Enzyme Kinetics

9/1/2004

General Properties of Enzymes

•Increased reaction rates sometimes 106 to 1012 increase

Enzymes do not change G just the reaction rates.

•Milder reaction conditions

•Great reaction specificity

•Can be regulated

Substrate specificityThe non-covalent bonds and forces are maximized to bind substrates with considerable specificity

•Van der Waals forces

•electrostatic bonds (ionic interactions)

•Hydrogen bonding

•Hydrophobic interaction

•Stereospecificity

•Geometrically specific

A + B P + Q

Substrates Products

enz

Enzymatic catalysis and mechanisms

•A. Acid - Base catalysis•B. Covalent catalysis•C. Metal ion aided catalysis•D. Electrostatic interactions•E. Orientation and Proximity effects•F. Transition state bindingGeneral Acid BaseRate increase by partial proton abstraction by a Bronsted base orRate increase by partial proton donation by a Bronsted Acid

Enzyme Kinetics

Rates of Enzyme Reactions How fast do reactions take place

•Reaction rates

Thermodynamics says I know the difference between state 1 and state 2 and G = (Gf - Gi)

But Changes in reaction rates in response to differing

conditions is related to path followed by the reaction and

is indicative of the reaction mechanism!!

Chemical kinetics and Elementary Reactions

A simple reaction like A B may proceed through several elementary reactions like A I1 I2 B Where I1 and I2 are intermediates.

The characterization of elementary reactions comprising an overall reaction process constitutes its mechanistic description.

Rate Equations

Consider aA + bB + • • • + zZ. The rate of a reaction is proportional to the frequency with which the reacting molecules simultaneously bump into each other

zba ZBAk Rate

The order of a reaction = the sum of exponentsGenerally, the order means how many molecules have to bump into each other at one time for a reaction to occur.

A first order reaction one molecule changes to another

A B

A second order reaction two molecules react

A + B P + Q

or

2A P

3rd order rates A + B + C P + Q + R rarely occur

and higher orders are unknown.

Let us look at a first order rate

A B

dtPd

dtAd v

= velocity of the reaction

in Molar per min.

or

moles per min per volume

k = the rate constant of the reaction

AdtAd kv

Instantaneous rate: the rate of reaction at any specified time point that is the definition of the derivative.

We can predict the shape of the curve if we know the order of the reaction.

A second order reaction: 2A P

2AA kdtdv

Or for A + B P + Q

BABA kdtd

dtdv

Percent change in A (ratio ) versus time in first and second order reactions

It is difficult to determine if the reaction is either first or second order by directly plotting changes in

concentration.

AdtAd k

dtd kAA

t

0

A

A

dtk-AA

o

d to kAlnAln

-kto eA A

However, the natural log of the concentration is directly proportional to the time.

- for a first order reaction-

The rate constant for the first order reaction has units of s-1 or min-1 since velocity = molar/sec

and v = k[A] : k = v/[A]

Gather your data and plot ln[A] vs time.

The half-life of a first order reaction

2

A A oPlugging in

to rate equation

21

o

A2

A

ln kt

kk693.02lnt

21

The half-life of a first order reaction is the time for half of the reactant which is initially present to decompose or react.32P, a common radioactive isotope, emits an energetic particle and has a half-life of 14 days. 14C has a half life of 5715 years.

A second order reaction such like 2A P

t

dtk0

A

oA2o

AAd-

ktoA

1A1

When the reciprocal of the concentration is plotted verses time a second order reaction is characteristic of a straight line.

The half-life of a second order reaction is

and shows a dependents on the initial concentration o2

1 A1t

k

The Transition State

A bimolecular reaction A + B C A B + C at some point in the reaction coordinate an intermediate ternary complex will exist

A B C

This forms in the process of bond formation and bond breakage and is called a transition state.

Ha + Hb Hc Ha Hb + Hc

This is a molecule of H2 gas reforming by a collision

An energy contour of the hydrogen reaction as the three molecules approach the transition state at location c. This is called a saddle point and has a higher energy than the starting or ending point.

Energy diagrams for the transition state using the hydrogen molecule

Transition state diagram for a spontaneous reaction. X‡ is the symbol for the species in the transition state

Xk'BAkdtPd

For the reaction

‡ Where [X] is the concentration of the transition state species

BA

X K ‡‡

G RTlnK - ‡ ‡

G‡ is the Gibbs free energy of the activated complex.

k' = rate constant for the decom-position of the activated complex

BAek'tP RT

G-

dd

‡The greater the G‡, the more unstable the transition

state and the slower the reaction proceeds.

This hump is the activation barrier or kinetic barrier for a reaction.

The activated complex is held together by a weak bond that would fly apart during the first vibration of the bond and can be expressed by k' =where is the vibrational frequency of the bond that breaks the activated complex and is the probability that it goes towards the formation of products.

Now we have to define . E = h and = E/h where h is Planks constant relating frequency to Energy. Also through a statistical treatment of a classical oscillator E= KbT where Kb is Boltzmann constant.

By putting the two together

hTK k b

RTG

b

hTK k

eAnd

The rate of reaction decreases as its free energy of activation, G‡ increases

or the reaction speeds up when thermal energy is added

‡

Multi-step reactions have rate determining steps

Consider PA 21 kk IIf one reaction step is much slower than all the rest this step acts as a “bottleneck” and is said to be the rate-limiting step

Catalysis lowers the activation energy

Preferential transition state bindingBinding to the transition state with greater affinity to

either the product or reactants.

