A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

Memory-Induced Bifurcation and Oscillations in the Chemical Brusselator Model

Previous studies assumed that the reaction processes in the chemical Brusselator model are memoryless or Markovian. However, as long as a reactant interacts with its environment, the reaction kinetics cannot be described as a memoryless process. This raises a question: how do we predict the behavior of the chemical Brusselator system with molecular memory characterized by nonexponential waiting-time distributions? Here, a novel technique is developed to address this question. This technique converts a non-Markovian question to a Markovian one by introducing effective transition rates that explicitly decode the memory effect. Based on this conversion, it is analytically shown that molecular memory can induce bifurcations and oscillations. Moreover, a set of sufficient conditions are derived, which can guarantee that the system of the rate equations for the Markovian reaction system generates oscillations via memory index-induced bifurcation. In turn, these conditions can guarantee that the original non-Markovian reaction system generates stochastic oscillations. Numerical simulation verifies the theoretical prediction. The overall analysis indicates that molecular memory is not a negligible factor affecting a chemical system’s behavior.

Stochastic resonance as the averaged response to random broadband excitation and its possible applications

Within the framework of vibrational mechanics, a stochastic analog of the Stephenson–Kapitza pendulum with random two-dimensional oscillations of the suspension point was considered and the dynamic properties of its averaged motion were studied. It is shown that, unlike the ordinary Stephenson–Kapitsa pendulum with deterministic vertical oscillations of the suspension point, both an increase and a decrease in the effective natural frequency are possible under the influence of high-frequency stochastic oscillations. A formula is derived for the amplitude of low-frequency oscillations as a function of the intensity of high-frequency stochastic oscillations and the possibility of a stochastic resonance in this system is shown. The dependence of the stochastic resonance on the mass and the damping coefficient is analyzed. It is shown that the points of the stochastic resonance lie in the plane of parameters “intensity of stochastic excitation” and “amplitude of low-frequency oscillation” on a universal curve that is independent of the mass of the pendulum. Peculiar self-oscillations in a system for which stochastic oscillations are produced by a technological load and, therefore, depend monotonically on the amplitude of low-frequency oscillations are discussed. A schematic diagram of these phenomena is proposed. The motion of the machine is described by the same equations as the stochastic analog of the Stephenson–Kapitza pendulum with random two-dimensional oscillations of the suspension point. A strategy of control for such a vibro-machine is proposed with the aim of maintaining it at resonance and providing an energetically efficient mode of operation.

Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle

In the mitotic spindle microtubules attach to kinetochores via catch bonds during metaphase. We investigate the cooperative stochastic microtubule dynamics in spindle models consisting of ensembles of parallel microtubules, which attach to a kinetochore via elastic linkers. We include the dynamic instability of microtubules and forces on microtubules and kinetochores from elastic linkers. We start with a one-sided model, where an external force acts on the kinetochore. A mean-field approach based on Fokker-Planck equations enables us to analytically solve the one-sided spindle model, which establishes a bistable force-velocity relation of the microtubule ensemble. All results are in agreement with stochastic simulations. We derive constraints on linker stiffness and microtubule number for bistability. The bistable force-velocity relation of the one-sided spindle model gives rise to oscillations in the two-sided model, which can explain stochastic chromosome oscillations in metaphase (directional instability). We also derive constraints on linker stiffness and microtubule number for metaphase chromosome oscillations. We can include poleward microtubule flux and polar ejection forces into the model and provide an explanation for the experimentally observed suppression of chromosome oscillations in cells with high poleward flux velocities. Chromosome oscillations persist in the presence of polar ejection forces, however, with a reduced amplitude and a phase shift between sister kinetochores. Moreover, polar ejection forces are necessary to align the chromosomes at the spindle equator and stabilize an alternating oscillation pattern of the two kinetochores. Finally, we modify the model such that microtubules can only exert tensile forces on the kinetochore resulting in a tug-of-war between the two microtubule ensembles. Then, induced microtubule catastrophes after reaching the kinetochore are necessary to stimulate oscillations.Author summaryThe mitotic spindle is responsible for proper separation of chromosomes during cell division. Microtubules are dynamic protein filaments that actively pull chromosomes apart during separation. Two ensembles of microtubules grow from the two spindle poles towards the chromosomes, attach on opposite sides, and pull chromosomes by depolymerization forces. In order to exert pulling forces, microtubules attach to chromosomes at protein complexes called kinetochores. Before the final separation, stochastic oscillations of chromosomes are observed, where the two opposing ensembles of microtubules move chromosome pairs back an forth in a tug-of-war.Using a a combined computational and theoretical approach we quantitatively analyze the emerging chromosome dynamics starting from the stochastic growth dynamics of individual microtubules. Each of the opposing microtubule ensembles is a bistable system, and coupling two such systems in a tug-of-war results in stochastic oscillations. We can quantify constraints on the microtubule-kinetochore linker stiffness and the microtubule number both for bistability of the one-sided system and for oscillations in the full two-sided spindle system, which can rationalize several experimental observations. Our model can provide additional information on the microtubule-kinetochore linkers whose molecular nature is not completely known up to now.