CHAPTER 1:

ENZYME KINETICS AND

APPLICATIONS

ERT 317 BIOCHEMICAL ENGINEERING SEM 1 2012/13

Course details

Credit hours/Units : 4

Contact hours : 3 hr (L), 3 hr (P) and 1 hr (T) per week

Evaluations

Final Exam – 50%

Midterm Tests – 20%

Course works – 30%

Laboratories – 15%

Assignments – 15%

CARRY MARKS – 50%

Course details

Course Outcome (COs) will be covered:

CO1 – Ability to develop enzyme reactions based on

its kinetics study and applied catalysis

Course works (Overall evaluations)

Assignments - 2

Quizzes -1

Midterm test – 1

Class participations – Max. of 3 points

Important reminder

Attendance should not less than 80%, or else you will be barred from taking final examination.

Plagiarism and copying other students’ work is strictly prohibited especially in doing assignments and lab reports, or else both parties will get zero.

Cheating in quizzes and examinations is also prohibited, or else both parties will get zero.

Therefore, study hard and smart. Take note of the important chapters or things that will be highlighted throughout lectures.

Week 1 (10 - 21 Sept 2012)

Reading assignment:

1. Chapter 3, Bioprocess Engineering basic

Concepts. Shuler and Kargi (Main)

Kinetics of Enzyme Catalyzed

Reactions C1.1

Outline

Introduction to enzymes

Enzyme structure

Enzyme function

Enzyme kinetics

Michaelis-Menten Kinetics

The Rapid Equilibrium Assumption

The Quasi-Steady-State Assumption

Enzymes

Enzymes are usually proteins

Typically high molecular weight (15kDa – several

million kDa)

Over 2000 enzymes have been identified

Often named by adding the suffix ‘ase’ to the name of

substrate acted upon, or the reaction catalyzed such as

urease, alcohol dehydrogenase

Catalytic function – very specific and effective

The majority of cellular reactions are catalyzed by

enzymes

Enzyme Specificity

Absolute specificity – the enzyme will catalyze only one reaction

Group specificity – the enzyme will act only on molecules that have specific functional groups, such amino, phosphate or methyl groups

Linkage specificity – the enzyme will act on a particular type of chemical bond regardless of the rest of the molecular structure

Stereochemical specificity – the enzyme will act on a particular steric or optical isomer

Enzyme Structure

Some enzymes require a non-protein group for their

activity

Co-factors: metal and other chemical ions, such as

Mg2+, Zn 2+, Mn2+, Fe2+, Fe3+, Ca2+, K+

Co-enzymes: complex organic molecules such as NAD,

FAD, CoA, or some vitamins

Enzyme that contains a non-protein group is called

holoenzyme, the protein part of the holoenzyme is

called apoenyzme:

Apoenzyme + Co-factor = Holoenzyme

Enzyme Function

Enzymes lower the activation energy of reaction

catalyzed

They do this by binding to the substrate of the

reaction, and forming an enzyme-substrate (ES)

complex

Substrate binds to a specific site on the enzyme

called the active site

Multi-substrate reactions possible

‘Lock and key’ model

Lysozyme - Structure

The first enzyme structure to solved by X-ray

crystallography

Monomer of 14.9kDa

5 helices and a 3 stranded antiparallel sheet

Deep, long binding cleft, sufficient for

hexasaccaride – open at the ends

Catalytic residue Glu35 & Asp52

X-Ray structure of HEW

lysozyme.

a) The polypeptide chain

with a bound (NAG)6

substrate (green).

b) A ribbon diagram

highlighting the protein’s

secondary structure.

X-Ray structure of HEW lysozyme.

A computer-generated model showing the protein’s molecular

envelope (purple) and Ca backbone (blue).

Note: catalytic residue

Glu35 (yellow)

Asp52 (yellow)

Lysozyme - Function

Substrate

Products

Lysozyme catalyzes the hydrolysis of the b (1->4) glycosidic

bonds in bacterial cell wall peptidoglycans and chitin (fungal

cell walls)

Found in egg white, tears and mucus membranes, bacterial

viruses

Enzyme-Substrate Complex

Activation Energy

Potential-energy curves for the reaction of

substrate, S, to products, P.

