Reginald H. GarrettCharles M. Grisham

Chapter 13Enzyme Kinetics

Outline

• What characteristic features define enzymes ?• Can the rate of an enzyme-catalyzed reaction be

defined in a mathematical way ?• What equations define the kinetics of enzyme-

catalyzed reactions ?• What can be learned from the inhibition of enzyme

activity ?• What is the kinetic behavior of enzymes catalyzing

bimolecular reactions ?• How can enzymes be so specific ? • Are all enzymes proteins ?• Is it possible to design an enzyme to catalyze any

desired reaction ?

Virtually All Reactions in Cells Are Mediated by Enzymes

• Living systems use enzymes to accelerate and control the rates of vitally important biochemical reactions (see Figure 13.1).

• Enzymes provide cells with the ability to exert kinetic control over thermodynamic potentiality. (glucose pyruvate)

• Enzymes are the agents of metabolic function.• Most enzymes are proteins.• Some enzymes require cofactors or coenzymes.

Virtually All Reactions in Cells Are Mediated by Enzymes

Figure 13.1 Reaction profile showing the large free energy of activation for glucose oxidation. Enzymes lower ΔG‡, thereby accelerating rate.

13.1 What Characteristic Features Define Enzymes ?

• Catalytic power is defined as the ratio of the enzyme-catalyzed rate of a reaction to the uncatalyzed rate.

• Specificity is the term used to define the selectivity of enzymes for their substrates.

• Regulation of enzyme activity ensures that the rate of metabolic reactions is appropriate to cellular requirements.

• Coenzymes and cofactors are nonprotein components essential to enzyme activity.

13.1 What Characteristic Features Define Enzymes ?

• Enzymes can accelerate reactions as much as 1016 over uncatalyzed rates.

• Urease is a good example: • Catalyzed rate Uncatalyzed rate Ratio

3x104/sec 3x10-10/sec 1x1014 • Enzymes selectively recognize proper substrates

over other molecules. • Enzymes produce products in very high yields -

often much greater than 95%. • Specificity is controlled by structure - the unique

fit of substrate with enzyme controls the selectivity for substrate and the product yield.

Enzymes are the Agents of Metabolic Function

Figure 13.2 The breakdown of glucose by glycolysis provides a prime example of a metabolic pathway with many sequential steps.

All pathways are regulated but not all enzymes are regulated .Regulation of glycolysis occurs at several points, the major control point is through the enzyme phosphofructokinase I.

90% yield in each step = 35% over 10 steps

Figure 13.3 Yields in biological reactions must be substantially greater than 90%.

Enzymes Nomenclature

• As in organic, there is a system of nomenclature devised for enzymes as well as a lot of commonly used names.

• There are six main classes used for naming enzymes. Know these six classes.

• There is also a numerical assignment for each enzyme developed by the Enzyme Commission.

• E.g.: EC 1.1.1.1 denotes alcohol dehydrogenase• Its systematic name is:

alcohol:NAD+ oxidoreductasedonor:acceptor main class

The Six Classes of Enzymes

Class# Class and reaction catalyzed

1. Oxidoreductases (dehydrogenases) oxidation-reduction reactions

2. Transferases group transfer reactions

3. Hydrolases hydrolysis reactions

4. Lyases lysis, forming a double bond

5. Isomerases isomerization reactions

6. Ligases (synthetases) joining of two substrates, uses ATP

Enzyme Nomenclature Provides a Systematic Way of Naming Metabolic Reactions

Know the six main classes.

Coenzymes and Cofactors Are Nonprotein Components Essential to Enzyme Activity

Coenzymes are organic cofactors.

13.2 A Mathematical statement of the Rate of an Enzyme-Catalyzed Reaction

• Kinetics is the branch of science concerned with the rates of reactions.

• Enzyme kinetics seeks to determine the maximum reaction velocity that enzymes can attain and binding affinities for substrates and inhibitors.

• Analysis of enzyme rates yields insights into enzyme mechanisms and metabolic pathways.

• This information can be exploited to control and manipulate the course of metabolic events.

Several kinetics terms to understand

• rate or velocity of reaction• rate constant • rate law • order of a reaction • molecularity of a reaction

Chemical Kinetics Provides a Foundation for Exploring Enzyme Kinetics• Consider a reaction of overall stoichiometry as

shown:

• The rate is proportional to the concentration of A

[ ] [ ]

[ ][ ]

A P

d P d Av

dt dtA

v k Adt

Chemical Kinetics Provides a Foundation for Exploring Enzyme Kinetics

• The simple elementary reaction of A→P is a first-order reaction, v = k[A] and exponent on [A] is 1.

