Allosteric reactions, glycolytic oscillations

Part I of the book is devoted to glycolytic oscillations. A two-variable allosteric model is analysed in chapter 2 for the phosphofructokinase reaction, which is responsible for the oscillations. The autocatalytic regulation of this reaction, which results from the cooperative activation of the multisubunit enzyme by one of its products, is at the core of the mechanism that produces the nonequilibrium instability beyond which... 

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 mechanism of glycolytic oscillations can thus be closely related to the periodic alternation of the allosteric enzyme between its two conformational states. Such an alternation occurs in an autonomous manner, driven by the constant substrate input and by the autocatalytic regulation of the enzyme by its reaction product. 

In the two-variable models studied for glycolytic oscillations and birhythmicity, periodic behaviour originates from a unique instability mechanism based on the autocatalytic regulation of an allosteric enzyme by its reaction product. The question arises as to what happens when two instabiUty-generating mechanisms are present and coupled within the same system can new modes of dynamic behaviour arise from such an interaction ... 

The coupling in series of two enzyme reactions with autocatalytic regulation (fig. 4.1) permits the construction of a three-variable biochemical prototype containing two instability-generating mechanisms (Decroly, 1987a,b Decroly Goldbeter, 1982). As in the model for glycolytic oscillations, the substrate S of the first enzyme is introduced at a constant rate into the system this substrate is transformed by enzyme Ej into product Pi, which serves as substrate for a second enzyme E2 that transforms Pj into P2. The two allosteric enzymes are both activated by their reaction product Pj and P2 are thus positive effectors of enzymes Ej and E2, respectively. 

This two-variable system (Goldbeter et al, 1978) presents the additional advantage of being formally identical with the system of eqns (2.7) studied in chapter 2 for glycolytic oscillations. This similarity stems from the basic structure common to the two models a substrate, injected at a constant rate, is transformed in a reaction catalysed by an allosteric enzyme activated by the reaction product. In the cAMP-synthesizing system in D. discoideum, activation is indirect as extracellular cAMP enhances the synthesis of intracellular cAMP, which is then transported into the extracellular medium. However, the hypothesis of a quasi-steady state for intracellular cAMP is tantamount to considering that the variation of )8 is so fast that the enzyme is, de facto, activated directly by its apparent product, extracellular cAMP. 

The detailed analysis of the model for the product-activated allosteric enzyme allowed a precise quantification of the role played by enzyme cooperativity in glycolytic oscillations (see chapter 2). However, the analysis of a slightly modified model in which the sink of the product becomes Michaelian - i.e. saturable - instead of linear, showed (see section 2.7) that oscillations can occur even if the allosteric enzyme contains a single subunit existing in two conformational states. Enzyme cooperativity is therefore not a condition sine qua non for oscillations to occur weaker nonlinearities, of the Michaelian type, distributed over several reactions of the system, can thus cooperate to raise its global... 

Glycolytic oscillations (Hess Boiteux, 1971 Goldbeter, Caplan, 1976) occur due to the allosterically controlled phosphofructokinase (PFK, EC 2.7.1.11) reaction. The simplest (perhaps oversimplified) model of glycolysis contains two variables only, namely the substrate (ATP) and the product (ADP) of the enzyme (Goldbeter Nicolis, 1976). The hierarchical regulatory mechanism of glycolysis can be described by a more detailed model (Boiteux Hess, 1984). 

The enzyme phosphofructokinase is allosteric, that is, it is made up of equivalent units that possess specific reaction sites for the fixation of the substrate and product. Each unit exists in two conformational states one active with more affinity for the substrate, and one inactive. The reaction products of phosphofructokinase (FDP and ADP) displace the conformational equilibrium in favor of the active form of the enzyme. This may create a destabilizing effect on the excess entropy production. In the glycolytic cycle, the allosteric properties of the phosphofructokinase may lead to oscillations. Consider the following simple model... 

In the model studied so far, the only source of nonlinearity arises from the allosteric nature of PFK kinetics and from the regulation of the enzyme by positive feedback. The sink of product was supposed to remain linear. This assumption, which greatly simplifies the calculations, corresponds to the fact that ADP is used as substrate by several glycolytic enzymes and therefore does not accumulate within the system. This would not occur if the sink reaction(s) were saturated by ADP. It is nevertheless instructive to consider the case of a saturable sink, of the Michaelian type, in order to better understand the role of nonlinearities in the mechanism of oscillations. Maybe the degree of cooperativity of the allosteric enzyme required for oscillations might diminish once an additional nonlinearity is present within the system ... 

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