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.     

哺乳动物生理振荡中的最佳内噪声效应

基于哺乳动物生理振子模型，构造了相应的介观随机模型，研究了该系统中内噪声对基因振荡的影响．结果发现通过内噪声随机共振的机制，随机的基因振荡可以在最佳内噪声水平处达到最佳状态．同时，还发现存在一个中间的系统尺度使得随机模型表现出比确定性模型更宽的有效振荡区域，这说明了内噪声增强了体系的鲁帮性．讨论了这些效应可能的生理意义．     

Circadian rhythms and molecular noise

Circadian rhythms, characterized by a period of about 24 h, are the most widespread biological rhythms generated autonomously at the molecular level. The core molecular mechanism responsible for circadian oscillations relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models account for the occurrence of autonomous circadian oscillations, for their entrainment by light-dark cycles, and for their phase shifting by light pulses. Stochastic versions of these models take into consideration the molecular fluctuations that arise when the number of molecules involved in the regulatory mechanism is low. Numerical simulations of the stochastic models show that robust circadian oscillations can already occur with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively. Various factors affect the robustness of circadian oscillations with respect to molecular noise. Besides an increase in the number of molecules, entrainment by light-dark cycles, and cooperativity in repression enhance robustness, whereas the proximity of a bifurcation point leads to less robust oscillations. Another parameter that appears to be crucial for the coherence of circadian rhythms is the binding/unbinding rate of the inhibitory protein to the promoter of the clock gene. Intercellular coupling further increases the robustness of circadian oscillations.     

LED照明系统的光谱组成对糖脂代谢的影响

哺乳动物昼夜节律系统对不同光谱组成敏感性不同。时差光照模式可导致昼夜节律紊乱进而增加患代谢性疾病的风险。然而,光谱组成是否影响时差光照模式的代谢效应尚不明确。采用昼夜节律系统敏感性显著不同的窄带LED光照（蓝光和红光波段）和宽带LED白光,分析光谱组成对时差光照模式下小鼠糖脂代谢功能的影响,并与正常光照模式进行比较。光照强度均采用120 μW·cm-2。结果显示白光时差组小鼠体重增加最多。红光时差组小鼠出现严重脂质代谢紊乱,并伴有肝功能受损。白光下时差光照会显著降低葡萄糖耐量和胰岛素敏感性,而红光和蓝光能阻碍时差光照引起空腹血糖升高。研究表明调整光谱组成可能改善时差光照模式对糖脂代谢的不良影响。     

The role of auditory feedback in sustaining vocal vibrato

Vocal vibrato and tremor are characterized by oscillations in voice fundamental frequency (F0). These oscillations may be sustained by a control loop within the auditory system. One component of the control loop is the pitch-shift reflex (PSR). The PSR is a closed loop negative feedback reflex that is triggered in response to discrepancies between intended and perceived pitch with a latency of approximately 100 ms. Consecutive compensatory reflexive responses lead to oscillations in pitch every approximately 200 ms, resulting in approximately 5-Hz modulation of F0. Pitch-shift reflexes were elicited experimentally in six subjects while they sustained /u/ vowels at a comfortable pitch and loudness. Auditory feedback was sinusoidally modulated at discrete integer frequencies (1 to 10 Hz) with +/- 25 cents amplitude. Modulated auditory feedback induced oscillations in voice F0 output of all subjects at rates consistent with vocal vibrato and tremor. Transfer functions revealed peak gains at 4 to 7 Hz in all subjects, with an average peak gain at 5 Hz. These gains occurred in the modulation frequency region where the voice output and auditory feedback signals were in phase. A control loop in the auditory system may sustain vocal vibrato and tremorlike oscillations in voice F0.     

