Enzyme Kinetic Equations

Enzyme Kinetic Equations

The rate of a chemical reaction is usually expressed in terms of a change in the concentration of a reactant or of a product in a given time interval. Any convenient experimental method can be used to monitor changes in concentration. In a reaction of the form A + B - >  P, where A and B are reactants and P is the product, the rate of the reaction can be expressed either in terms of the rate of disappearance of one of the reactants or in terms of the rate of appearance of the product. The rate of disappearance of A is - ∆[A]/ ∆ t, where symbolizes change, [A] is the concentration of A in moles per liter, and t is time. Likewise, the rate of disappearance of B is - ∆ [B]/ ∆t, and the rate of appearance of P is ∆ [P]/ ∆t. The rate of the reaction can be expressed in terms of any of these changes because the rates of appearance of product and disappearance of reactant are related by the stoichiometric equation for the reaction

The negative signs for the changes in concentration of A and B indicate that A and B are being used up in the reaction, while P is being produced.

It has been established that the rate of a reaction at a given time is pro-portional to the product of the concentrations of the reactants raised to the appropriate powers,

where k is a proportionality constant called the rate constant. The exponents f and g must be determined experimentally. They are not necessarily equal to thecoefÞcients of the balanced equation, but frequently they are. The square brackets, as usual, denote molar concentration. When the exponents in the rate equation have been determined experimentally, a mechanism for the reactionÑa description of the detailed steps along the path between reactants and productsÑcan be proposed.

The exponents in the rate equation are usually small whole numbers, such as 1 or 2. (There are also some cases in which the exponent 0 occurs.) The values of the exponents are related to the number of molecules involved in the detailed steps that constitute the mechanism. The overall order of a reaction is the sum of all the exponents. If, for example, the rate of a reaction A 3 P is given by the rate equation

where k is the rate constant and the exponent for the concentration of A is 1, then the reaction is first order with respect to A and Þrst order overall. The rate of radioactive decay of the widely used tracer isotope phosphorus 32 (³²P; atomic weight = 32) depends only on the concentration of ³²P present. Here we have an example of a Þrst-order reaction. Only the ³²P atoms are involved in the mechanism of the radioactive decay, which, as an equation, takes the form

where k is the rate constant, the exponent for the concentration of A is 1, and the exponent for the concentration of B is 1, then the reaction is said to be Þrst order with respect to A, Þrst order with respect to B, and second order overall. In the reaction of glycogen_n (a polymer of glucose with n glucose residues) with inorganic phosphate, P_i, to form glucose 1-phosphate + glycogen_n-1, the rate of reaction depends on the concentrations of both reactants.

where k is the rate constant. Both the glycogen and the phosphate take part in the reaction mechanism. The reaction of glycogen with phosphate is Þrst order with respect to glycogen, Þrst order with respect to phosphate, and second order overall.

Many common reactions are first or second order. After the order of the reaction is determined experimentally, proposals can be made about the mechanism of a reaction.

Is the rate of a reaction always based on the concentration of reactants?

Exponents in a rate equation may be equal to zero, with the rate for a reaction A - > B given by the equation

Such a reaction is called zero order, and its rate, which is constant, depends not on concentrations of reactants but on other factors, such as the presence of catalysts. Enzyme-catalyzed reactions can exhibit zero-order kinetics when the concentrations of reactants are so high that the enzyme is completely saturated with reactant molecules. This point will be discussed in more detail later but, for the moment, we can consider the situation analogous to a trafÞc bottleneck in which six lanes of cars are trying to cross a two-lane bridge. The rate at which the cars cross is not affected by the number of waiting cars, only by the number of lanes available on the bridge.

Summary

The rate of a chemical reaction is measured by the rate of appearance of the products or the rate of disappearance of the substrates.

The rate of a reaction is mathematically equal to a rate constant, k, multi-plied by the concentration of substrate(s) raised to an exponent.

The order of a reaction is described by the exponent in the rate equation. Common reaction orders are zero order, first order, and second order.

The rate constant, k, and the exponents must be measured experimen-tally for each reaction.