Reaction kinetics: 1 st order reactions [A] t Decay reactions, like radio-activity; S N 1 reactions Rate: - Rewriting: - Integration gives: So: ln[A] t.

Reaction kinetics: 1st order reactions•

•••

• • • • • • •

[A]

t

k1A B (+ C)

Decay reactions, like radio-activity;SN1 reactions

[A]k=dt

d[A]1Rate: -

Rewriting: - dtk=[A]

d[A]1

Integration gives: t

0

t

0

ktdtd[A][A]

1

So: ln[A]t – ln[A]0 = -kt or: =]A[

]A[ln

0

t -kt

t 1/2 = ln2/k = 0.693/k

The time for half of the reactant initially present to decompose, its half-time or half-life, t1/2 , is a constant and hence independent of the initial concentration of reactant.

By substituting the relationship [A] = [A0] / 2 when t = t1/2 into ln [A]=ln [A]0 - ktand rearranging:

The half-time for a second order reaction is expressed t 1/2 = 1/k [A]0 and therefore, in contrast to a

first order reaction depends on the initial reactant concentration.

Second-order reaction 2A P

A+B P

Here, the reaction is said to be first order in A and first order in B.

Unimolecular and bimolecular reactions are common. Termolecular reactions are unusual because the simultaneous collision of three molecules is a rare event. Fourth and higher order reactions are unknown.

Sucrose + H2O glucose + fructose

Enzyme Kinetics

ß-fructofuranosidase:

When [S] » [E] : the rate is zero order with respect to sucrose

The Michaelis-Menten Equation

This equation cannot be explicitly integrated, however, without simplifying assumptions, two possibilities are

1. Assumption of equilibrium. Leonor Michaelis and Maud Menten, building on the work of Victor Henri, assumed that k-1 » k2, so that the first step of the reaction reaches equilibrium.

Ks is the dissociation constant of the first step in the enzymatic reaction

The Michaelis-Menten Equation

1. Assumption of steady-state. Figure illustrates the progress curves of the various participants in reaction

under the physiologically common conditions that substrate is in great excess over Enzyme ([S] » [E]).

ES maintains a steady state and [ES] can be treated as having a constant value:

The so called steady state assumption, a more general condition than that of equilibrium, was first proposed in 1925 by G. E. Briggs and B. S. Haldane

The Michaelis constant, KM , is defined as

Letting [E] = [E]T - [ES] and rearranging yields

The Michaelis-Menten Equation

Solving for [ES],

The Michaelis-Menten Equation

The expression of the initial velocity (v0) of the reaction, the velocityat t=0, thereby becomes

The maximal velocity of a reaction, Vmax occurs at high substrate concentrations when the enzyme is saturated, that is, when it is entirely in the ES form

Therefore, combining the last two equations, we obtain:

This expression, the Michaelis-Menten equation, is the basic equation of enzyme kinetic.

Significance of the Michaelis Constant

The Michaelis constant, KM, has a simple operational definition. At the substrate concentration at which [S] = KM, this equation

yields v0 = Vmax/2 so that

KM is the substrate concentration at which the reaction velocity is half maximal

Analysis of Kinetic Data

Lineweaver-Burk or double-reciprocal plot

S >> Kmvi=VmaxVmax= k2Et

Vmax= 10 M/seg Km=10 x10-5 MSi en el ensayo se usaron 5mg/L de preparación enzimática, entonces:v= Vmax = k2 ET k2= 10/5 = 2 moles/mg seg

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Significance of the Michaelis Constant

The magnitude of KM varies widely with the identity of the enzyme and the nature of the substrate. It is also a function of temperature and pH. The Michaelis constant can be expressed as

Since Ks is the dissociation constant of the Michaelis complex, as Ks decreases, the enzyme’s affinity for substrate increases. KM in therefore also a measure of the affinity of the enzyme for its substrate, provided k2/k1 is small compared to Ks, that is k2 ‹ k-1 so that the ES P reaction proceeds more slowly than ES reverts to E + S

kcat/KM Is a Measure of Catalytic Efficiency

We can define the catalytic constant, kcat, of an enzyme as

This quantity is also known as the turnover number of an enzyme because it is the number of reaction processes (turnovers) that each active site catalyzes per unit time.

