From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.     

Cytoskeletal turnover and Myosin contractility drive cell autonomous oscillations in a model of Drosophila Dorsal Closure

Oscillatory behaviour in force-generating systems is a pervasive phenomenon in cell biology. In this work, we investigate how oscillations in the actomyosin cytoskeleton drive cell shape changes during the process of Dorsal Closure (DC), a morphogenetic event in Drosophila embryo development whereby epidermal continuity is generated through the pulsatile apical area reduction of cells constituting the amnioserosa (AS) tissue. We present a theoretical model of AS cell dynamics by which the oscillatory behaviour arises due to a coupling between active myosin-driven forces, actin turnover and cell deformation. Oscillations in our model are cell-autonomous and are modulated by neighbour coupling, and our model accurately reproduces the oscillatory dynamics of AS cells and their amplitude and frequency evolution. A key prediction arising from our model is that the rate of actin turnover and Myosin contractile force must increase during DC in order to reproduce the decrease in amplitude and period of cell area oscillations observed in vivo. This prediction opens up new ways to think about the molecular underpinnings of AS cell oscillations and their link to net tissue contraction and suggests the form of future experimental measurements.     

Molecular-scale density oscillations in water adjacent to a mica surface

High-resolution specular x-ray reflectivity of the mica(001)-water interface under ambient conditions reveals oscillations in water oxygen density in the surface-normal direction, giving evidence of interfacial water ordering. The spacings between neighboring water layers in the near-surface, strongly oscillatory region are 2.5(2)-2.7(2) A, approximately the size of the water molecule. The density oscillations extend to about 10 A above the surface and do not strictly maintain a solvent-size periodicity as that in interfacial liquid metal and hard-sphere molecular liquids. We interpret this oscillatory density profile of the interfacial water as due to the "hard-wall" effect of the molecularly smooth mica surface.     

Quantum oscillations in confined and degenerate Fermi gases. II. The phase diagram and applications of half-vicinity model

For part I see DOI: 10.1016/j.physleta.2018.02.006. Size and density dependent quantum oscillations appear in Fermi gases under strong confinement and degeneracy conditions. We provide a universal recipe that explicitly separates oscillatory regime from non-oscillatory (stationary) one. A phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space is proposed. Analytical expressions of phase transition interfaces are derived. The critical point, which separates entirely stationary and oscillatory regions, is determined and its dependencies on aspect ratios are examined for anisometric domains. Accuracy of the half-vicinity model and the phase diagram are verified through the quantum oscillations in electronic heat capacity and its ratio to entropy.     

Discrete coherent amplification of oscillations by nonresonant forcing

A new mechanism of generation of oscillations in a linear forced oscillatory system is found. Natural oscillations may be generated at a "sharp" pulse (rapid variation) of the natural frequency. In this process oscillations are generated by nonresonant forcing, e.g., by the action of a constant, nonperiodic or periodic force (with driving frequency much less than the natural one). Repetitive pulses of the natural frequency result in emergence of oscillations that interfere and may give a powerful resultant output. These phenomena relate to a basis of the theory of open linear oscillatory systems.     

Population dynamics on random networks: simulations and analytical models

We study the phase diagram of the standard pair approximation equations for two different models in population dynamics, the susceptible-infective-recovered-susceptible model of infection spread and a predator-prey interaction model, on a network of homogeneous degree k. These models have similar phase diagrams and represent two classes of systems for which noisy oscillations, still largely unexplained, are observed in nature. We show that for a certain range of the parameter k both models exhibit an oscillatory phase in a region of parameter space that corresponds to weak driving. This oscillatory phase, however, disappears when k is large. For k = 3, 4, we compare the phase diagram of the standard pair approximation equations of both models with the results of simulations on regular random graphs of the same degree. We show that for parameter values in the oscillatory phase, and even for large system sizes, the simulations either die out or exhibit damped oscillations, depending on the initial conditions. We discuss this failure of the standard pair approximation model to capture even the qualitative behavior of the simulations on large regular random graphs and the relevance of the oscillatory phase in the pair approximation diagrams to explain the cycling behavior found in real populations.     

Kinetic oscillations in the rate of CO oxidation on Pd(110)

The oxidation of carbon monoxide on a Pd(110) single crystal plane has been studied using work function changes (Δφ) and mass spectrometric measurements. The rate of reaction showed oscillatory behaviour for oxygen pressures greater than 10⁻³ Torr. The existence region for oscillations was determined for pressures ranging from 10⁻³ to 1.0 Torr and depended on the pressures of oxygen and carbon monoxide and the sample temperature (P_co, PO₂ T). Transitions from regular oscillation to chaos via period doubling have been observed in certain areas of the existence region. A comparison between Pd(110) and platinum single crystal surfaces that exhibit oscillations showed that similar but not identical oscillatory behaviour and existence regions exist in each case. Our results indicate that oscillations can occur on other metal single crystal surface that are less likely than platinum to reconstruct under reaction conditions. The extension of oscillations from UHV conditions to the 1.0 Torr pressure region indicates that the mechanism responsible for isothermal oscillations is basically independent of the reactant pressure up to several Torr.     

Oscillatory spin polarization and magneto-optical Kerr effect in Fe₃O₄ thin films on GaAs(001)

The spin dependent properties of epitaxial Fe?O? thin films on GaAs(001) are studied by the ferromagnetic proximity polarization (FPP) effect and magneto-optical Kerr effect (MOKE). Both FPP and MOKE show oscillations with respect to Fe?O? film thickness, and the oscillations are large enough to induce repeated sign reversals. We attribute the oscillatory behavior to spin-polarized quantum well states forming in the Fe?O? film. Quantum confinement of the t(2g) states near the Fermi level provides an explanation for the similar thickness dependences of the FPP and MOKE oscillations.     

