Deterministic models theorem

It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model. 

Although there are several techniques for estimating the reaction rate constants based upon the deterministic model, these methods are usually rather complicated, and the results cannot be statistically characterised. That is why from time to time estimates based upon one or another stochastic model are suggested. Such a suggestion has earlier been described under the name fluctuation-dissipation theorem , and similar methods have been presented by Mulloolly (1971, 1972, 1973), Hilden (1974), and Matis and Hartley (1971). 

As it has turned out that consistency in the mean does not hold in general, several people have presented a proof of the fact that the stochastic model of a certain simple special reaction tends to the corresponding deterministic model in the thermodynamic limit. This expression means that the number of particles and the volume of the vessel tend to infinity at the same time and in such a way that the concentration of the individual components (i.e. the ratio of the number and volume) tends to a constant and the two models will be close to each other. In addition to this the fluctuation around the deterministic value is normally distributed as has been shown in a special case by Delbriick (later head of the famous phage group) almost fifty years ago (Delbriick, 1940). To put it into present-day mathematical terms the law of the large numbers, the central limit theorem, and the invariance principle all hold. These statements have been proved for a large class of reactions for those with conservative, reversible mechanisms. Kurtz used the combinatorial model, and the same model was used by L. Arnold (Arnold, 1980) when he generalised the results for the cell model of reactions with diffusion. 

The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, i) of the deterministic model a law of large numbers and a central limit theorem hold (Arnold Theodoso-pulu, 1980 Arnold 1980). 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

Some possibilities for defining stochastic models of reaction-diffusion systems will be shortly studied. The connection between deterministic and certain stochastic models that can be established by the extension of Kurtz theorems will be mentioned (Arnold, 1980 Arnold Theodosopulu, 1980). 

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