The Selkov model

While other models of forced glycolysis (e.g. Richter (1984 based on the Selkov model)) demonstrates the occurrence of entrainment (the frequency locking of an autonomous oscillator by externally imposed frequencies) Hess and Markus emphasised the failure to see entrainment leading to quasi-periodic or even chaotic dynamics, and the coexistence of complicated periodic attractors. Currently the biological function of complex temporal patterns are not well-understood. Some neurobiological examples will illustrate this problem. 

The fiuctuational trajectory away from a stationary state to a given point in concentration space (X, Y) in general differs from the deterministic path from that point back to the stationary state for systems without detailed balance. We show this in some calculations for the Selkov model in (3.38) we take m = n = r = l,s = 3 other parameters are given in [1], p. 4555. Figure 3.1 gives some results of these calculations. 

In Fig. 4.3, [2], we plot a cut through the stationary solution of the master equation for selected parameters of the Selkov model vs. the variable Y the dotted line is a numerical solution of the probability distribution and the solid line is that distribution calculated from (4.15-4.18) with the approximation described in this paragraph and the same parameters for the Selkov model. The approximation gives a reasonable estimate. A different impression is gathered from the plot shown in Fig. 4.4 A most probable fluctuational trajectory obtained from numerical integration of the stationary solution of the master... 

Fig. 4.3. Plot of the stationary probability distribution of the Selkov model a cut at constant X vs. Y. The parameters used are given in [2], see the caption to Fig. 8 in that reference. From [2]...

Fig. 4.4. Plot in the concentration space of the variables x,y of the Selkov model (a) optimal fluctuational trajectory from a stable stationary state (x,y)st to a given point (x,y)p and (b) the deterministic return to the stationary state. Prom [3]...

As the system approaches a stationary state the starred variables approach their values of the stationary state. For the Selkov model the stationary state of the linear equivalent system is... 

Calculation of Relative Stability in a Two-Variable Example, the Selkov Model... 

In this section we compare the predictions of the thermod50iamic theory for relative stability in a two-variable example, the Selkov model, with the results obtained from numerical integration of the reaction diffusion equation. The model, constructed for early studies of glycolysis, has two variables, X and Y, and two constant concentrations. The reaction mechanism is... 

Fig. 5.4. Plots of concentration profiles of X and Y vs. distance 2 during the front propagation to the right in the Selkov model. The solid line is the initial concentration profile the dotted lines are concentration profiles with a spacing of 500 in arbitrary time units. For values of parameters see the caption to Fig. 4 in [1]. Prom [1]...

The predictions of the thermodynamic theory presented in this chapter for equistability of two stable stationary states for the Selkov model is shown in Fig. 5.6. The results of the theory run parallel to the calculations and approach them as the length of the interface region is increased. To show this quantitatively we define the relative error... 

Fig. 5.5. The solid line is a plot of zero velocity of the interface between phase 1 and 3 calculated for the Selkov model. Above the solid line the interface moves to the right, below the solid line to the left. From [1]...

A system such as the Selkov model may have many Liapunov functions. We note that any Liapunov function of a system with multiple stationary states may serve as a criterion of relative stability [2]. Moreover the derivative of the Liapunov function with respect to L, the length of the interphase region, may also serve as such a criterion. 

Many chemical and biochemical reactions can be in an oscillatory regime in which the concentrations of intermediates and products vary in a regular oscillatory way in time the oscillations may be sinusoidal but usually are not. Sustained oscillations require an open system with a continuous influx of reactants in a closed system oscillations may occur initially when the sjretem is far from equilibrium, but disappear as the system approaches equilibrium. A simple example of an oscillatory reaction is the Selkov model [1]... 

In Fig. 16.2 we show a plot of the dissipation calculated for the Selkov model, (16.1), in which the concentration of the reactant A is perturbed sinusoidally with a small amplitude of 5% on the abscissa we show the variation of the ratio of frequency of perturbation to that of the frequency of the autonomous reaction. We see that within the regions of entrainment bands, at 0.5, 1.0, 2.0, and barely visible at 1.5, the dissipation varies significantly. The variations are small in magnitude because of the small amplitude of the perturbation. Again, later in the chapter we show a similar calculation for a part of glycolysis. 

The problem of fluctuation lissipation relations in multivariable systems is analyzed in [15] the mathematics needed for that task goes beyond the level chosen for this book, and hence only a brief verbal smnmary is presented. A statistical ensemble is chosen, which consists of a large number of replicas of the system, such as for example the Selkov model, each characterized by different composition vectors. There exists a master equation for this probability distribution of this ensemble, which serves as a basis for this approach an analytical solution of this master equation is given in [15]. 

Fig. 19.1. Monte Carlo results for the stationary probability distribution for the Selkov model with the shape of a volcanic crater. The parameters are fci = 1.0, k2 = 0.2, fcs = 1.0, ki = 0.1, fcs = 1.105 and fee = 0.1. The system has a stable deterministic limit cycle located on the ridge of the center. The symbol O denotes the effective dimensionless volume which scales the total number of molecules, taken to be C = 50, 000...

Fig. 19.2. Plot of the probability distribution in a cross section, tranverse to the ridge, for the Selkov model, (a) results of the Monte Carlo calculation (b) solution of the linearized Fokker-Planck equation. Taken from [3]...

Fig. 19.3. Comparison of the analytical results with the numericai calculations on the Selkov model. The parameters are the same as in Fig. 19.1. Cmve a x, the concentration of X vs. time curve b y, the concentration of Y vs. time curve c numerical result of the probability density in the cross section along the limit cycle with the maximum value normalized to unity curve d anal5dical result for the same as in curve c curve e numerical result for the product of the area of the cross section times the velocity, which is almost constant curve fi analytical result the same plotted in quantity as curve e. From [3]...

These saturation and selection mechanisms depend on the nonlinearities of the model. Most analytical and numerical nonlinear studies have been performed on model chemical schemes like the Brussellator or variants [1,9, 19-26], the Selkov model [27], the Schnackenberg model [28-30], the Gray-Scott model [31], or the Lengyel-Epstein model [32-34]. Only the last one... 

Application of the Thermodynamic Theory to the Selkov Model and Comparison with Numerical Results... 

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