• Figure 13.4 shows the course of a first-order reaction as a function of time.

• This is a unimolecular reaction.• For a second order reaction, the rate law is:• v = k[A][B]. This reaction may be bimolecular or

unimolecular depending on the mechanism.• Kinetics cannot prove a reaction mechanism.• Kinetics can only rule out various alternative

hypotheses because they don’t fit the data.

The Time-Course of a First-Order Reaction

Figure 13.4 Plot of the course of a first-order reaction. The half-time, t1/2 is the time for one-half of the starting amount of A to disappear.

Catalysts Lower the Free Energy of Activation for a Reaction• A typical enzyme-catalyzed reaction must pass

through a transition state.• The transition state sits at the apex of the energy

profile in the energy diagram.• The reaction rate is proportional to the

concentration of reactant molecules with the transition-state energy.

• This energy barrier is known as the free energy of activation, ΔG‡.

• Decreasing ΔG‡ increases the reaction rate.• The activation energy is related to the rate

constant by: /G RTk Ae

Catalysts Lower the Free Energy of Activation for a Reaction

Figure 13.5 Energy diagram for a chemical reaction (A→P) and the effects of (a) raising the temperature from T1 to T2, or (b) adding a catalyst.

The Transition State

Understand the difference between G and G‡

• The overall free energy change for a reaction is related to the equilibrium constant.

• The free energy of activation for a reaction is related to the rate constant.

• It is extremely important to appreciate this distinction.

13.3 What Equations Define the Kinetics of Enzyme-Catalyzed Reactions ?

• Simple first-order reactions display a plot of the reaction rate as a function of reactant. concentration that is a straight line (Figure 13.6)

• Enzyme-catalyzed reactions are more complicated.

• At low concentrations of the enzyme substrate, the rate is proportional to S, as in a first-order reaction.

• At higher concentrations of substrate, the enzyme reaction approaches zero-order kinetics.

• This behavior is a saturation effect.

13.3 What Equations Define the Kinetics of Enzyme-Catalyzed Reactions ?

Figure 13.6 A plot of v versus [A] for the unimolecular chemical reaction, A→P, yields a straight line having a slope equal to k.

As [S] increases, kinetic behavior changes from 1st order to zero-order kinetics

Figure 13.7 Substrate saturation curve for an enzyme-catalyzed reaction.

The Michaelis-Menten Equation is the Fundamental Equation of Enzyme Kinetics

• Louis Michaelis and Maud Menten's theory. • It assumes the formation of an enzyme-substrate

complex. • It assumes that the ES complex is in rapid

equilibrium with free enzyme. • Breakdown of ES to form products is assumed to

be slower than 1) formation of ES and 2) breakdown of ES to re-form E and S.

• Briggs and Haldane later introduced the steady state assumtion.

The Michaelis-Menten Equation is the Fundamental Equation of Enzyme Kinetics

E = enzyme concentration. S = Substrate concentration. ES = Enzyme-substrate complex concentration (noncovalent). P = product concentration. k1 =

rate constant for formation of ES from E + S. k-1 = rate constant for decomposition of ES to E +

S. k2 = rate constant for decomposition of ES to E

+ P.

E + S ES E + P k1 k2

k-1

Development of the Michaelis-Menton Equation

1. The overall rate of product formation: v = k2 [ES]

2. Rate of formation of [ES]: vf = k1[E][S]

3. Rate of decomposition of [ES]:

vd = k-1[ES] + k2 [ES]

4. The steady state assumption requires that: Rate of ES formation = Rate of ES decomposition

5. So: k1[E][S] = k-1[ES] + k2 [ES]

E + S ES E + P k1 k2

k-1

6. In solving for [ES], use the enzyme balance to eliminate [E]. ET = [E] + [ES]

7. k1 (ET - [ES])[S] = k-1[ES] + k2 [ES]

k1 ET[S] - k1[ES][S] = k-1[ES] + k2 [ES]

8. Rearrange and combine [ES] terms:

k1 ET[S] = (k-1 + k2 + k1 [S])[ES]

k1 ET[S] 9. Solve for [ES] = -----------------------

(k-1 + k2 + k1 [S])

Michaelis-Menton Derivation

ET[S]10. Divide through by k1: [ES] = -----------------------

(k-1 + k2)/k1 + [S]

11. Defined Michaelis constant: KM = (k-1 + k2) / k1

12. Substitute KM into the equation in step 10.

13. Then substitute [ES] into v = k2 [ES] from step1 and replace Vmax with k2 ET to give:

Vmax[S] vo = ----------- KM + [S]

Michaelis-Menton Derivation

[ES] Remains Constant Through Much of the Enzyme Reaction Time Course

Figure 13.8 Time course for a typical enzyme-catalyzed reaction obeying the Michaelis-Menten, Briggs-Haldane models for enzyme kinetics. The early state of the time course is shown in greater magnification in the bottom graph.

