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.     

Entrainment Versus Chaos in a Model for a Circadian Oscillator Driven by Light-Dark Cycles

Circadian rhythms occur in nearly all living organisms with a period close to 24 h. These rhythms constitute an important class of biological oscillators which present the characteristic of being naturally subjected to forcing by light-dark (LD) cycles. In order to investigate the conditions in which such a forcing might lead to chaos, we consider a model for a circadian limit cycle oscillator and assess its dynamic behavior when a light-sensitive parameter is periodically forced by LD cycles. We determine as a function of the forcing period and of the amplitude of the light-induced changes in the light-sensitive parameter the occurrence of various modes of dynamic behavior such as quasi-periodicity, entrainment, period-doubling and chaos. The type of oscillatory behavior markedly depends on the forcing waveform; thus the domain of entrainment grows at the expense of the domain of chaos as the forcing function progressively goes from a square wave to a sine wave. Also studied is the dependence of the phase of periodic or aperiodic oscillations on the amplitude of the light-induced changes in the control parameter. The results are discussed with respect to the main physiological role of circadian rhythms which is to allow organisms to adapt to their periodically varying environment by entrainment to the natural LD cycle.     

How coupling determines the entrainment of circadian clocks

Autonomous circadian clocks drive daily rhythms in physiology and behaviour. A network of coupled neurons, the suprachiasmatic nucleus (SCN), serves as a robust self-sustained circadian pacemaker. Synchronization of this timer to the environmental light-dark cycle is crucial for an organism’s fitness. In a recent theoretical and experimental study it was shown that coupling governs the entrainment range of circadian clocks. We apply the theory of coupled oscillators to analyse how diffusive and mean-field coupling affects the entrainment range of interacting cells. Mean-field coupling leads to amplitude expansion of weak oscillators and, as a result, reduces the entrainment range. We also show that coupling determines the rigidity of the synchronized SCN network, i.e. the relaxation rates upon perturbation. Our simulations and analytical calculations using generic oscillator models help to elucidate how coupling determines the entrainment of the SCN. Our theoretical framework helps to interpret experimental data.     

Amplified Biochemical Oscillations in Cellular Systems

We describe a mechanism for pronounced biochemical oscillations, relevant to microscopic systems, such as the intracellular environment. This mechanism operates for reaction schemes which, when modeled using deterministic rate equations, fail to exhibit oscillations for any values of rate constants. The mechanism relies on amplification of the underlying stochasticity of reaction kinetics within a narrow window of frequencies. This amplification means that fluctuations have a dominant effect, even though the number of molecules in the system is relatively large. The mechanism is quantitatively studied within simple models of self-regulatory gene expression, and glycolytic oscillations.     

Regulation of prokineticin 2 expression by light and the circadian clock

Background  

The suprachiasmatic nucleus (SCN) contains the master circadian clock that regulates daily rhythms of many physiological and behavioural processes in mammals. Previously we have shown that prokineticin 2 (PK2) is a clock-controlled gene that may function as a critical SCN output molecule responsible for circadian locomotor rhythms. As light is the principal zeitgeber that entrains the circadian oscillator, and PK2 expression is responsive to nocturnal light pulses, we further investigated the effects of light on the molecular rhythm of PK2 in the SCN. In particular, we examined how PK2 responds to shifts of light/dark cycles and changes in photoperiod. We also investigated which photoreceptors are responsible for the light-induced PK2 expression in the SCN. To determine whether light requires an intact functional circadian pacemaker to regulate PK2, we examined PK2 expression in cryptochrome1,2-deficient (Cry1-/-Cry2-/-) mice that lack functional circadian clock under normal light/dark cycles and constant darkness.     

Circadian rhythms in the pineal organ persist in zebrafish larvae that lack ventral brain

Background  

The mammalian suprachiasmatic nucleus (SCN), located in the ventral hypothalamus, is a major regulator of circadian rhythms in mammals and birds. However, the role of the SCN in lower vertebrates remains poorly understood. Zebrafish cyclops (cyc) mutants lack ventral brain, including the region that gives rise to the SCN. We have used cyc embryos to define the function of the zebrafish SCN in regulating circadian rhythms in the developing pineal organ. The pineal organ is the major source of the circadian hormone melatonin, which regulates rhythms such as daily rest/activity cycles. Mammalian pineal rhythms are controlled almost exclusively by the SCN. In zebrafish and many other lower vertebrates, the pineal has an endogenous clock that is responsible in part for cyclic melatonin biosynthesis and gene expression.     

Light, Reactive Oxygen Species, and Magnetic Fields Activating ERK/MAPK Signaling Pathway in Cultured Zebrafish Cells

To guarantee that organism’s biological rhythms remain tied to the rhythms of its environment, the circadian clock must be able to reset itself in response to environmental cues. The main environmental stimulus for organisms is light, which is provided by day–night cycles. Cultured lines of zebrafish cells have been established as an attractive vertebrate cell-based model suitable for the examination of the light signaling pathway for entraining the circadian clock. Studies using these cell lines have revealed critical roles for the mitogen-activated protein kinase (MAPK) signaling pathways in light-dependent circadian entrainment. Here, we show in cultured zebrafish cells that artificial magnetic fields induce extracellular signal-regulated kinase (ERK)/MAPK activation with kinetics analogous to those elicited by light, suggesting that magnetic fields may influence circadian regulation in zebrafish. Our findings indicate that cultured zebrafish cells represent a valuable system for investigating the links between magnetic fields and the signaling pathways responsible for the synchronization of vertebrate circadian clocks under laboratory conditions.     