RACK MECHANISM

Strain promotes faster rates

The strained reaction more closely resembles the transition state and interactions that preferentially

bind to the transition state will have faster rates

EP ES

P S

E

N

k

k

kN for uncatalyzed reaction

and

kE for catalyzed reaction

EP ES ES

K K E P E SS E

E

N

KTR

K

‡ ‡

‡ ‡

SEES

KR SE

ES KT

‡

‡

‡‡

ES

ES KE

‡ ‡ N

E

R

TKK

ES S ESS

KK

‡

‡

‡

‡

RTGG EN

expkk

N

E106 rate enhancement requires a 106 higher affinity which is 34.2 kJ/mol

The enzyme binding of a transition state (ES‡ ) by two hydrogen bonds that cannot form in the Michaelis Complex (ES) should result in a rate enhancement of 106 based on this effect alone

Catalysts act to lower the activation barrier of the reaction being catalyzed by the enzyme.

Where G‡cat = G‡

uncat- G‡cat

The rate of a reaction is increased by RTGcate

G‡cat = 5.71 kJ/mol is a ten fold increase in rate.

This is half of a hydrogen bond!!

G‡cat = 34.25 kJ/mol produces a million fold

increase in rate!!

Rate enhancement is a sensitive function of G‡cat

Kinetics of EnzymesEnzymes follow zero order kinetics when substrate

concentrations are high. Zero order means there is no increase in the rate of the reaction when more substrate is added.

Given the following breakdown of sucrose to glucose and fructose

Sucrose + H20 Glucose + Fructose

O

H

HO

H

HO

H

OH

OHHH

OH

OH

HOH

H OH

O

HH

HO

H

H

H

OH

P EES S E 21

1-

kk

k

E = Enzyme S = Substrate P = Product

ES = Enzyme-Substrate complex

k1 rate constant for the forward reaction

k-1 = rate constant for the breakdown of the ES to substrate

k2 = rate constant for the formation of the products

When the substrate concentration becomes large enough to force the equilibrium to form completely all ES the second step in the reaction becomes rate limiting because no more ES can be made and the enzyme-substrate complex is at its maximum value.

ESP2kdt

dv [ES] is the difference between the rates of ES formation minus the rates of its disappearance.

ESESSEES211 kkk

dtd

1

Assumption of equilibriumk-1>>k2 the formation of product is so much

slower than the formation of the ES complex. That we can assume:

ES

SE

1

1

kkK s

Ks is the dissociation constant for the ES complex.

Assumption of steady stateTransient phase where in the course of a reaction the concentration of ES does not change

0ES

dtd

2

ES E E T 3

Combining 1 + 2 + 3

ESk k SES-Ek 21-T1

SEk Sk k kES T1121-

S KSE ES T

M1

21-

kk k K

M

rearranging

Divide by k1 and solve for [ES] Where

SKSEESP T2

20

Mto

kkdtdv

vo is the initial velocity when the reaction is just starting out.

And is the maximum velocity T2max Ek V

SKSVmax

Mov

The Michaelis - Menten equation

The Km is the substrate concentration where vo equals one-half Vmax

The KM widely varies among different enzymes

The KM

can be expressed as:1

2

1

2

1

1 KKkk

kk

kk

sM

As Ks decreases, the affinity for the substrate increases. The KM can be a measure for substrate affinity if k2<k-1

There are a wide range of KM, Vmax , and efficiency seen in enzymes

But how do we analyze kinetic data?

The double reciprocal plot

maxmax

M

V1

S1

VK1

ov

Lineweaver-Burk plot: slope = KM/Vmax,

1/vo intercept is equal to 1/Vmax

the extrapolated x intercept is equal to -1/KM

For small errors in at low [S] leads to large errors in 1/vo

Tmax

EV

catkkcat is how many reactions an enzyme can catalyze per second

The turnover number

For Michaelis -Menton kinetics k2= kcat

When [S] << KM very little ES is formed and [E] = [E]T

and SEKkSE

Kk

M

catT

M

2 ov

Kcat/KM is a measure of catalytic efficiency

What is catalytic perfection?

When k2>>k-1 or the ratio 21

21

kkkk

is maximum

Then1

MKkkcat

Or when every substrate that hits the enzyme causes a reaction totake place. This is catalytic perfection.

Diffusion-controlled limit- diffusion rate of a substrate is in the range of 108 to 109 M-1s-1. An enzyme lowers the transition state so there is no activation energy and the catalyzed rate is as fast as molecules collide.

Kinetic data cannot unambiguously establish a reaction mechanism.

Although a phenomenological description can be obtained the nature of the reaction intermediates remain indeterminate and other independent measurements are needed.

Reaction MechanismsA: Sequential Reactions

• All substrates must combine with enzyme before reaction can occur

Bisubstrate reactions

Random Bisubstrate Reactions

Ping-Pong Reactions

• Group transfer reactions• One or more products released before all

substrates added

Inhibition kinetics

There are three types of inhibition kinetics competitive, mixed and uncompetitive.•Competitive- Where the inhibitor competes with the substrate.

Competitive Inhibition

SK

SV

M

max

ov

IKI1

EI

IEK I

HIV protease inhibitors

Competitive Inhibition: Lineweaver-Burke Plot

Uncompetitive Inhibition

Uncompetitive Inhibition: Lineweaver-Burke Plot

Mixed inhibition

Mixed inhibition is when the inhibitor binds to the enzyme at a location distinct from the substrate binding site. The binding of the inhibitor will either alter the KM or Vmax or both.

EI

IEK I ESI

IESK I

SK

SV

M

max

ov

IK

I1

The effect of pH on kinetic parameters