Enzyme regulation simple models

The carboxyl proteases are so called because they have two catalytically essential aspartate residues. They were formerly called acid proteases because most of them are active at low pH. The best-known member of the family is pepsin, which has the distinction of being the first enzyme to be named (in 1825, by T. Schwann). Other members are chymosin (rennin) cathepsin D Rhizopus-pepsin (from Rhizopus chinensis) penicillinopepsin (from Penicillium janthinel-lum) the enzyme from Endothia parasitica and renin, which is involved in the regulation of blood pressure. These constitute a homologous family, and all have an Mr of about 35 000. The aspartyl proteases have been thrown into prominence by the discovery of a retroviral subfamily, including one from HIV that is the target of therapy for AIDS. These are homodimers of subunits of about 100 residues.156,157 All the aspartyl proteases contain the two essential aspartyl residues. Their reaction mechanism is the most obscure of all the proteases, and there are no simple chemical models for guidance. 

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. 

A simple form of eqns (2.7-2.8), which is particularly useful in analysing the dynamics of models based on allosteric regulation with positive feedback, is obtained in the limit case where the enzyme is a dimer and the substrate binds exclusively to the R state. When incorporating parameter e into the normalization of the substrate concentration, one obtains the simplified expression (2.11) for the rate function appearing in eqns (2.7) ... 

Similar complex oscillatory phenomena have been observed in a closely related model containing two regulated enzyme reactions coupled in a different manner (Li, Ding Xu, 1984). An additional indication of the generality of the results obtained in the multiply regulated biochemical system is given by the study of the model for the synthesis of cAMP in Dictyostelium amoebae. In addition to simple periodic oscillations and excitability (see chapter 5), this realistic model based on experimental observations also predicts the appearance of more complex oscillatory phenomena in the form of birhythmicity, bursting and chaos (chapter 6). 

Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism they can become quite involved and may contain a large number of p>arameters. Rate laws and enzyme parameters are still unknown for most enzymes. Convenience kinetics is used to translate a biochemical network - manually into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimization. In general, the convenience kinetics applies to arbitrary reaction stoichiometries and captures biologically relevant behavior such as saturation, activation, inhibition with a small number of free parameters. It represents a simple molecular reaction mechanism in which substrates bind rapadly and in random order to the enzyme. 

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