Turn Over Numbers of Enzymes

Catalase H2O2

Carbonic anhydrase HCO3-

Acetylcholinesterase Acetylcholine

40,000,000

400,000

140,000

-Lactamase Benzylpenicillin 2,000

Fumarase Fumarate 800

RecA protein (ATPase) ATP 0.4

Enzymes Substrate kcat (s-1)

The number of product transformed from substrate by one enzyme molecule in one second

Adapted from Nelson & Cox (2000) Lehninger Principles of Biochemistry (3e) p.263

kcat/KM Is a Measure of Catalytic Efficiency

When [S] « KM, very little ES is formed. Consequently, [E] ≈ [E]T, so

reduces to a second-order rate equation:

The quantity kcat/KM is a measure of an enzyme’s catalytic efficiency.

There is an upper limit to the value of kcat/KM : It can be not greater than k1; that is, the decomposition of ES to E + P can occur no more frequently than E and S come together to form ES. The most efficient enzymes have kcat/KM values near to the diffusion-controlled limit of 108 to 109 M-1.s-1

Chymotrypsin Has Distinct kcat / Km to Different Substrates

O R O

H3C–C–N–C–C–O–CH3

H H

= – =––

–HGlycine

kcat / Km

1.3 ╳ 10-1

–CH2–CH2–CH3Norvaline 3.6 ╳ 102

–CH2–CH2–CH2–CH3Norleucine 3.0 ╳ 103

–CH2–Phenylalanine 1.0 ╳ 105

(M-1 s-1)

R =

Adapted from Mathews et al (2000) Biochemistry (3e) p.379

- dS/dt = vi = So dX/dt

Al iniciar: t = 0, S = So

A cualquier tiempo:T = t S = S X = (So-S)/So

• As is the case with most reactions, an increase in temperature will result in an increase in kcat for an enzymatic reaction.

• From general principles, it can be determined that the rate of any reaction will typically double for every 10°C increase in temperature.

• Many enzymes display maximum temperatures around 40°C, which is relatively close to body temperature.

• There are enzymes that are isolated from thermophilic organisms that display maxima around 100°C, and some that are isolated from psychrophilic organisms that display maxima around 10°C.

Temperature Dependence of Enzymes

Enzyme Inhibition (Mechanism)

I

I

S

S

S I

I

I II

S

Competitive Non-competitive Uncompetitive

EE

Different siteCompete for

active siteInhibitor

Substrate

Ca

rtoo

n G

uid

eEq

uatio

n an

d De

scrip

tion

[II] binds to free [E] only,and competes with [S];increasing [S] overcomesInhibition by [II].

[II] binds to free [E] or [ES] complex; Increasing [S] cannot overcome [II] inhibition.

[II] binds to [ES] complex only, increasing [S] favorsthe inhibition by [II].

E + S → ES → E + P + II↓EII

←

↑

E + S → ES → E + P + + II II↓ ↓EII + S →EIIS

←

↑ ↑

E + S → ES → E + P + II ↓ EIIS

←

↑

EI

S X

Juang RH (2004) BCbasics

Competitive Inhibition

Succinate Glutarate Malonate Oxalate

Succinate Dehydrogenase

Substrate Competitive InhibitorProduct

Adapted from Kleinsmith & Kish (1995) Principles of Cell and Molecular Biology (2e) p.49

C-OO-

C-H C-H C-OO-

C-OO-

H-C-H H-C-H C-OO-

C-OO-

H-C-H H-C-H H-C-H C-OO-

C-OO-

C-OO-

C-OO-

H-C-H C-OO-

Sulfa Drug Is Competitive Inhibitor

-COOHH2N-

-SONH2H2N-

PrecursorFolicacid

Tetrahydro-folic acid

SulfanilamideSulfa drug (anti-inflammation)

Para-aminobenzoic acid (PABA)

Bacteria needs PABA for the biosynthesis of folic acid

Sulfa drugs has similar structure with PABA, andinhibit bacteria growth.

