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

Abstract

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.

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

“…S1b,d,f) indicates that this oscillation originates from the instability, given by coupling the cross-bridge dynamics with the external boundary condition that allows sarcomeric length change in response to the active tension, as analysed mathematically in our previous work 8 . We suggest that the avalanche of reversal power-strokes and its propagation along the myofibril also occurs in the natural physiological condition of the cardiac muscle, and is important for the quick relaxation during the diastolic phase in the cardiac cycle, as we previously simulated 4 . Note that the SPOC in the present study should be distinguished from the forced oscillation subject to small (0.23%) sinusoidal length change 20 .…”

“…For the factor R 0 , the different values 1 and 30 are applied to the standard SPOC and insect flight muscle models, respectively. If different parameters were applied to the standard SPOC and insect flight muscle models, both values are shown with an (S) or (I), respectively.ParameterValueUnitReferenceParameterValueUnitReference ATP hydrolysis energy Sarcomere Geometry E ATP 76.5pN⋅nm 4 SL 0 1.9μm 32 k B T 4.05pN⋅nm T = 293°C LM 1.65μm 31 Stroke sizes and Free energies LB 0.16μm 30 s 1 , s 2 5.0nm 35 LA 1.0μm 30 E wb E ATP pN/nm 19 SA 0 693nm 2 11 E 0 1.05 E wb pN/nm 19 Force regulation through T/T-unit E 1 0.82 E ATP pN/nm 19 50 R 0 1/sadjusted E 2 0pN/nm 19 5 R 0 1/sadjusted Rod strain energy W rod K on 90 R 0 1/sadjusted k xb 2.0pN/nm 19 K off 60 R 0 1/sadjusted b xb 0.05unitless 19 K basic 201/s 19 ξ 1 4.35nm 19 Q 2unitless 19…”

“…Here, δ OV = 1 if the myosin head was located at the single overlapping region with the thin filament, otherwise δ OV = 0. The factors γ ng and γ − ng ( γ = 60) represent the nearest-neighbour cooperativity of the myosin heads, as reported by Rice 34 , which plays an important role for the force-pCa relationship 4 . We assumed that one thin filament in the three-dimensional arrangement corresponds to two thin filaments in our half-sarcomere model.…”

“…However, muscles can relax more rapidly than can be explained by the decay rate of free Ca 2+ ions. For example, in a healthy beating heart, the left ventricular pressure falls much more rapidly than the free Ca 2+ ion concentration, which is important for the quick filling of blood, especially with fast heart rates 4 . In the flapping asynchronous muscles of insect wings, the sarcomeres must relax in tandem without Ca 2+ -ion regulation.…”

Stochastic modeling of chemical–mechanical coupling in striated muscles

“…On the other hand, the case for a mechanical vision of the myosin head with two dofs -rather than two (or more) different chemical states -mainly lies in the ability to account for two significantly different values of stiffness parameters experimentally measured for different time constants, namely, before and after power stroke can occur. We point out that a similar strategy was recently proposed in (Marcucci et al 2016), see also (Washio et al 2018). Beyond a few technical differences in the microscopic modeling assumptions -three local minima for the power stroke dof in (Marcucci et al 2016) instead of two in our case, and chemical transition within the unbound state instead of mechanical -our main fundamental contribution lies in carrying over the agentbased Langevin dynamics to Fokker-Planck type equations describing the evolution of probability densities.…”