Washio Takumi, Hisada Toshiaki, Shintani Seine A, Higuchi Hideo
Graduate School of Frontier Sciences, The University of Tokyo, 178-4 Wakashiba, Kashiwa, Chiba 277-0871, Japan.
Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
Phys Rev E. 2017 Feb;95(2-1):022411. doi: 10.1103/PhysRevE.95.022411. Epub 2017 Feb 21.
Our study considers the mechanism of the spontaneous oscillations of molecular motors that are driven by the power stroke principle by applying linear stability analysis around the stationary solution. By representing the coupling equation of microscopic molecular motor dynamics and mesoscopic sarcomeric dynamics by a rank-1 updated matrix system, we derived the analytical representations of the eigenmodes of the Jacobian matrix that cause the oscillation. Based on these analytical representations, we successfully derived the essential conditions for the oscillation in terms of the rate constants of the power stroke and the reversal stroke transitions of the molecular motor. Unlike the two-state model, in which the dependence of the detachment rates on the motor coordinates or the applied forces on the motors plays a key role for the oscillation, our three-state power stroke model demonstrates that the dependence of the rate constants of the power and reversal strokes on the strains in the elastic elements in the motor molecules plays a key role, where these rate constants are rationally determined from the free energy available for the power stroke, the stiffness of the elastic element in the molecular motor, and the working stroke size. By applying the experimentally confirmed values to the free energy, the stiffness, and the working stroke size, our numerical model reproduces well the experimentally observed oscillatory behavior. Furthermore, our analysis shows that two eigenmodes with real positive eigenvalues characterize the oscillatory behavior, where the eigenmode with the larger eigenvalue indicates the transient of the system of the quick sarcomeric lengthening induced by the collective reversal strokes, and the smaller eigenvalue correlates with the speed of sarcomeric shortening, which is much slower than lengthening. Applying the perturbation analyses with primal physical parameters, we find that these two real eigenvalues occur on two branches derived from a merge point of a pair of complex-conjugate eigenvalues generated by Hopf bifurcation.
我们的研究通过围绕稳态解应用线性稳定性分析，来考虑由动力冲程原理驱动的分子马达自发振荡的机制。通过用一个秩 - 1 更新矩阵系统表示微观分子马达动力学和细肌丝节段动力学的耦合方程，我们推导了导致振荡的雅可比矩阵本征模的解析表达式。基于这些解析表达式，我们成功地根据分子马达动力冲程和反向冲程转变的速率常数，推导了振荡的必要条件。与两态模型不同，在两态模型中，脱离速率对马达坐标的依赖性或作用在马达上的力对振荡起关键作用，我们的三态动力冲程模型表明，动力冲程和反向冲程的速率常数对马达分子中弹性元件应变的依赖性起关键作用，其中这些速率常数是根据动力冲程可用的自由能、分子马达中弹性元件的刚度以及工作冲程大小合理确定的。通过将实验确认的值应用于自由能、刚度和工作冲程大小，我们的数值模型很好地再现了实验观察到的振荡行为。此外，我们的分析表明，具有正实特征值的两个本征模表征了振荡行为，其中具有较大特征值的本征模表示由集体反向冲程引起的快速细肌丝节段延长系统的瞬态，而较小的特征值与细肌丝节段缩短的速度相关，细肌丝节段缩短的速度比延长慢得多。通过对原始物理参数进行微扰分析，我们发现这两个实特征值出现在由霍普夫分岔产生的一对复共轭特征值的合并点导出的两个分支上。
