CHAPTER 1:
ENZYME KINETICS AND
APPLICATIONS
SEM 1 2012/13 ERT 317 BIOCHEMICAL ENGINEERING
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
 Quizzes
-2
-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.
Kinetics of Enzyme Catalyzed
C1.1
Reactions
Week 1 (10 - 21 Sept 2012)
Reading assignment:
1. Chapter 3, Bioprocess Engineering basic
Concepts. Shuler and Kargi (Main)
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

a)
b)
X-Ray structure of HEW
lysozyme.
The polypeptide chain
with a bound (NAG)6
substrate (green).
A ribbon diagram
highlighting the protein’s
secondary structure.
Note: catalytic residue
Glu35 (yellow)
Asp52 (yellow)
X-Ray structure of HEW lysozyme.
A computer-generated model showing the protein’s molecular
envelope (purple) and Ca backbone (blue).
Lysozyme - Function


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
Substrate
Products
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:


Orientation 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
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

LOCK-ANDKEY
INDUCED FIT
The Conformational Change Induced in
Hexokinase by the Binding of a substrate, DGlucose
BINDING
CLEFT
CLEFT
CLOSES
Enzyme Kinetics


Mathematical models of single-substrate, enzymecatalyzed 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
k1
k2
E  S 
ES 
EP

It is assumed that:



k 1
(3.1)
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
Rate of Reaction as a Function of
Substrate Concentration
Mechanistic Models for Simple Enzyme
Kinetics

The rate of product formation is:
d P
v
 k 2 ES 
dt
 Where
(3.2)
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
Mechanistic Models (cont’d)


The rate of variation of the ES complex is:
d ES 
 k1 E S   k1 ES   k2 ES 
dt
And since the enzyme is not consumed:
E   E0   ES 

(3.3)
(3.4)
At this point, an assumption is required in order to
achieve an analytical solution
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: ' k 1 E S 
Km 
k1

ES 
(3.5)
Since E   E0   ES  if the enzyme is conserved

E0 S 
ES  
(3.6)
k1 k1   S 

E0 S 
ES   '
K m  S 
(3.7)
The Rapid Equilibrium Assumption


Substitution Eq 3.7 into Eq 3.2 yields:

E0 S 
Vm S 
d P
v
 k2 '
 '
(3.8)
dt
K m  S  K m  S 
Where Vm  k2 E0 and Vm is the maximum forward
rate of the reaction

Vm changes with the addition of additional enzyme, but not
additional substrate
'
 K m is called the Michaelis-Menten constant, and the prime(‘)
indicates that it was derived assuming rapid equilibrium
 A low value of K ' suggests that the enzyme has a high
m
affinity for the substrate
'
V
 K m corresponds to the [S], such that K '  m
m
2
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 S0   E0 , then d ES 
dt


0
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:
k1 E S 
ES  
k 1  k2
(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
k1 E0   ES S 
(3.10)
ES  

k1  k2
Solving Eq 3.10 for [ES]

E0 S 
ES   k  k
1
2
 S 
(3.11)
k1

Substituting Eq 3.11 into Eq 3.2
d P 
k 2 E S 
v

k 1  k 2
dt
 S 
k1
(3.12a)
The Quasi-Steady-State Assumption


Therefore:
Vm S 
v
K m  S 
Where:
Km

k 1  k 2 

,
k1
(3.12b)
and
Vm  k 2 E0 

Eq 3.12b is the classic Michaelis-Menten equation
for single-substrate enzyme kinetics
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 initialrate 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
Vm S 
v
K m  S 
(3.12b)
1 1 Km 1



v Vm Vm S 
(3.13)
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]

Lineweaver-Burk Plot
Lineweaver-Burk Plot with Actual
Experimental Data Sets
Eadie-Hofstee Plot

Eq 3.12b can be arranged as:
v
v  Vm  K m
S 


(3.14)
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
Eadie-Hofstee Plot
Eadie-Hofstee Plot with Actual
Experimental Data Sets
Hanes-Woolf Plot

Rearrangement of Eq 3.12b yields:
S   K m 
v


Vm
1
S 
Vm
(3.15)
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
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:

S0 
Vmt  S 0   S   K m ln
S 

S 0   S  K m S 0 
Vm 

ln
S 
t
t

(3.16)
(3.17)
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
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:
Vm S 
 d S 
v
 "
n
dt


Km  S
n



(3.18)
Where n = cooperativity coefficient and n>1 indicates
positive cooperativity (=activator; n<1=inhibitor)
The cooperativity coefficient can be determined by
rearranging 3.18:
v
(3.19)
ln
 n lnS   ln K m"
Vm  v
And by plotting ln v/(Vm-v) versus ln [S]
Allosteric Enzymes
Graphical Determination of the
Cooperativity Coefficient, n