拟南芥AtGrp7RRM结构域的纯化及其结构与结合的初步分析(英文)

富含甘氨酸的RNA结合蛋白AtGrp7是拟南芥（Arabidopsis thaliana）调节生物钟负反馈回路的组分.在使用常规方法纯化AtGrp7 RRM结构域的初始试验中，观察到烟草蚀纹病毒（TEV）酶切后AtGrp7 RNA识别基序（RRM）结构域的紫外吸收峰为蛋白和杂质的混合信号峰.为解决常规纯化中的杂质问题，对AtGrp7_1-90应用了变性-复性两步纯化方法.AtGrp7 RRM结构域的¹H-¹⁵N HSQC指纹谱和CS-Rosetta模型结构表明快速稀释重折叠后其结构完全恢复.等温滴定量热法（ITC）和核磁共振（NMR）滴定实验进一步证实，重折叠后AtGrp7_1-90 RRM结构域具有正确结合RNA/DNA的功能.     

Sustainment and Controlment of Noise-Induced Circadian Oscillations in <Emphasis Type="Italic">Neurospora</Emphasis>: Noise and External Signal Effects

The effect of light noise on a Neurospora circadian clock system in the steady states is investigated. It is found that the circadian oscillations could be induced by light noise, leading to various resonance phenomena including internal signal stochastic resonance (ISSR) and ISSR without tuning in the system. The strength of ISSR could be significantly reinforced with the decrease of the distance of the control parameter to the Hopf bifurcation point of the system. The fundamental frequency of noise-induced circadian oscillations almost does not change with the increment of light noise intensity, which implies that the Neurospora system could sustain intrinsic circadian rhythms. In addition, the ISSR and ISSR without tuning could be both amplified, suppressed or destroyed by tuning the frequency or amplitude of external signal.     

Mutual synchronization in a system of two bistable oscillators executing regular and chaotic oscillations

A numerical analysis of a new model describing two coupled modified Chua??s oscillators is conducted. Equations of a partial oscillator differ from classical equations in that the former contain additional delayed feedback in another writing of dimensionless time. Changeover from regular oscillations in the absence of additional feedback to additional-feedback-induced (switchable) chaotic oscillations is studied. It is shown that, when normal regular oscillations, as well as additional-feedback-induced chaotic oscillations, are synchronized, difference oscillations are left. They are absent only when the control parameters of partial oscillators are identical. The application of a harmonic signal allows one to control the oscillations of a chaotic system of coupled modified bistable oscillators.     

Analysis of a random self-oscillating system with a varying structure

The Rossler equations are transformed to the oscillatory form and are used for constructing a new mathematical model of a controllable system with a varying structure based on the randomizing feedback algorithm. Numerical methods are employed for analyzing randomization of oscillations with the help of self-switching in the case when only regular movements can take place without it. Apart from analysis of a single system with self-switching, randomization of oscillations in a complex system with a varying structure is considered on the basis of two models. It is found that partial synchronization of randomized oscillations of subsystems, which is associated with the existence of a common controlling feedback for these subsystems, takes place.     

Theoretical modeling of the dynamics of a semiconductor laser subject to double-reflector optical feedback

We theoretically model the dynamics of semiconductor lasers subject to the double-reflector feedback. The proposed model is a new modification of the time-delay rate equations of semiconductor lasers under the optical feedback to account for this type of the double-reflector feedback. We examine the influence of adding the second reflector to dynamical states induced by the single-reflector feedback: periodic oscillations, period doubling, and chaos. Regimes of both short and long external cavities are considered. The present analyses are done using the bifurcation diagram, temporal trajectory, phase portrait, and fast Fourier transform of the laser intensity. We show that adding the second reflector attracts the periodic and perioddoubling oscillations, and chaos induced by the first reflector to a route-to-continuous-wave operation. During this operation, the periodic-oscillation frequency increases with strengthening the optical feedback. We show that the chaos induced by the double-reflector feedback is more irregular than that induced by the single-reflector feedback. The power spectrum of this chaos state does not reflect information on the geometry of the optical system, which then has potential for use in chaotic (secure) optical data encryption.     

