Allosteric regulation equations

Aiming at a computer-based description of cellular metabolism, we briefly summarize some characteristic rate equations associated with competitive and allosteric regulation. Starting with irreversible Michaelis Menten kinetics, the most common types of feedback inhibition are depicted in Fig. 9. Allowing all possible associations between the enzyme and the inhibitor shown in Fig. 9, the total enzyme concentration Er can be expressed as... 

To account for positive cooperativity and sigmoidal rate equations, a number of theoretical models for allosteric regulation have been developed. Common to most models is the assumption (and requirement) that enzymes act as multimers and exhibit interactions between the units. We briefly mention the most... 

In the following, we mainly use variants of the functional form given in Eq. (56) to describe cooperativity and allosteric regulation in metabolic systems. In particular, within Section VII.C, we discuss a general functional form of rate equations, including allosteric interaction... 

Figure 18. A simple bistable pathway [96], Left panel The metabolite A is synthesized with a constant rate vi and consumed with a rate vcon V2(A) + V3(A), with the substrate A inhibiting the rate V3 at high concentrations (allosteric regulation). Right panel The rates of vsyn vi const. and vcon V2 (A) + V3(A) as a function of the concentration A. See text for explicit equations. The steady state is defined by the intersection of synthesizing and consuming reactions. For low and high influx v, corresponding to the dashed lines, a unique steady state A0 exists. For intermediate influx (solid line), the pathway gives rise to three possible solutions of A0. The rate equations are specified in Eq. 67, with parameters 0.2, 3 2.0, Kj 1.0, and n 4 (in arbitrary units).

In the above equation, E denotes the enzyme, S denotes the substrate, ES denotes the enzyme-substrate complex, P denotes the product, and kf, k, and k at denote the rate constants. Monod, Wyman, and Changeux (Changeux and Edelstein, 2005) added to the hypothesis of allosteric regulations. Allosteric sites are binding sites on catalysts, which shape feeble noncovalent bonds with the coupling substrates. Allosteric collaborations... 

In a metabolic pathway, the initial enzyme is regulated by allosteric or other mechanisms as discussed below. The subsequent enzymes in the pathway obey Michaelis-Menten kinetics. If the initial enzyme is down-regulated (its reaction rate falls), the concentration of its product falls. Since this product is the substrate for the second enzyme, its reaction rate will fall according to the Michaelis-Menten equation. Hence, its production concentration will fall, and so on down the whole pathway. 

The inhibition of certain enzymes by specific metabolites is an important element in the regulation of intermediary metabolism and most often occurs with cooperative enzymes that are regulated allosterically. Inhibition of enzymes that obey the Michaelis-Menten equation, noncooperative enzymes, is more commonly used by pharmacists to alter a patient s metabolism. Reversible inhibition of noncooperative enzymes is classified into three groups which can be distinguished kinetically and which have different mechanisms and effects when administered. The classes are called competitive, uncompetitive, and noncompetitive inhibition. Mixed inhibition also occurs. In all these types of inhibition, the inhibitor (usually a small molecule) binds reversibly and rapidly with the enzyme. 

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