Self quadratic dependence

The phenomenon of self organization occurs at nonstabHities of the sta tionary state and leads to the formation of temporal and spatio temporal dissipative structures. Remember that oscillating instabilities of stationary states of dynamic systems can be observed for the intermediate nonlinear stepwise reactions only, when no fewer than two intermediates are involved (see Section 3.5) and at least one of the elementary steps is kinet icaUy irreversible. The minimal sufficient requirements for the scheme of a process with temporal instabilities are not yet strictly formulated. However, in aU known examples of such reactions, the rate of the kineti caUy irreversible elementary reaction at one of the intermediate steps is at least in a quadratic dependence on the intermediate concentrations. Among these reactions are autocatalytic steps. 

The difficulties in this example arise from the self-inhibition of the enzyme catalysis by Xg. The rate coefficient kA first increases with increasing concentration of Xg and then decreases. The hypersurface formed by eliminating the time dependence from the set of deterministic rate equations, by dividing the equation for each but one of the species by the equation for that one species, is folded over due to the quadratic dependence of kA on Xg. In the simulation the concentration of Xg is varied randomly and the responses of the other species are calculated to give time series of 2,000 data points these series are the starting point for both the EMC and the CMC analysis. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Fig. 6.6. Dependences of reaction rate R on extent of reaction y typical of self-accelerating (autocatalytic) systems (a) prototype quadratic autocatalysis (b) prototype cubic autocatalysis.

R. Cammi and J. Tomasi, Nonequilibrium solvation theory for the polarizable continuum model - a new formulation at the SCF level with application to the case of the frequency-dependent linear electric-response function, Int. J. Quantum Chem., (1995) 465-74 B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798-807 R. Cammi, L. Frediani, B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, A second-order, quadratically... 

This form is self-consistent for harmonic diabatic potentials in that it has the same Q-dependence as the potentials for large Q and automatically fulfills the requirements of the variation principle and the noncrossing rule. If the linear coupling term vanishes by symmetry and the quadratic term does not (as in the pseudo-Renner-Teller effect) a coupling operator of a comparable degree of sophistication is... 

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