Oscillatory reactions other methods

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

Oscillatory reactions require a separate analysis, which is presented in detail. Responses of nonlinear systems to pulses or other perturbations are treated in some generality. The concluding chapter gives a brief introduction to bioinformatics, including several methods for determining reaction mechanisms. 

Most of the methods outlined above are suitable for obtaining information on oscillatory reaction networks. As pointed out in several other chapters in this book, related methods can be used for determination of causal connectivities of species and deduction of mechanims in general nonoscillatory networks. Pulses of species concentration by an arbitrary amount have been proposed (see chapter 5) and experimentally applied to glycolysis (see chapter 6). Random perturbation by a species can be used and the response evaluated by means of correlation functions (see chapter 7) this correlation metric construction method has also been tested (see chapter 8). Another approach to determining reaction mechanisms by finding Jacobian matrix elements is described in Mihaliuk et al. [69]. 

Finally, the quest to develop mechanistic explanations for these varied and fascinating phenomena can succeed only if more data become available on the component processes. Kinetics studies of the reactions which make up a complex oscillatory system are essential to its understanding. In some cases, traditional techniques may be adequate, though in many others, fast reaction methods will be required. There also appears to be some promise in developing an analysis of the relaxation of flow systems in non-equilibrium steady states as a technique to complement equilibrium relaxation techniques. 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

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