Advanced analysis of a cryptochrome mutation's effects on the robustness and phase of molecular cycles in isolated peripheral tissues of Drosophila

Background  

Previously, we reported effects of the cry ^b mutation on circadian rhythms in period and timeless gene expression within isolated peripheral Drosophila tissues. We relied on luciferase activity driven by the respective regulatory genomic elements to provide real-time reporting of cycling gene expression. Subsequently, we developed a tool kit for the analysis of behavioral and molecular cycles. Here, we use these tools to analyze our earlier results as well as additional data obtained using the same experimental designs.     

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.     

Non-random fluctuations and multi-scale dynamics regulation of human activity

We investigate if known extrinsic and intrinsic factors fully account for the complex features observed in recordings of human activity as measured from forearm motion in subjects undergoing their regular daily routine. We demonstrate that the apparently random forearm motion possesses dynamic patterns characterized by robust scale-invariant and nonlinear features. These patterns remain stable from one subject to another and are unaffected by changes in the average activity level that occur within individual subjects throughout the day and on different days of the week, since they persist during daily routine and when the same subjects undergo time-isolation laboratory experiments designed to account for the circadian phase and to control the known extrinsic factors. Further, by modeling the scheduled events imposed throughout the laboratory protocols, we demonstrate that they cannot account for the observed scaling patterns in activity fluctuations. We attribute these patterns to a previously unrecognized intrinsic nonlinear multi-scale control mechanism of human activity that is independent of known extrinsic factors such as random and scheduled events, as well as the known intrinsic factors which possess a single characteristic time scale such as circadian and ultradian rhythms.     

Nonadiabatic Van der Pol oscillations in molecular transport

The force exerted by the electrons on the nuclei of a current-carrying molecular junction can be manipulated to engineer nanoscale mechanical systems. In the adiabatic regime a peculiarity of these forces is negative friction, responsible for Van der Pol oscillations of the nuclear coordinates. In this work we study the robustness of the Van der Pol oscillations against high-frequency sources. For this purpose we go beyond the adiabatic approximation and perform full Ehrenfest dynamics simulations. The numerical scheme implements a mixed quantum-classical algorithm for open systems and is capable to deal with arbitrary time-dependent driving fields. We find that the Van der Pol oscillations are extremely stable. The nonadiabatic electron dynamics distorts the trajectory in the momentum-coordinate phase space but preserves the limit cycles in an average sense. We further show that high-frequency fields change both the oscillation amplitudes and the average nuclear positions. By switching the fields off at different times one obtains cycles of different amplitudes which attain the limit cycle only after considerably long times.     

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．     

Signal analysis of behavioral and molecular cycles

Background  

Circadian clocks are biological oscillators that regulate molecular, physiological, and behavioral rhythms in a wide variety of organisms. While behavioral rhythms are typically monitored over many cycles, a similar approach to molecular rhythms was not possible until recently; the advent of real-time analysis using transgenic reporters now permits the observations of molecular rhythms over many cycles as well. This development suggests that new details about the relationship between molecular and behavioral rhythms may be revealed. Even so, behavioral and molecular rhythmicity have been analyzed using different methods, making such comparisons difficult to achieve. To address this shortcoming, among others, we developed a set of integrated analytical tools to unify the analysis of biological rhythms across modalities.     

生物钟基因网络中分子噪声诱导的日夜节律振荡及相干共振

研究了果蝇细胞内生物钟基因调节网络的分子噪声,特别讨论了生物钟系统处于略微远离振荡区域的稳态时分子噪声对于时钟蛋白的日夜节律振荡的影响.结果表明：(1)虽然时钟蛋白合成或者衰减的生化反应事件是在随机的时间间隔里随机发生的,但系统可以依赖自身固有的调节机理诱导出明显的日夜节律振荡;(2)分子噪声诱导的日夜节律振荡的相干性可以在合适的分子噪声水平下到达最佳,说明了相干共振现象的发生. 关键词： 生物钟 分子噪声 日夜节律振荡 相干共振     

生物钟基因网络中分子噪声诱导的日夜节律振荡及相干共振

研究了果蝇细胞内生物钟基因调节网络的分子噪声，特别讨论了生物钟系统处于略微远离振荡区域的稳态时分子噪声对于时钟蛋白的日夜节律振荡的影响.结果表明：(1)虽然时钟蛋白合成或者衰减的生化反应事件是在随机的时间间隔里随机发生的，但系统可以依赖自身固有的调节机理诱导出明显的日夜节律振荡；(2)分子噪声诱导的日夜节律振荡的相干性可以在合适的分子噪声水平下到达最佳，说明了相干共振现象的发生.     

Robust Statistical Inference in Generalized Linear Models Based on Minimum Renyi’s Pseudodistance Estimators

Minimum Renyi’s pseudodistance estimators (MRPEs) enjoy good robustness properties without a significant loss of efficiency in general statistical models, and, in particular, for linear regression models (LRMs). In this line, Castilla et al. considered robust Wald-type test statistics in LRMs based on these MRPEs. In this paper, we extend the theory of MRPEs to Generalized Linear Models (GLMs) using independent and nonidentically distributed observations (INIDO). We derive asymptotic properties of the proposed estimators and analyze their influence function to asses their robustness properties. Additionally, we define robust Wald-type test statistics for testing linear hypothesis and theoretically study their asymptotic distribution, as well as their influence function. The performance of the proposed MRPEs and Wald-type test statistics are empirically examined for the Poisson Regression models through a simulation study, focusing on their robustness properties. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the superior performance of the robust MRPEs as well as Wald-type tests.