A Model for the Control of Metabolic Reaction Chains

The model that we have discussed in Section 2.2 for enzyme-catalyzed reactions showed saturation of the reaction flux J at high values of the substrate concentration S. Clearly, this saturation phenomenon is due to the limited total amount of enzyme. It may also be considered as some effective control of the reaction flux which prevents an overstrain of the system with increasing substrate concentrations. 

Quite a lot of reaction chains within the network of biochemical pathways show a much more efficient control mechanism in that some intermediate product after n steps of the chain blocks or inhibits an enzyme E which catalyzes the first step of the chain  

S and P are the initial substrate and the final product of the chain, respectively, which are kept at fixed concentrations. In the inhibition reaction, a numer of p molecules of the intermediate product is assumed to transfer the enzyme from its active configuration E to its inactived configuration E.  

The model in (2.56) has been investigated by GRIFFITH (1968), HUNDING (1974) and TYSON (1975). In contrast to the excitation models in Sections 2.4 and 2.5, it shows a negative feedback a sudden increase of S leads to stepwise delayed increases of the chain products S 2. .. until finally S reduces the enzyme E involved in the first step such that now the chain products Sp S2,. .. S will decrease, etc. From this qualitative consideration we expect that the system will be able to perform oscillatory changes of its state. We shall convince ourselves by a mathematical analysis that this will really be possible however, we will also learn that these oscillatory motions are basically different from those of the undamped Volterra-Lotka model. 

In our analysis we follow the work of HUNDING (1974) and first make three approximations which are not essential for the qualitative behaviour of the model but make it mathematically tractable. The first approximation is the assumption that the inhibition reaction, i.e., the third line of (2.56), is so rapid compared with the chain reactions that it may be evaluated at its steady state, which in this particular case means vanishing inhibition reaction flux, such that 

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