5-oscillator phase space model

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

Thus the mechanism formed by steps (l)-(4) can be called the simplest catalytic oscillator. [Detailed parametric analysis of model (35) was recently provided by Khibnik et al. [234]. The two-parametric plane (k2, k 4/k4) was divided into 23 regions which correspond to various types of phase portraits.] Its structure consists of the simplest catalytic trigger (8) and linear "buffer , step (4). The latter permits us to obtain in the three-dimensional phase space oscillations between two stable branches of the S-shaped kinetic characteristics z(q) for the adsorption mechanism (l)-(3). The reversible reaction (4) can be interpreted as a slow reversible poisoning (blocking) of... 

The Lotka-Volterra type of equations provides a model for sustained oscillations in chemical systems with an overall affinity approaching infinity. Perturbations at finite distances from the steady state are also periodic in time. Within the phase space (Xvs. Y), the system produces an infinite number of continuous closed orbits surrounding the steady state... 

In this model, a two-level system is coupled to a classical nonlinear oscillator with mass Mo and phase space coordinates (Pq,Pd)- This coupling is given by hyoRo- The nonlinear oscillator, which has a quartic potential energy function Vn Ro) = aPg/4 — Moco Ro/", is then bilinearly coupled... 

The phase space structures for two identical coupled anharmonic oscillators are relatively simple because the trajectories lie on the surface of a 2-dimensional manifold in a 4-dimensional phase space. The phase space of two identical 2-dimensional isotropic benders is 8-dimensional, the qualitative forms of the classifying trajectories are far more complicated, and there is a much wider range of possibilities for qualitative changes in the intramolecular dynamics. The classical mechanical polyad 7feff conveys unique insights into the dynamics encoded in the spectrum as represented by the Heff fit model. 

By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in the following chapters. Similarly, the phase space analysis of two-variable models indicates that the oscillatory dynamics of neurons corresponds to the evolution towards a limit cycle (Fitzhugh, 1961 Kokoz Krinskii, 1973 Krinskii Kokoz, 1973 Hassard, 1978 Rinzel, 1985 Av-Ron, Parnas Segel, 1991). A similar evolution is predicted (May, 1972) by models for predator-prey interactions in ecology. 

Fig. 4.17. Phase space trajectory associated with bursting in the biochemical model with multiple regulation. The curve corresponds to the complex periodic oscillations of fig. 4.3d (Decroly Goldbeter, 1982).

The examples of rhythmic behaviour analysed in this book all belong to dynamics of the limit cycle type. In the case of phosphofructokinase, like in that of cAMP synthesis in Dictyostelium or signal-induced Ca oscillations, the analysis of models based on experimental data indeed shows that these systems admit a nonequilibrium steady state that becomes unstable beyond a critical value of some control parameter. It is in these conditions that sustained oscillations occur, in the form of a limit cycle in the phase space. 

More recently, Fisher information has been studied as an intrinsic accuracy measure for concrete atomic models and densities [43, 44] and also for quantum mechanics central potentials [45]. Also, the concept of phase space Fisher information, where position and momentum variables are included, was analyzed for hydrogenlike atoms and the isotropic harmonic oscillator [46]. The net Fisher information measure is found to correlate well with the inverse of the ionization potential and dipole polarizability [44]. 

The spectra from the 4-mode model coupled to 0, 5, 10, and 20 bath modes are plotted in Figs. 9(a)-(d). The addition of the bath clearly results in the structure of the spectrum being washed out. The experimental spectrum is, however, not obtained. The effect of the bath modes is made clear in Figs. 10(a)-(d), in which the absolute values of the autocorrelation function for the 4-mode system with 0, 5, 10, and 20 bath modes is plotted. Just the strongest 5 bath modes lead to a significant damping of the oscillations in the function. This is simply due to the extra volume of phase-space available... 

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