Designer gene networks: Towards fundamental cellular control

The engineered control of cellular function through the design of synthetic genetic networks is becoming plausible. Here we show how a naturally occurring network can be used as a parts list for artificial network design, and how model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics. We first review the relevant work on synthetic gene networks, highlighting the important experimental findings with regard to genetic switches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistability in the steady-state protein concentration arises naturally as a consequence of autoregulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the effect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formulation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how the addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency dependence on the tunable parameter values, and discuss design plausibility. We emphasize how the model equations can be used to develop design criteria for robust oscillations, and illustrate this point with parameter plots illuminating the oscillatory regions for given parameter values. We then turn to the utilization of an intrinsic cellular process as a means of controlling the oscillations. We consider a network design which exhibits self-sustained oscillations, and discuss the driving of the oscillator in the context of synchronization. Then, as a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscillations and amplification of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with those of a network constructed using negative regulation. Our results demonstrate the utility of model analysis in the construction of synthetic gene regulatory networks. (c) 2001 American Institute of Physics.     

Noise-induced synchronous stochastic oscillations in small scale cultured heart-cell networks

This paper reports that the synchronous integer multiple oscillations of heart-cell networks or clusters are observed in the biology experiment.The behaviour of the integer multiple rhythm is a transition between super-and subthreshold oscillations,the stochastic mechanism of the transition is identified.The similar synchronized oscillations are theoretically reproduced in the stochastic network composed of heterogeneous cells whose behaviours are chosen as excitable or oscillatory states near a Hopf bifurcation point.The parameter regions of coupling strength and noise density that the complex oscillatory rhythms can be simulated are identified.The results show that the rhythm results from a simple stochastic alternating process between super-and sub-threshold oscillations.Studies on single heart cells forming these clusters reveal excitable or oscillatory state nearby a Hopf bifurcation point underpinning the stochastic alternation.In discussion,the results are related to some abnormal heartbeat rhythms such as the sinus arrest.     

Thermally assisted current-driven bistable precessional regimes in asymmetric spin valves

Spin-transfer torque in asymmetric spin valves can destabilize both parallel and antiparallel configurations and can lead to precessional modes also in the absence of an external magnetic field. We find a bistable precessional regime in such systems and show that thermal fluctuations can excite transitions (telegraph noise) between the corresponding oscillatory regimes that are well separated by irreversible paths at low temperatures. Because of the thermally induced transitions, the frequency of the resulting current-driven oscillations is different from that obtained at very low temperatures. We also show that the power spectrum in the bistable region is dominated by the out-of-plane oscillatory mode.     

Parametric excitation of oscillatory states and symmetry in phase space

We study the excitation of nonlinear dissipative oscillator under influence of a monochromatic force at the level of a few quanta. With this purpose we consider an optical parametric oscillator combined with phase-modulation in which the oscillatory mode is excited through down-conversion process under a monochromatic laser field. The temporal Rabi oscillations of Fock states as well as the properties of oscillatory mode in phase space are studied with use of the Wigner functions.     

A bifurcation analysis of two coupled calcium oscillators

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. (c) 2001 American Institute of Physics.     

Electrothermal Oscillatory Instability in Spherical Ferroelectric Resonators. Three-Mode Case

We consider the equations of interaction between electromagnetic oscillations and the temperature in a nonlinear dielectric resonator and study the dynamics of the oscillatory instability in the system. The threshold conditions (power and self-modulation frequency) of electrothermal excitation are calculated for microwave potassium-tantalate resonators for the case of three-mode interaction. The conditions for observing electrothermal excitation in the three-mode case are found to be quite favorable. In this case, the threshold power of excitation of temperature oscillations is smaller than that in the two-mode case and can amount to a few microwatts.     

Oscillatory behavior of magneto-optical interband emissions in asymmetric quantum well structures

We report on the oscillatory behavior of the photoluminescence intensity from asymmetric AlGaAs/InGaAs/GaAs quantum well structures in the presence of a perpendicular magnetic field. Two distinct photoluminescence peaks originating from transitions from the ground (e₁) and the first excited (e₂) electronic states to the heavy hole state (hh₁) are observed. The opposite phase of the oscillations shows clearly the competitive process between the transitions from the ground and first excited states. Electron transfer mechanisms cannot explain the origin of these oscillations. The optical oscillations emerge from changes in the effective electron–hole interaction.     

Oscillatory critical amplitudes in hierarchical models

We study the oscillatory critical amplitudes of theq-states Potts model on a diamond hierarchical lattice. We consider an example of the generic case (finite critical index), as well as the degenerate case (essential singularity). In both cases, we compare the magnitude of the oscillations with geometrical characteristics of the Julia set of zeroes of the partition function.     

Probable observation of unusual local vibrational modes in the oscillatory photoconductivity of semi-insulating GaAs

We have observed oscillatory photoconductivity with a period of 20.9 meV in semi-insulating GaAs at T = 18–23°K. These oscillations are associated with the 0.75 eV impurity level in GaAs which has often been attributed to O. It is suggested that the oscillations are due to a cascade process involving multiple emission of a local vibrational mode, of energy 20.9 meV, associated with the 0.75 eV defect.     

Third order corrections and finite conduction band effects on the indirect exchange interaction in the Bloembergen-Rowland approximation

We have investigated third order corrections and finite conduction band effects on the indirect exchange interaction assuming a large energy gap compared to the valence band width. We regain an oscillatory expression in which as opposed to the Bloembergen-Rowland formula the energy gap as well as the conduction band width modifies both the magnitude and phase of the oscillations.