The dual nature of the Michaelis-Menten equation

Combination of 0-order and 1st-order kinetics

• When S is low, the equation for rate is 1st order in S.

• When S is high, the equation for rate is 0-order in S.

• The Michaelis-Menten equation describes a rectangular hyperbolic dependence of v on S.

• The relation of the “rectangular hyperbola” to the enzyme kinetics profile is described in references at the end of the chapter.

Understanding Vmax

The theoretical maximal velocity

• Vmax is a constant.

• Vmax is the theoretical maximal rate of the reaction - but it is NEVER achieved in reality.

• To reach Vmax would require that ALL enzyme molecules are tightly bound with substrate.

• Vmax is asymptotically approached as substrate is increased.

Understanding Km

The "kinetic activator constant"

• Km is a constant.

• Km is a constant derived from rate constants.

• Km is, under true Michaelis-Menten conditions, an estimate of the dissociation constant of E from S.

• A measure of ES binding.

• Small Km means tight binding; large Km means weak binding. Where k2 is small then Km ≈ Kd.

Table 13.3 gives the Km values for some enzymes and their substrates

Table 13.3 gives the Km values for some enzymes and their substrates

The Turnover Number Defines the Activity of One Enzyme Molecule

A measure of catalytic activity

• kcat, the turnover number, is the number of substrate molecules converted to product per enzyme molecule per unit of time, when E is saturated with substrate. A measure of rate of enzyme activity.

• If the M-M model fits, k2 = kcat = Vmax/Et.

• Values of kcat range from less than 1/sec to many millions per sec.

The Turnover Number Defines the Activity of One Enzyme Molecule

The Ratio kcat/Km Defines the Catalytic Efficiency of an Enzyme

The catalytic efficiency: kcat/Km

An estimate of "how perfect" the enzyme is

• kcat/Km is an apparent second-order rate constant.

• It measures how the enzyme performs when S is low.

• The upper limit for kcat/Km is the diffusion limit - the rate at which E and S diffuse together.

• The maximum rate of diffusion for small molecules is 109 M-1-sec-1.

The Ratio kcat/Km Defines the Catalytic Efficiency of an Enzyme

Superoxide dismutase O2¯· (radical) 1 x 106 5 x 10-4 2 x 109

α-Chymotrypsin Acetyl-Phe-amide 1.4 x 10-1 1.5 x 10-2 9.3

Units of Enzyme Activity

Terms in discussing enzyme activity

Units of enzyme activity:mol S/min mol S/sec = katal

Specific activity: (to follow purification, see p 99) mol S/min/mg E mol S/sec/kgm E

Molecular activity: (turn-over number, TON = kcat )

mol S/min/mol E mol S/sec/mol E

Turnover Number

Example calculation

• An enzyme (1.84 gm, MW 36800), in presence of excess substrate catalyzes at a rate of 4.2 mol substrate/min. Calculate the TON. (mol S/min/mol E)

1.84 gm• mol E: = --------------------- = 5 x 10-5 mol E

36800 gm/mol

Vmax 4.2mol S/min • TON = ------ = ----------------------- = 84000 min-1

Et 5 x 10-5 mol E

Michaleis-Mention Equation

Example calculation • The rate of an enzyme catalyzed reaction is 35 mol/min

at [S] = 10-4 M. KM for this substrate is = 2 x 10-5 M.

Calculate the rate where [S] = 2 x 10-6 mol/min.

VM [S] VM (10-4)• v = ------------- so 35 = ---------------------

KM + [S] (2 x 10-5) + (10-4)

And VM = 1.2(35) = 42 mol/min

(42)(2 x 10-6) (84 x 10-6)• v = -------------------------- = ------------- = 3.8 mol/min-1

(2 x 10-5) + (2 x 10-6) (22 x 10-6)

Table 5-1, p. 99

Specific Activity

• Assume that an assay of 0.8 ml of the crude extract gives 0.518 activity units.

• (3800/0.8)(0.518) = 2460 units in the 3.8 l. • 2460 units/22800 mg protein = 0.108 units/mg • As purification proceeds, the specific activity

increases due to loss of extraneous protein.• By purifying to a constant specific activity, one has

reached a limit in purification.

