An Analysis of a Circadian Model Using The Small-Gain Approach to Monotone Systems

The theory of monotone dynamical systems has been found very useful in the modeling of some gene, protein, and signaling networks. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. This paper shows that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone syste...

New trajectory-based small-gain results are obtained for nonlinear feedback systems under relaxed assumptions. Specifically, during a transient period, the solutions of the feedback system may not satisfy some key inequalities that previous small-gain results usually utilize to prove stability properties. The results allow the application of the small-gain perspective to various systems which satisfy less demanding stability notions than the Input-to-Output Stability property...

Some biological systems operate at the critical point between stability and instability and this requires a fine-tuning of parameters. We bring together two examples from the literature that illustrate this: neural integration in the nervous system and hair cell oscillations in the auditory system. In both examples the question arises as to how the required fine-tuning may be achieved and maintained in a robust and reliable way. We study this question using tools from nonline...

Circadian rhythms are archetypical examples of nonlinear oscillations. While these oscillations are usually attributed to circuits of biochemical interactions among clock genes and proteins, recent experimental studies reveal that they are also affected by the cell's mechanical environment. Here we extend a standard biochemical model of circadian rhythmicity to include mechanical effects in a parametric manner. Using experimental observations to constrain the model, we sugges...

Circadian rhythms are biological rhythms of approximately 24 h that persist even under constant conditions without environmental daily cues. The molecular circadian clock machinery generates the physiological rhythms, which can be transmitted into the downstream output system. Owing to the stochastic nature of the biochemical reactions, the oscillation period of circadian rhythms exhibited by individual organisms or cells is not constant on a daily basis with variations as hi...

The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades...

Rhythms in electrical activity in the membrane of cells in the suprachiasmatic nucleus (SCN) are crucial for the function of the circadian timing system, which is characterized by the expression of the so-called clock genes. Intracellular Ca$^{2+}$ ions seem to connect, at least in part, the electrical activity of SCN neurons with the expression of clock genes. In this paper, we introduce a simple mathematical model describing the linking of membrane activity to the transcrip...

The circadian clock is the molecular mechanism responsible for the adaptation to daily rhythms in living organisms. Oscillations and fluctuations in environmental conditions regulate the circadian clock through signaling pathways. We study the response to continuous photic perturbations in a minimal molecular network model of the circadian clock, composed of 5 nonlinear delay differential equations with multiple feedbacks. We model the perturbation as a stationary stochastic ...

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic...

We analyze a class of network motifs in which a short, two-node positive feed- back motif is inserted in a three-node negative feedback loop. We demonstrate that such networks can undergo a bifurcation to a state where a stable fixed point and a stable limit cycle coexist. At the bifurcation point the period of the oscillations diverges. Further, intrinsic noise can make the system switch between oscillatory state and the stationary state spontaneously. We find that this swit...