Positively cooperative binding

Prove that for n > 1, the Hill equation conforms to a positively cooperative binding or enzyme system. 

This equation embodies all the features of positive cooperative binding of substrate and the effect of this upon biocatalytic rate. Given the power terms, this equation is decidedly not identical with the Michaelis-Menten equation (8.8) However, an alternative and equally important biocatalytic equation can be derived from Equation (8.30) with a little more work and a few simple assumptions. First, let us assume that cooperative binding is sufficiently positive that terms in [S] are eliminated from Equation (8.30). Second, let us define Vmax according to... 

Hemoglobin is a classic example of protein quaternary structure. The protein has 4 subunits, two a-chains and two p-chains, and it exhibits positive cooperativity. Binding of oxygen to one subunit makes it easier for oxygen to bind to other subunits. 

If nn = 1, this can (but does not have to) be due to the simple mechanism I, in that (for example) rnechanism III also yields nH = 1 at 4Kdi = Kd2- Similarly, mechanism IV at Kd3 -I- Kd4 = 2-7Kdi, Kd3, which can be a positive cooperative binding (i.e., with Kdi > Kd4)-The Scatchard plots for III and IV are linear at nH = 1, which means that linear Scatchard plots are no proof of the existence of only one binding site. In addition, it is difficult to distinguish between nn = 1 and nn 1, because the difference of nn and 1 is often small and indistinguishable from the experimental error. 

The simplest model that can describe allosteric interactions at GPCRs is the ternary complex allosteric model [9], As shown in Figure 1, according to this model two parameters define the actions of allosteric agent (X) its affinity for the unoccupied receptor (Kx) and its cooperativity (a) with the ligand (A) that interacts at the primary binding site a < 1 represents negative cooperativity a = 1, no cooperativity a > 1, positive cooperativity. 

Tucek S, Musilkova J, Neduoma J, Proska J, Shelkovnikov SW, Vorlicek J. Positive cooperativity in the binding of alcuronium and A-methy lscopolamine to muscarinic acetylcholine receptors. Mol Pharmacol 1990 38 674-680. 

Lazareno S, Gharagozloo P, Kuonen D, Popham A, Birdsall NJ. Subtype-selective positive cooperative interactions between brucine analogs and acetylcholine at muscarinic receptors radioligand binding studies. Mol Pharmacol 1998 53 573-589. 

COOPERATIVE ENZYMES do not show a hyperbolic dependence of the velocity on substrate concentration. If the binding of one substrate increases the affinity of an oligomeric enzyme for binding of the next substrate, the enzyme shows positive cooperativity. If the first substrate makes it harder to bind the second substrate, the enzyme is negatively cooperative. 

PHOSPHOFRUCTOKINASE shows positive cooperativity with fructose-e-phosphate as the substrate. ATP, an allosteric inhibitor, binds to the T state and decreases the velocity. AMP, a signal for low energy, binds to the R state and increases the velocity of the reaction. 

The formation of ER dimers can be favored once the first monomer has bound to the DNA, since this presents positive cooperation in binding the next monomer. In any case, DNA binding creates a greater compaction of the dimer that results in a subsequent spatial restructuring of the receptor molecules. 

Boyer, J. L Martinez-Carcamo, M Monroy-Sanchez, J. A., Posadas, C., and Garcia-Sainz, J. A. (1986) Guanine nucleotide-induced positive cooperativity in muscarinic-cholinergic antagonist binding. Biochem. Biophys. Res. Commun. 134,172-177. 

In some earlier publications the term cooperativity is used for positive cooperativity and anticoop-erativity is used for negative cooperativity. In this book cooperativity is used whenever g 1. Sometimes, when there is positive cooperativity one says that a ligand at a supports or favors the binding of a ligand at b, and vice versa. 

The effect of f/( 1,1) on the conditional probabilities [see Eq. (4.2.13)] is easily understood. When there exists attraction between the ligands, a ligand occupying one site attracts the second ligand, and hence increases the probability of binding the second ligand—hence positive cooperativity. The reverse holds when the two ligands repel each other. 

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