Allosteric/cooperative kinetics

For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively (Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1 1 function in both A and B (i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2 2 function (i.e., second order in both A and B). The 2 2 function rate equation results in nonlinear kinetics that should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966 Tsai, 1978). 

The nonlinear 2 2 function kinetics should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Bardsley and Waight, 1978) and the formation of the abortive substrate complex (Dalziel and Dickinson, 1966). The cooperative kinetics (of the double reciprocal plots) can either concave up (positive cooperativity) or... 

The enzymatic activities are also subjected to allosteric/cooperative regulations (Monad et al., 1965 Perutz, 1990 Ricard and Cornish-Bowden, 1987). A regulatory molecule (effector) binds to the regulatory site (allosteric site) distinct from the catalytic site to affect one or more kinetic parameters of an enzymatic reaction. 

The previous section illustrated how allosteric cooperativity can result in a sigmoidal relationship between binding saturation and substrate concentration. In this section, we demonstrate how a sigmoidal relationship between product concentration and time can arise from enzyme kinetics with time lags. 

Enzyme, oscillatory, 31,91,118 see also Allosteric enzyme kinetics Biochemical oscillations Cooperativity Michaelian enzyme kinetics... 

The basic kinetic properties of this allosteric enzyme are clearly explained by combining Monod s theory and these structural results. The tetrameric enzyme exists in equilibrium between a catalytically active R state and an inactive T state. There is a difference in the tertiary structure of the subunits in these two states, which is closely linked to a difference in the quaternary structure of the molecule. The substrate F6P binds preferentially to the R state, thereby shifting the equilibrium to that state. Since the mechanism is concerted, binding of one F6P to the first subunit provides an additional three subunits in the R state, hence the cooperativity of F6P binding and catalysis. ATP binds to both states, so there is no shift in the equilibrium and hence there is no cooperativity of ATP binding. The inhibitor PEP preferentially binds to the effector binding site of molecules in the T state and as a result the equilibrium is shifted to the inactive state. By contrast the activator ADP preferentially binds to the effector site of molecules in the R state and as a result shifts the equilibrium to the R state with its four available, catalytically competent, active sites per molecule. 

This expression is called a linked function and indicates how the binding of ligands at nearby sites can influence each other. See also Basic Regulatory Kinetics Cooper-ativity Allosterism Feedback Effectors Bohr Effect Hemoglobin Le Chatelier s Principle Adair Equation... 

Figure 1. Plot of v/V ax versus the millimolar concentration of total substrate for a model enzyme displaying Michaelis-Menten kinetics with respect to its substrate MA (i.e., metal ion M complexed to otherwise inactive ligand A). The concentrations of free A and MA were calculated assuming a stability constant of 10,000 M k The Michaelis constant for MA and the inhibition constant for free A acting as a competitive inhibitor were both assumed to be 0.5 mM. The ratio v/Vmax was calculated from the Michaelis-Menten equation, taking into account the action of a competitive inhibitor (when present). The upper curve represents the case where the substrate is both A and MA. The middle curve deals with the case where MA is the substrate and where A is not inhibitory. The bottom curve describes the case where MA is the substrate and where A is inhibitory. In this example, [Mfotai = [Afotai at each concentration of A plotted on the abscissa. Note that the bottom two curves are reminiscent of allosteric enzymes, but this false cooperativity arises from changes in the fraction of total "substrate A" that has metal ion bound. For a real example of how brain hexokinase cooperatively was debunked, consult D. L. Purich H. J. Fromm (1972) Biochem. J. 130, 63.

The activity of allosteric enzymes is adjusted by reversible binding of a specific modulator to a regulatory site. Modulators may be the substrate itself or some other metabolite, and the effect of the modulator may be inhibitory or stimulatory. The kinetic behavior of allosteric enzymes reflects cooperative interactions among enzyme subunits. 

The enzyme from B. stearothermophilus is an a4 tetramer of subunit Mr 33 900. Early kinetic studies indicated that the enzyme acts in a manner that is qualitatively consistent with an MWC two-state model. The enzyme acts as a A system i.e., both states have the same value of kcal but different affinities for the principle substrate. In the absence of ligands, the enzyme exists in the T state that binds fructose 6-phosphate more poorly than does the R state. In the absence of ADP, the binding of fructose 6-phosphate is highly cooperative, and h = 3.8. The positive homotropic interactions are lowered on the addition of the allosteric effector ADP, with h dropping to 1.4 at 0.8-mM ADP.52 ADP thus binds preferentially to the R state. The allosteric inhibitor phosphoenolpyruvate binds preferentially to the T... 

The enzyme has been partially purified (70-fold) from 38,000 X 9 supernatant fluid from sheep brain homogenates by Ipata (55-58). Thq enzyme (MW 140,000) is reported to be specific for 5 -AMP and 5 -IMP although the substrate specificity does not appear to have been examined closely. 2 - and 3 -AMP are not hydrolyzed (56). Unlike the enzyme from many sources the brain enzyme does not require divalent cations and indeed Co2+, which stimulates several other 5 -nucleotidases, was inhibitory at 5 mM. The enzyme is strongly inhibited by very low concentrations of ATP, UTP, and CTP (50% inhibition by 0.3 pM ATP) but not by GTP. 2 -AMP, 3 -AMP, and a variety of other nucleoside monophosphates, nucleosides, and sugar phosphates do not inhibit. A kinetic examination of ATP, UTP, and CTP inhibition (56-58) revealed that inhibition curves were sigmoidal, indicating cooperativity between inhibitor molecules and an allosteric type of interaction between inhibitor and protein. The metabolic significance of ATP inhibition is... 

The allosteric properties of FDPases present an interesting subject for future study. In the case of the liver enzyme the substrate shows positive cooperativity in binding, but no evidence for cooperativity in catalytic activity has been obtained. Perhaps this is because of the high affinity of the enzyme for the substrate, which prevents precise kinetic measurement at low substrate concentration. On the other hand, the substrate has been shown to increase the affinity of the enzyme for AMP, the allosteric inhibitor. 

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

Phosphofructokinase was one of the first enzymes to which Monod and his colleagues applied the symmetry model of allosteric transitions. It contains four identical subunits, each of which has both an active site and an allosteric site. The cooperativity of the kinetics suggests that the enzyme can adopt two different conformations (T and R) that have similar affinities for ATP but differ in their affinity for fructose-6-phosphate. The binding for fructose-6-phosphate is calculated to be about 2,000 times tighter in the R conformation than in T. When fructose-6-phosphate binds to any one of the subunits, it appears to cause all four subunits to flip from the T conformation to the R conformation, just as the symmetry model specifies. The allosteric effectors ADP, GDP, and phosphoenolpyruvate do not alter the maximum rate of the reaction but change the dependence of the rate on the fructose-6-phosphate concentration in a manner suggesting that they change the equilibrium constant (L) between the T and R conformations. 

It was noted earlier that Michaelis-Menten kinetics and its linear transformations are not valid for allosteric enzymes. Instead, the Hill equation, an equation originally empirically developed to describe the cooperative binding of Oz to hemoglobin (Chapter 7), is used. The expression describing such a straight-line plot is... 

Allosteric enzymes constitute an important class of enzymes whose catalytic activity can be regulated. These enzymes, which do not conform to Michaelis-Menton kinetics, have multiple active sites. These active sites display cooperativity, as evidenced by a sigmoidal depen-dence of reaction velocity on substrate concentration. 

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