enzyme kinetics chapter

The six classes of enzymes
1. Oxidoreductases catalyze oxidation-reduction reactions. Most of these enzymes are referred to as dehydrogenases, but some are called oxidases, peroxidases, oxygenases, or reductases.

2. Transferases catalyze group-transfer reactions, and many require the presence of coenzymes. In group-transfer reactions, a portion of the substrate molecule usually binds covalently to the enzyme or its coenzyme. This group includes the kinases.

3. Hydrolases catalyze hydrolysis. They are a special class of transferases, with water serving as the acceptor of the group transferred.

4. Lyases catalyze nonhydrolytic and nonoxidative elimination reactions, or lysis of a substrate, generating a double bond. In the reverse direction, lyases catalyze addition of one substrate to a double bond of a second substrate. A lyase that catalyzes an addition reaction in cells is often termed a synthase.

5. Isomerases catalyze structural change within one molecule, that is, isomerization reactions. Because these reactions have only one substrate and one product, they are among the simplest enzymatic reactions.

6. Ligases catalyze ligation, or joining, of two substrates. These reactions require the input of the chemical potential energy of a nucleoside triphosphate such as ATP. Ligases are usually referred to as synthetases.

Enzyme kinetics
Enzymes catalyze chemical reactions by transiently binding and specifically activating substrates. This binding of specific substrates to a given enzyme to form a enzyme-substrate complex was first proposed by Emil Fisher in 1894. A simple enzymatic reaction for the conversion of a substrate to a product can be expressed by:
E + S ® ES ® E + P
Rate of reaction depends on both substrate and enzyme concentration. Most often, the [enzyme] is much less than the [substrate], and the reaction is pseudo first-order. In this situation, the reaction velocity (n) is linearly related to the enzyme concentration:

Effect of enzyme concentration, [E], on the velocity, n, of an enzyme-catalyzed reaction at a fixed, saturating [S]. Because the reaction rate is affected by the concentration of enzyme but not by the concentration of the other reactant, S, the bimolecular reaction is pseudo first order.

Determination of order and rate constant of a reaction
Rate of a first order reaction:

The rate of a single reaction over its entire course, with time expressed as multiples of the half-life (t_1/2 ) of the reactant. Note that for each interval of t_1/2 the reactant concentration is halved. A graph of [A] versus t shows that the rate, defined as the slope of the line, decreases as the reaction continues.

The initial rates of a reaction at different starting concentrations:

The initial reaction rate is determined for three values of [A]₀.

Enzyme kinetics (continued)
In the initial period of the enzyme-catalyzed reaction, the amount of product formed is negligible and can be described by:

The rate constants, k₁ and k_-1, govern the rates of association of S with E and dissociation of S from ES, respectively. The rate constant for the second step is k_cat, the catalytic constant (turnover number), which is the number of catalytic events per second per enzyme molecule. This step is essentially one-way in the initial period when P negligible, thus, initial velocities (n_o) should be measured:

Progress curve for an enzyme-catalyzed reaction in which a substrate is converted to product. [P], the concentration of product, increases as the reaction proceeds. The initial velocity of the reaction (n_o) is the slope of the initial linear portion of the curve. Note that the rate of the reaction doubles when twice as much enzyme (2 E, the upper curve) is added to an otherwise identical reaction mixture.

Michaelis-Menten Kinetics
Assumptions made for “steadystate kinetic conditions”:
[S] and [P] are changing but [ES] does not change (constant flux of S)

Also [E]_tot = [E]_free + [ES]
K₁ and K _-1 >> k₂ P is small at the beginning
S >> E and [ES] formation does not influence [S]

Vo= K₂[ES] forward reaction / rate limiting step
formation *[ES] = k1[E][S]*
breakdown *[ES] = (k -1+k2[ES]**

A steady state occurs when the rates of formation and breakdown of the ES complex are equal, this gives the formula:
*k1[E][S] = (k-1+k2)[ES] which then gives [E][S] / [ES]=(k-1+k2) / k1**
defining Km
Km = (k-1+k2) / k1

The Michaelis-Menten Equation
Rate equations for an enzyme catalyzed reaction support a theory involving the formation of ES complexes. At high [S], S saturates E, and the reaction rate is independent of the [S]. The value of n_o under this condition is called the maximum velocity, V_max. At low [S], the reaction is first-order with respect to S. The plot of n_o versus [S] from low to high [S] is a rectangular hyperbola. The rate equation (Michaelis-Menten equation) that describes this relationship is:
	where K_m is the Michaelis constant and V_max is the maximum velocity.
Plots of initial velocity, n_o, versus substrate concentration, [S], for an enzyme-catalyzed reaction.
	Each experimental point is obtained from a separate progress curve using the same concentration of enzyme. The shape of the curve is hyperbolic. At low substrate concentrations, the curve approximates a straight line that rises steeply. In this region of the curve, the reaction is first order with respect to substrate. At high concentrations of substrate, the enzyme is saturated, and the reaction is zero order with respect to substrate.

	The concentration of substrate that corresponds to half-maximum velocity is called the Michaelis constant, K_m. The enzyme is half-saturated when [S] = K_m.

