Hyperbolicity breakdown

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

The third stage of our strategy is discussed in Sections IX and X. Our discussion is speculative, since quantitative analysis is lacking at present. In Section IX, we point out that, in reaction dynamics, breakdown of normal hyperbolicity would also play an important role. Such cases would include phase transitions in systems with a finite number of degrees of freedom. In Section X, we will discuss the possibility of bifurcation in the skeleton of reaction paths, and we point out that it corresponds to crisis in multidimensional chaos. This approach offers an interesting mechanism for chemical evolution. 

In this section, we consider the breakdown of the condition of normal hyperbolicity. First, we explain a simple example where breakdown of normal hyperbolicity leads to a bifurcation in reaction processes. In the Belousov-Zhabotinsky (BZ) reaction [40], the bifurcation from the stable fixed point to the limit cycle takes place through the breakdown of normal hyperbolicity. This is the simplest case where mathematical analyses are in progress [41]. 

Second, we point out the possibility that normal hyperbolicity breaks down for NHIMs with saddles as the energy of the vibrational modes increases at saddles. These cases seem to be much more difficult than that in the BZ reaction. At present, no attempt to analyze these cases has been made. However, considering that we face these cases frequently in reactions, the study of the breakdown of normal hyperbilicity is urgent. 

Moreover, the breakdown of normal hyperbolicity leads to the bifurcation from the fixed point to the limit cycle. Suppose that under a smooth variation of parameters we change the flow from the one in Fig. 30 to the one in Fig. 31. Then, in order for the fixed point PI in Fig. 30 to shift to P2 in Fig. 31, it should go through the point where normal hyperbolicity breaks down. 

These situations take place when we raise the energy of clusters. Then, clusters would frequently change their structures, crossing over multiple saddles with considerable vibrational energy [42,43]. This leads to a phase transition of the cluster from the solid to the liquid state. Thus, a phase transition in systems with finite degrees of freedom belongs to the class of cases where the breakdown of normal hyperbolicity plays a crucial role. 

In the previous section, we explained the bifurcation on NHIMs which would result from breakdown of normal hyperbolicity. Here, we speculate on bifurcation in the connections among NHIMs. In Fig. 33, we schematically display how the connections among NHIMs would change. As parameters of the system vary, transverse intersections between the unstable manifold VT of a NHIM Ma and the stable manifold W of a NHIM Mb [see Fig. 33(i)] disappear. Instead, transverse intersections between the unstable manifold W of the NHIM Ma and the stable manifold of another NHIM M [see Fig. 33(ii)] appear. 

The fact that so many transport systems (cf. Figs. 1-3) display the same characteristic form for the dependence of velocity on substrate concentration as do enzyme systems strongly suggests that a formal analysis of transport kinetics along the lines of that of enzyme kinetics might be valuable. The simple Michaelis-Menten or hyperbolic velocity vs. substrate concentration curve for enzymes has traditionally been interpreted as arising from the combination between enzyme and substrate, with the subsequent breakdown of this complex to product and free enzyme. One writes... 

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