Graphical Determination of KM and VM

The Michaleis-Menton plot only permits an estimate of Vmax, so KM is also an estimate at VM/2.

There are several graphical methods which provide a better determination of VM and KM.

• We will focus on the Lineweaver-Burk equation which is obtained by taking the reciprocal of the Michaleis-Menton equation (See the next slide).

• A Lineweaver-Burke plot is frequently referred to as a double reciprocal plot since one plots 1/v vs 1/[S].

• The plot gives a straight line which has a slope of KM/VM, a y-intercept of 1/VM and an x-intercept of -1/KM.

Linear Plots Can Be Derived from the Michaelis-Menten Equation

Be able to develop this equation

• Lineweaver-Burk:

• Begin with v = Vmax[S]/(Km + [S]) and take the reciprocal of both sides.

• Rearrange to obtain the Lineweaver-Burk equation:

• A plot of 1/v versus 1/[S] should yield a straight line.

max max

1 1 1

[ ]mK

v V S V

Linear Plots Can Be Derived from the Michaelis-Menten Equation

Figure 13.9 The Lineweaver-Burk double-reciprocal plot.

The Lineweaver-Burk equation.

Enzymatic Activity is Strongly Influenced by pH

• Enzyme-substrate recognition and catalysis are greatly dependent on pH.

• Enzymes have a variety of ionizable side chains that determine its secondary and tertiary structure and also affect events in the active site.

• Substrate may also have ionizable groups.• Enzymes are usually active only over a limited

range of pH.

• The effects of pH may be due to effects on Km or Vmax or both.

Enzymatic Activity is Strongly Influenced by pH

Figure 13.11 The pH activity profiles of four different enzymes.

The Response of Enzymatic Activity to Temperature is Complex

• Rates of enzyme-catalyzed reactions generally increase with increasing temperature.

• However, at temperatures above 50° to 60° C, enzymes typically show a decline in activity.

• Two effects here:• Enzyme rate typically doubles in rate for ever

10°C as long as the enzyme is stable and active.

• At higher temperatures, the protein becomes unstable and denaturation occurs.

The Response of Enzymatic Activity to Temperature is Complex

Figure 13.12 The effect of temperature on enzyme activity.

13.4 Inhibition of Enzyme Activity

• Enzymes may be inhibited reversibly or irreversibly.

• Reversible inhibitors may bind at the active site or at some other site.

• Reversible inhibitors typically change VM, KM or both.

• There are three common types of reversible inhibition:• Competitive• Non-competitive• Uncompetitive

13.4 Inhibition of Enzyme Activity

• Enzymes may also be inhibited in an irreversible manner.

• Iodoacetate is an irreversible inhibitor.• Penicillin is an irreversible inhibitor.• Irreversible inhibitors are also called suicide

inhibitors.

Enzyme Inhibition

CompetitiveI binds at the active siteVM does not changeKM changes

UncompetitiveI binds with ES onlyVM changesKM changesSlope does not change

Noncompetitive (Pure & Mixed)I does not effect binding of SVM changesKM does not change

Uninhibited: Competitive:

Vmax[S] Vmax[S] v = ----------- v = ------------------------

KM + [S] KM(1 + [I]/Ki) + [S]

Noncompetitive: Uncompetitive:

(Vmax/(1 + [I]/Ki)) [S] (Vmax/(1 + [I]/Ki))[S] v = ------------------------- v = -------------------------

KM + [S] KM(1 + [I]/Ki) + [S]

Michaelis-Menton Equations

Competitive Inhibitors Compete With Substrate for the Same Site on the Enzyme

Figure 13.13 Lineweaver-Burk plot of competitive inhibition, showing lines for no I, [I], and 2[I].

Succinate Dehydrogenase – a Classic Example of Competitive Inhibition

Figure 13.14 Structures of succinate, the substrate of succinate dehydrogenase (SDH), and malonate, the competitive inhibitor. Fumarate is also shown.

Pure Noncompetitive Inhibition – where S and I bind to different sites on the enzyme

Figure 13.15 Lineweaver-Burk plot of pure noncompetitive inhibition. Note that I does not alter Km but that it decreases Vmax.

Mixed Noncompetitive Inhibition: binding of I by E influences binding of S by E

Figure 13.16 Lineweaver-Burk plot of mixed noncompetitive inhibition. Note that both intercepts and the slope change in the presence of I.

Uncompetitive Inhibition, where I combines only with E, but not with ES

Figure 13.17 Lineweaver-Burk plot of uncompetitive inhibition. Note that both intercepts change but the slope (Km/Vmax) remains constant in the presence of I.

