardiomyocytes. First, we developed an experimental system for simultaneous nanoscale analysis of single sarcomere dynamics and [Ca(2+)]i changes via the expression of AcGFP in Z discs. We found that the averaging of the lengths of sarcomeres along the myocyte, a method generally used in today's myocardial research, caused marked underestimation of sarcomere lengthening speed because of the superpositioning of different timings for lengthening between sequentially connected sarcomeres. Then, we found that after treatment with ionomycin, neonatal myocytes exhibited spontaneous sarcomeric oscillations (cell-SPOCs) at partial activation with blockage of sarcoplasmic reticulum functions, and the waveform properties were indistinguishable from those obtained in electric field stimulation. The myosin activator omecamtiv mecarbil markedly enhanced Z-disc displacement during cell-SPOC. Finally, we interpreted the present experimental findings in the framework of our mathematical model of SPOCs. The present experimental system has a broad range of application possibilities for unveiling single sarcomere dynamics during excitation-contraction coupling in cardiomyocytes under various settings.