Self-sustained oscillations in systems with delay under irregularly varying excitation conditions

A mathematical model of a system consisting of two coupled chaotic delay subsystems is presented. Instead of constant initial conditions in the form of a single impetus to excite the subsystems, continuous irregular oscillations are used that simulate intrinsic noise and continue acting on self-sustained oscillations after their excitation. An equation of an autonomous subsystem with regard to feedback variation is derived. It is shown that, when an autonomous subsystem is excited by irregular oscillations, chaotic motions become stochastic. In this case, the intensity of oscillations simulating intrinsic noise increases, suppressing self-sustained oscillations and providing the regenerative amplification of irregular oscillations. Interaction of coupled oscillations for identical and nonidentical subsystems is considered for the case of different noiselike initial conditions. It is found that interacting oscillations are not completely identical even if the parameters of the subsystems are the same.     

A reduced-order deterministic model describing an intermittency route to combustion instability

Recently, there has been a growing interest in understanding and characterising intermittent burst oscillations that presage the onset of combustion instability. We construct a deterministic model to capture this intermittency route to instability in a bluff-body stabilised combustor by coupling the equations governing vortex shedding and the acoustic wave propagation in a confinement. A feedback mechanism is developed wherein the sound generated due to unsteady combustion affects the vortex shedding. This feedback leads to a variation in the time of impingement of the vortices with the bluff body causing the system to exhibit chaos, intermittency, and limit cycle oscillations. Experimental validation of the model is provided using various precursor measures that quantify the observed intermittent states.     

Statistical analysis of calcium oscillations

Calcium is an important and versatile second messenger in eukaryotic cells. External signals are often transmitted to intracellular targets by oscillations of the cytosolic calcium concentration. Recently, we have experimentally shown that these oscillations consist of sequences of random spikes and depend on spatial characteristics of cells. Here, we apply further statistical analysis to experimental data in order to describe the spike generating process and compare spontaneous and agonist-induced oscillations. It turns out that these oscillations exhibit a non-resonant behavior, and, consequently, regular spiking originates by array-enhanced coherent resonance. Moreover, we present a heuristic model based on the probability density of intervals between spikes that takes a positive feedback of a spike to its successor into account. The extended model is analyzed with respect to statistical properties and information content.     

Collective behavior of coupled nonuniform stochastic oscillators

Theoretical studies of synchronization are usually based on models of coupled phase oscillators which, when isolated, have constant angular frequency. Stochastic discrete versions of these uniform oscillators have also appeared in the literature, with equal transition rates among the states. Here we start from the model recently introduced by Wood et al. [K. Wood, C. Van den Broeck, R. Kawai, K. Lindenberg, Universality of synchrony: critical behavior in a discrete model of stochastic phase-coupled oscillators, Phys. Rev. Lett. 96 (2006) 145701], which has a collectively synchronized phase, and parametrically modify the phase-coupled oscillators to render them (stochastically) nonuniform. We show that, depending on the nonuniformity parameter 0≤α≤1, a mean field analysis predicts the occurrence of several phase transitions. In particular, the phase with collective oscillations is stable for the complete graph only for α≤α^′<1. At α=1 the oscillators become excitable elements and the system has an absorbing state. In the excitable regime, no collective oscillations were found in the model.     

Contagion effects in a chartist-fundamentalist model with time delays

In this paper two models of speculative markets are developed to study the effects of feedback mechanisms in financial markets. In the first model, a crash market model couples a linear chartist-fundamentalist model with time delays with a log-periodic market index I(t) through direct coupling. Numerical solutions to the model show that asset prices exhibit significant persistence as a result of the coupling to the log-periodic market index. An extension to include endogenous wealth dynamics shows that the chartists benefit from the persistent dynamics induced by the coupling. The second model is a two-asset model represented by a 2-dimensional delay-differential equation. Asset one price exhibits limit cycle dynamics while in the second market asset prices follow stable damped oscillations. The markets are coupled through a diffusive coupling term. Solutions to the coupled model show that the dynamics of asset two changes fundamentally with the price now exhibiting a limit cycle. The stable converging dynamics is replaced with limit cycle oscillations around the fundamental.