Enzymes Can Be Inhibited Irreversibly

Figure 13.18 Penicillin is an irreversible inhibitor of the enzyme glycoprotein peptidease, the enzyme which catalyzes an essential step in bacterial cell wall synthesis.It covalently blocks the active site serine.

13.5 - Kinetic Behavior of Enzymes Catalyzing Bimolecular Reactions

Enzymes often catalyze reactions involving two (or more) substrates.

• Reactions may be sequential (single-displacement) reactions.• These can be of two distinct classes:• Random, where either substrate may bind

first, followed by the other substrate.• Ordered, where a leading substrate binds

first, followed by the other substrate.• Reactions may be ping-pong (double

displacement) reactions.

Conversion of AEB to PEQ is the Rate-Limiting Step in Random, Single-Displacement Reactions

In this type of sequential reaction, all possible binary enzyme-substrate and enzyme-product complexes are formed rapidly and reversibly when enzyme is added to a reaction mixture containing A, B, P, and Q. This is a random sequential reaction.

This symbolism is known as Cleland notation.

Creatine Kinase Acts by a Random, Single-Displacement Mechanism

The overall direction of the reaction will be determined by the relative concentrations of ATP, ADP, Cr, and CrP and the equilibrium constant for the reaction.

An example of a random sequential reaction.

Creatine Kinase Acts by a Random, Single-Displacement Mechanism

Figure 13.21 The structures of creatine and creatine phosphate, guanidinium compounds that are important in muscle energy metabolism.

In an Ordered, Single-Displacement Reaction, the Leading Substrate Must Bind First

The leading substrate (A) binds first, followed by B. Reaction between A and B occurs in the ternary complex and is usually followed by an ordered release of the products, P and Q. This is an ordered sequential reaction.

An Alternative way of Portraying the Ordered, Single-Displacement Reaction

This is another view of ordered sequential.

Figure 13.19 Single-displacement bisubstrate mechanism.

13.5 - Kinetic Behavior of Enzymes Catalyzing Bimolecular Reactions

The Double Displacement (Ping-Pong) Reaction

Double-Displacement (Ping-Pong) reactions proceed via formation of a covalently modified enzyme intermediate. Reactions conforming to this kinetic pattern are characterized by the fact that the product of the enzyme’s reaction with A (called P in the above scheme) is released prior to reaction of the enzyme with the second substrate, B.

An Alternative Presentation of the Double-Displacement (Ping-Pong) Reaction

Other views of the ping-pong mechanism.

The Double Displacement (Ping-Pong) Reaction

Figure 13.22 Double-displacement (ping-pong) bisubstrate mechanisms are characterized by parallel lines.

Glutamate:aspartate Aminotransferase

Figure 13.23 An enzyme conforming to a double-displacement bisubstrate mechanism (ping-pong).

13.7 – How Can Enzymes Be So Specific ?

• The “Lock and key” hypothesis was the first explanation for specificity.

• The “Induced fit” hypothesis provides a more accurate description of specificity.

• Induced fit favors formation of the transition-state.• Specificity and reactivity are often linked. In the

hexokinase reaction, binding of glucose in the active site induces a conformational change in the enzyme that causes the two domains of hexokinase to close around the substrate, creating the catalytic site.

13.7 – How Can Enzymes Be So Specific ?

Figure 13.24 A drawing, roughly to scale, of H2O, glycerol, glucose, and an idealized hexokinase molecule. Binding of glucose in the active site induces a conformational change in the enzyme that causes the two domains of hexokinase to close around the substrate, creating the catalytic site.

13.7 – Are All Enzymes Proteins ?

• Ribozymes - segments of RNA that display enzyme activity in the absence of protein.• Examples: RNase P and peptidyl transferase.

• Abzymes - antibodies raised to bind the transition state of a reaction of interest. • For a good review of abzymes, see Science,

Vol. 269, pages 1835-1842 (1995). • Transition states are covered in more depth in

in Chapter 14.

13.8 Is It Possible to Design An Enzyme to Catalyze Any Desired Reaction ?• A known enzyme can be “engineered” by in vitro

“site-directed” mutagenesis, replacing active site residues with new ones that might catalyze a desired reaction.

• Another approach attempts to design a totally new protein with the desired structure and activity.• This latter approach often begins with studies “in

silico” – i.e., computer modeling.• Protein folding and stability issues make this

approach more difficult.• And the cellular environment may provide

complications not apparent in the computer modeling.

End Chapter 13Enzyme Kinetics