Lecture – 3 The Kinetics Of Enzyme-Catalyzed Reactions Dr. AKM Shafiqul Islam

Our objective is to develop suitable mathematical expressions for the rates of enzyme-catalyzed reactions. Naturally, a reaction rate equation is comparison with experimentally determined rates.

Let us consider the following chemical reaction Where S is the substrate (reactant) and P is the product.

The reaction rate v, in the quasi-steady state approximation is defined by The reaction rate is an intensire quantity, defined at each point in the reaction mixture. Consequently, the rate will vary with position in a mixture if concentrations or other intensive state variables change from point to point.

A typical experiment to measure enzyme kinetics is as follows. – At time zero, solutions of substrate and of an appropriate purified enzyme are mixed in a well- stirred, closed isothermal vessel containing a buffer solution to control pH. – The substrate and/or product concentrations are monitored at various later times. – Analytical techniques include spectrophotometric, manometric, electrode, polarimetric, and sampling- analysis methods are used.

Since the reaction conditions including enzyme and substrate concentrations are known best at time zero, the initial slope of the substrate or product concentration vs. time curve is estimated from the data:

Rate (or velocity) is proportional to the instantaneous increase in P with time or to the decrease in substrate S with time

Michaelis-Menten Kinetics Now let us suppose that a good set of experimental rate data representing mathematically.

The following qualitative features can write: – The rate of reaction is first order in concentration of substrate at relatively low values of concentration. [if v = (const)(s n ), n is the order of the reaction rate.] – As the substrate concentration is continually decrease, the reaction order in substrate diminishes continuously from one to zero. – The rate of reaction is proportional to the total amount of enzyme present.

Henri observed this behavior, and in 1902 propsed the rate equation which exhibits all three features listed above. Notice that v = v max /2 when s is equal to K m. where s is the concentration of free substrate in the reaction mixure, while e, is the concentration of the total amount of enzyme present in both the free and combined forms.

Henri provided a theoretical explanation for this equation using a hypothesized reaction mechanism, Similar equation given by Michaelis and Menten in Michaelis-Menten treatment will find repeated useful application in derivation of more complicated kinetic models

Assumed that the enzyme E and substrate S combine to form a complex ES, which then dissociates into product P and free or uncombined enzyme E: This mechanism includes the intermediate complex formed as well as regeneration of catalyst in its original form upon completion of the reaction sequence.

Henri and Michaelis and Menten assumed that reaction is in equilibrium The mass-action law for the kinetics of molecular events, gives Here, s, e, and (es) denote the concentrations of S, E and ES, respectively.

Decomposition of the complex to product and free enzyme is assumed irreversible: Since all enzyme present is either free or complexed, we also have where eo is the total concentration of enzyme in the system.

This is known from the amount of enzyme initially charged into the reactor. By eliminating (es) and e from the three previous equations. v max = k 2 e 0 The reaction described by Henri is commonly referred to as having Michaelis-Menten kinetics,

The parameter v max is called the maximum or limiting velocity, and K m is known as the Michaelis constant.

Briggs and Haldane later provided kinetic studies and mathematical derivation of Henri quation For reaction in a well-mixed closed vessel we can write the following mass balances for substrate and the ES complex: At the initial condition

These equations cannot be solved analytically but they can be readily integrated on a computer to find the concentrations of S, E, ES, and P as functions of time.

At the beginning Briggs and Haldane assumed this condition to be true. This assumption is called the quasi-steady-state approximation, It is valid for the enzyme-catalyzed reaction in a closed system provided that e 0 /s 0 is sufficiently small.

If the initial substrate concentration is not large compared with the total enzyme concentration, the assumption not true. Normally, the amount of catalyst present is considerably less than the amount of reactant, So that is an excellent approximation after start-up.

After starting, to eliminate e and (es) from We get,

the Michaelis-Menten form results K m is no longer can be interpreted as a dissociation constant

Michaelis-Menten rate expression equation can be determined analytically by integrating to obtain This equation is most easily exploited indirectly by computing the reaction time t required to reach various substrate concentrations.