Allosteric model for glycolytic oscillations

In contrast with Michaelian enzymes, which have hyperbolic kinetics, allosteric enzymes, thanks to their sigmoidal kinetics, possess an enhanced sensitivity towards variations in the concentration of an effector or of the substrate. This is the reason why many enzymes that play an important role in the control of metabolism are of the allosteric type. 

Several models have been proposed to account for the sigmoidal kinetics observed as a function of the concentration in substrate or effector. The first, developed by Hill (1910) for the binding of oxygen to 

The regulation of PFK by a large number of positive or negative effectors, which reflects the key role of this enzyme in cellular metabolism (Krebs, 1972 Hofmann, 1978), is achieved mainly through allosteric control (Mansour, 1972 Laurent et al, 1978 Laurent Seydoux, 1977 Nissler et al, 1977), although the protein can be phos-phorylated (Kitajima, Sakakibara Uyeda, 1983). As shown by the 

Phosphofructokinase possesses two substrates, ATP and F6P, which it transforms into ADP and FBP. A complete model for this reaction should therefore take into account the evolution of these four metabolites. However, studies carried out in yeast indicate that the couple ATP-ADP plays a more important role than the couple F6P-FBP in the control of oscillations. Indeed, the addition of ADP ehcits an immediate phase shift of the oscillations (fig. 2.8) while the effect of FBP is much weaker (Hess Boiteux, 1968b Pye, 1969). The predominant regulation is thus exerted by ADP. In order to keep the model as simple as possible and to limit the number of variables to only two, which allows us to resort to the powerful tools of phase plane analysis, the situation in which an allosteric enzyme is activated by its unique reaction product is considered (fig. 2.10). This monosubstrate, product-activated. 

The model is represented in fig. 2.11 in the general case of an enzyme formed by n subunits or protomers. The hypotheses of the model are as follows  

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Oscillatory enzymes simple periodic behaviour in an allosteric model for glycolytic oscillations... 

Fig. 2.11. Allosteric model for glycolytic oscillations. The enzyme is formed by n subunits existing in the states R and T. The substrate (S), injected at a constant rate, binds to the two forms of the enzyme with different affinities. The complexes thus formed in the two states decompose with different rates to yield the product (P). The latter binds in an exclusive manner to the the most active, R, form of the enzyme, and disappears from the reaction medium in an apparent first-order reaction (Goldbeter Lefever, 1972 Venieratos Goldbeter, 1979 Goldbeter, 1980).

The allosteric model for glycolytic oscillations thus shows how periodic behaviour originates from the peculiar regulation of PFK. The period... 

Fig. 2.29. Entrainment of the oscillations by a periodic source of substrate in the allosteric model for glycolytic oscillations. The period of the sinusoidal source and the resulting period of the oscillatory enzyme are denoted by T and T, respectively. Domains C, B and A correspond to entrainment by the fundamental frequency, and by the 1/2 and 1/3 harmonics of the forcing input (Boiteux et al, 1975).

Although the allosteric model for glycolytic oscillations is relatively simple, it provides a quahtative and, in significant measure, a... 

Fig. 2.33. Adenylate energy charge at steady state in the two-variable, allosteric model for glycolytic oscillations (Goldbeter, 1974).

The main properties of the two-variable allosteric model for glycolytic oscillations, in the absence of product recycling into substrate, are summarized in fig. 3.2. In the phase plane (a, y), the system governed by eqns (2.7) evolves toward a limit cycle (dashed line) when the steady state, located at the intersection of the nullclines (da/dr) = 0, (dy/dr) = 0, lies in a region of sufficiently negative slope (do/dy) on the latter nullcline. In fig. 3.2, this region extends, schematically, from A to B. 

Latency of Ca oscillations correlation with period, 372-4 in one-pool versus two-pool model, 385 Life cycle, of Dictyostelium, 164,165 Limit cycle(s), 4,5,495-8 in allosteric model for glycolytic oscillations, 51-3,65, 75... 

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