Comparison of activation energies in the uncatalyzed

and catalyzed decompositions of ozone.

Enzyme-Substrate Binding

Proximity effect:

In multi-substrate enzyme-catalyzed reactions, enzymes can hold substrates such that reactive regions of substrates are close to each other and to enzyme’s active site

Orientation effect:

Enzymes may hold the substrates at certain positions or angles to improve the reaction rate

Induced fit:

In some cases, formation of the ES complex causes slight changes in the 3D shape of the enzyme

May contribute to catalytic activity of the enzyme

INDUCED FIT

LOCK-AND-

KEY

The Conformational Change Induced in

Hexokinase by the Binding of a substrate, D-

Glucose

BINDING

CLEFT

CLEFT

CLOSES

Enzyme Kinetics

Mathematical models of single-substrate, enzyme-

catalyzed reactions were first developed by Henri in

1902 and Michaelis & Menten in 1913

Simple enzyme kinetics are now commonly referred to

as Michaelis-Menten or ‘saturation’ kinetics

At high substrate concentrations, all active sites on the

enzyme are occupied by substrate – enzyme is saturated

Models are based on data from batch reactors with

constant liquid volume in which the initial substrate, [S0],

and enzyme, [E0], concentrations are known

Single-Substrate Enzyme Kinetics

It is assumed that:

The ES complex is established very rapidly

The rate of the reverse reaction of the second step is negligible (i.e k-2~0)

Assumption 2 is typically only valid when product (P) accumulation is negligible, at the beginning of the reaction

PEESSEk

k

k

2

1

1(3.1)

Rate of Reaction as a Function of

Substrate Concentration

Mechanistic Models for Simple Enzyme

Kinetics

The rate of product formation is:

Where v is the rate of product formation or substrate

consumption in moles/L-s

The rate constant k2 is often denoted as kcat in

biological literature

ESk

dt

Pdv 2 (3.2)

Mechanistic Models (cont’d)

The rate of variation of the ES complex is:

And since the enzyme is not consumed:

At this point, an assumption is required in order to

achieve an analytical solution

ESkESkSEk

dt

ESd211

ESEE 0

(3.3)

(3.4)

The Rapid Equilibrium Assumption

Assuming equilibrium in the first part of the reaction

(E+S forms ES), we can use the equilibrium

coefficient to express [ES] in terms of [S]

The equilibrium constant is:

Since if the enzyme is conserved

ES

SE

k

kKm

1

1'

ESEE 0

SK

SEES

Skk

SEES

m

'

0

11

0

(3.5)

(3.6)

(3.7)

The Rapid Equilibrium Assumption

Substitution Eq 3.7 into Eq 3.2 yields:

Where and is the maximum forward rate of the reaction

changes with the addition of additional enzyme, but not additional substrate

is called the Michaelis-Menten constant, and the prime(‘) indicates that it was derived assuming rapid equilibrium

A low value of suggests that the enzyme has a high affinity for the substrate

corresponds to the [S], such that

SK

SV

SK

SEk

dt

Pdv

m

m

m

''

02

02 EkVm mV

mV

'

mK

'

mK

'

mK2

' mm

VK

(3.8)

The Quasi-Steady-State Assumption

The assumption of rapid equilibrium is often not valid

The QSSA assumes that if the initial substrate

concentration greatly exceeds the initial enzyme

concentration , then

Computer simulations show that the QSSA holds, in a

closed system, after a brief transition period while the

reaction is initiated and equilibrium achieved

Applying the QSSA to Eq 3.3 gives us:

00 ES 0

dt

ESd

21

1

kk

SEkES

(3.9)

Formation of [ES] and Initiation of

Steady State

The Quasi-Steady-State Assumption

Substituting the enzyme conservation Eq 3.4 into Eq

3.9 yields

Solving Eq 3.10 for [ES]