Derivation of the Michaelis-Menten equation

For a simple enzyme-catalyzed reaction (E + S ⇋ ES ® E + P), the graph shows how the concentrations of substrate [S], free enzyme [E], enzyme-substrate complex [ES], and product [P] vary with time. After a very brief initial period, [ES] reaches a steady state in which ES is consumed approximately as rapidly as it is formed, so d[ES]/dt ≅ 0. The amounts of E and ES are greatly exaggerated for clarity. Note that [E] + [ES] = [E]_t, or total enzyme concentration, and that [ES] actually falls very slowly as substrate is consumed, while [E] accordingly rises.

V_o vs. [S] for an enzyme-catalyzed reaction
In the cartoons below, the gray ellipses, labeled E, represent enzyme molecules. The smaller filled rectangles represent substrate molecules
The number of enzyme molecules with bound substrate is an indication of the reaction rate because those enzymes have the opportunity to "act" and convert the substrate to product. The initial reaction rate (y axis) is shown as a function of the substrate concentration (x axis) in the graphs below. The shadowed area indicates the region of the graph that corresponds to the cartoon at its left.
Low substrate concentration: There are few substrate molecules and they are all bound to the enzyme molecules


Medium substrate concentration There are more substrate molecules and some of them are free in solution, not bound to an enzyme molecule.


High substrate concentration There are many substrate molecules and all the enzyme molecules have a bound substrate

Meanings of K_cat and K_cat/K_M
Interpretation of K_M

Interpretation of k_cat

Reversible enzyme inhibition

Competitive inhibition

Classical competitive inhibition: The substrate (S) and the inhibitor (I) bind at the same site on the enzyme, the active site.

Nonclasssical competitive inhibition: With some allosteric enzymes, inhibitors bind at a different site than substrates, although the inhibition exhibits competitive characteritics.

Kinetic scheme illustrating the binding of I to E.

Effects of competitive inhibition on enzyme kinetics

The effect of a competitive inhibitor (I) on reaction velocity at different substrate concentrations. Addition of the inhibitor decreases the velocity but not the V_max. The apparent K_M is higher in the presence of inhibitor.

Lineweaver-Burk plots of the reactions. The lines cross the 1/V axis at the same V_max, showing that I is a competitive inhibitor.

Determination of K_M and K_I. If the measurement of K^app_M is repeated at different concentrations of I, K_I can be determined from the slope of the line, and the true K_M from the line's intercept where [I] = 0.

A substrate and its competitive inhibitor.

The substrate UpA and the very similar molecule UpcA are competitors for the enzyme ribonuclease. The single difference between the substrate and the inhibitor is shown in red.

Noncompetitive inhibition

Noncompetitive inhibitors can bind to E or ES, forming inactive EI or ESI complexes, respectively. These inhibitors are not substrate analogs and do not bind at the same site as S.

Kinetic scheme illustrating noncompetitive binding.

Effects of noncompetitlve inhibition on enzyme kinetics.

The effect of a non-competitive inhibitor (I) on reaction velocity at different substrate concentrations. In this simple example, K_M is not affected, but V_max is decreased because the enzyme is not as catalytically efficient in the presence of the inhibitor.

Lineweaver-Burk plots of the reactions. The lines cross the 1/V axis at different points, clearly distinguishing this situation from competitive inhibition

Determination of K_I and the uninhibited V_max (which will give k_cat).

Comparison of a substrate and a designed inhibitor of purine nucieoside phosphorylase.

The two substrates of this enzyme are guanosine and inorganic phosphate. (a) Guanosine. (b) The most potent known inhibitor of the enzyme. N-9 of guanosine has been replaced by a carbon atom. The chlorinated benzene ring binds to the sugar-binding site of the enzyme, and the acetate side chain binds to the phosphate-binding site

Uncompetitive inhibition

Uncompetitive inhibitors bind only to ES, not to free enzyme.

In uncompetitive inhibition, V_max is decreased (l/V_max is increased) by the conversion of some molecules of E to the inactive form ESI. Since it is the ES complex that binds I, the decrease in V_max is not reversed by the addition of more substrate.

Uncompetitive inhibitors also decrease the K_m because the equilibria for the formation of both ES and ESI are shifted toward the complexes by the binding of I.

Experimentally, the lines on a double-reciprocal plot representing varying concentrations of an uncompetitive inhibitor all have the same slope

Usually occurs only with multisubstrate reactions.

Kinetic scheme illustrating the binding of I to ES.

Double-reciprocal plot. In uncompetitive inhibition, both V_max and K_m decrease. The ratio of V_max/K_m remains unchanged.

Diagrams of reversible inhibition

Catalytically competent enzymes Inactive enzymes

The six classes of enzymes

Enzyme kinetics

Determination of order and rate constant of a reaction

Enzyme kinetics

Michaelis-Menten Kinetics

The Michaelis-Menten Equation

Derivation of the Michaelis-Menten equation

Vo vs. [S] for an enzyme-catalyzed reaction

Meanings of Kcat and Kcat/KM

Reversible enzyme inhibition

Competitive inhibition

Noncompetitive inhibition

V_o vs. [S] for an enzyme-catalyzed reaction

Meanings of K_cat and K_cat/K_M