Adapted from Bohinski (1987) Modern Concepts in Biochemistry (5e) p.197

Domagk (1939)

Enzyme Inhibition

Competitive Inhibition

Many substances alter the activity of an enzyme by reversibly combining with it in a way what influence the binding of substrate and/or its turnover number. Substances that reduce an enzyme’s activity in this way are known as inhibitors

A substance that competes directly with a normal substrate for an enzyme’s substrate-binding site is known as a competitive inhibitor.

Here it is assumed that I, the inhibitor, bind reversibly to the enzyme and is in a rapid equilibrium with it so that

And EI, the enzyme-inhibitor complex, is catalytically inactive. A competitive inhibitor therefore reduces the concentration of free enzyme available for substrate binding.

Enzyme Inhibition

This is the Michaelis-Menten equation that has been modified by a factor, , which is defined as

Competitive Inhibition

Is a function of the inhibitor’s concentration and its affinity for the enzyme. It cannot be less than 1.

Enzyme Inhibition

Competitive Inhibition

Recasting in the double-reciprocal form yields

A plot of this equation is linear and has a slope of KM/Vmax, a 1/[S] intercept of -1/ KM, and a 1/v0 intercept of 1/ Vmax

Enzyme Inhibition

Uncompetitive Inhibition

In uncompetitive inhibition, the inhibitor binds directly to the enzyme-substrate complex but not to the free enzyme

In this case, the inhibitor binding step has the dissociation constant

The uncompetitive inhibitor, which need not resemble the substrate, presumably distorts the active site, thereby rendering the enzyme catalytically inactive.

Enzyme Inhibition

Uncompetitive Inhibition

The double-reciprocal plot consists of a family of parallel lines with slope KM/Vmax, 1/v0 intercepts of ’/Vmax and 1/[S] intercept of -’/KM

Enzyme InhibitionMixed Inhibition (noncompetitive inhibition)

A mixed inhibitor binds to enzyme sites that participate in both substrate binding and catalysis. The two dissociation constants for inhibitor binding

Double-reciprocal plots consist of lines that have the slope KM/Vmax, with a 1/v0 intercept of ’/Vmax and 1/[S] intercept of -’/ KM

Km

Enzyme Inhibition (Plots)

I II Competitive Non-competitive Uncompetitive

Dir

ect

Plo

tsD

ou

ble

Rec

ipro

cal

Vmax Vmax

Km Km’ [S], mM

vo

[S], mM

vo

II II

Km [S], mM

Vmax

II

Km’

Vmax’Vmax’

Vmax unchangedKm increased

Vmax decreasedKm unchanged

Both Vmax & Km decreased

II

1/[S]1/Km

1/vo

1/ Vmax

II

Two parallellines

II

Intersect at X axis

1/vo

1/ Vmax

1/[S]1/Km 1/[S]1/Km

1/ Vmax

1/vo

Intersect at Y axis

= Km’

Juang RH (2004) BCbasics

Bisubstrate Reactions

Almost all of these so called bisubstrate reactions are either transferase reactions in which enzyme catalyzed the transfer of a specific functional group, X, from one of the substrates to the other:

or oxidation-reduction reactions in which reducing equivalents are transferred between two substrates.

Sequential Reactions

Reactions in which all substrates must combine with the enzyme before a reaction can occur and products be released are known as Sequential reactions

Sequential Reactions

Ordered bisubstrate reaction

Random bisubstrate reaction

A and B : substrates in order that they add to the enzymeP and Q : products in order that they leave the enzyme

Group-transfer reactions in which one or more products are released before all substrates have been added are known as Ping Pong reactions

Ping Pong Reactions

Bisubstrate Reactions