Substituting Eq 3.11 into Eq 3.2

21

01

kk

SESEkES

Sk

kk

SEES

1

21

0

Sk

kk

SEk

dt

Pdv

1

21

2

(3.10)

(3.11)

(3.12a)

The Quasi-Steady-State Assumption

Therefore:

Where:

Eq 3.12b is the classic Michaelis-Menten equation for single-substrate enzyme kinetics

SK

SVv

m

m

02

1

21 and ,

EkV

k

kkK

m

m

(3.12b)

Outline

Simple enzyme kinetics

Complex enzyme kinetics

Allosteric enzymes

Inhibited enzyme kinetics

Competitive

Noncompetitive

Uncompetitive

Course details

Course Outcome (COs) will be covered:

CO1 – Ability to develop enzyme reactions based on

its kinetics study and applied catalysis

Course works (Overall evaluations)

Assignments 1 (Due Wed, 19/09)

Quizzes 1 (Wed, 19/09)

Midterm test – 1

Class participations – Max. of 3 points

Experimental Determination of

Michaelis-Menten Parameters

Determination of values for Km and Vm with high precision can be difficult

Experimental data are typically obtained from initial-rate experiments

Batch reactor charged with a known amount of substrate [S0] and enzyme [E0]

Product and/or substrate concentration plotted against time

Create many plots at different [S0] and enzyme [E0] and use to generate a plot as Figure 3.1

Cumbersome method of determining Km and Vm, therefore after methods have been developed

Lineweaver-Burk Plot

Eq 3.12b can be linearized in double-reciprocal form

A plot of 1/v versus 1/[S] yields a line with a slope of Km/Vm and a y-intercept of 1/Vm

Give good estmates of Vm but not necessarily Km

Data points at low substrate concentrations influence the slope and intercept more than data points at high [S]

SV

K

Vv

SK

SVv

m

m

m

m

m

111

(3.12b)

(3.13)

Lineweaver-Burk Plot

Lineweaver-Burk Plot with Actual

Experimental Data Sets

Eadie-Hofstee Plot

Eq 3.12b can be arranged as:

A plot of v versus v/[S] results in a line with slope –Km,

and a y-intercept of Vm

Eadie-Hofstee plots can be subjected to large errors,

since both coordinates contain v, but there is less bias on

points at low [S] than with Lineweaver-Burk plots

S

vKVv mm

(3.14)

Eadie-Hofstee Plot

Eadie-Hofstee Plot with Actual

Experimental Data Sets

Hanes-Woolf Plot

Rearrangement of Eq 3.12b yields:

A plot of [S]/v versus [S] results in a line of slope

1/Vm with a y-intercept of Km/Vm

This plot is used to determine Vm more accurately

than the previous two plots

S

VV

K

v

S

mm

m 1

(3.15)

Hanes-Woolf Plot

Hanes-Woolf Plot with Actual

Experimental Data Sets

Batch Kinetics

The time course of variation of [S] in a batch enzymatic reaction can be determined by integrating equation 3.12b to yield:

A plot 1/t (ln[S0]/[S]) versus {[S0]-[S]}/t results in a line of slope -1/Km with a y-intercept of Vm/Km

S

S

t

K

t

SSV

S

SKSStV

mm

mm

00

00

ln

ln

(3.16)

(3.17)

Complex Enzyme Kinetics: Allosteric

Enzymes

Allosteric enzymes:

Some enzymes posses more than one substrate binding

site

The binding of one substrate molecule to the enzyme

facilitates binding of other substrate molecules

This is known as allostery or cooperative binding

Often seen in regulatory enzymes

Allosteric Enzymes

Allos -other, steros –shape

The rate expression for allosteric enzymes is:

Where n = cooperativity coefficient and n>1 indicates positive cooperativity (=activator; n<1=inhibitor)

The cooperativity coefficient can be determined by rearranging 3.18:

And by plotting ln v/(Vm-v) versus ln [S]

nm

n

m

SK

SV

dt

Sdv

"

"lnlnln m

m

KSnvV

v

(3.18)

(3.19)

Allosteric Enzymes

Graphical Determination of the

Cooperativity Coefficient, n