CP504 – Lecture 4
Enzyme kinetics and associated reactor design:
Determination of
the kinetic parameters of
enzyme-induced reactions
- learn about the meaning of kinetic parameters
- learn to determine the kinetic parameters
- learn the effects of pH and temperature on reaction rates
- learn about inhibited enzyme kinetics
- learn about allosteric enzymes and their kinetics
Prof. R. Shanthini
23 Sept 2011
Simple Enzyme Kinetics (in summary)
k1
E+S
k3
ES
E+P
k2
which is equivalent to
[E]
S
Prof. R. Shanthini
23 Sept 2011
P
S
for substrate (reactant)
E
for enzyme
ES
for enzyme-substrate complex
P
for product
Simple Enzyme Kinetics (in summary)
[E]
S
rP = - r S =
P
rmaxCS
KM + CS
where rmax = k3CE0 and KM = f(rate constants)
rmax is proportional to the initial concentration of the enzyme
KM is a constant
Prof. R. Shanthini
23 Sept 2011
Simple Enzyme Kinetics (in summary)
-rs
Catalyzed reaction
rmax
- rS =
rmax
2
KM
Prof. R. Shanthini
23 Sept 2011
rmaxCS
KM + CS
uncatalyzed reaction
Cs
How to determine the kinetic parameters rmax and KM ?
Carry out an enzyme catalysed experiment, and
measure the substrate concentration (CS) with time.
t
0
Cs
50
10
45
15
41
- rs
- rS =
rmaxCS
KM + CS
From the data, we could calculate the substrate utilization
rate (-rs) as follows:
Prof. R. Shanthini
23 Sept 2011
How to determine the M-M kinetics rmax and KM ?
Carry out an enzyme catalysed experiment, and
measure the substrate concentration (CS) with time.
t
0
Cs
50
- rs
(50-45)/10
10
45
(45-41)/5
15
41
- rS =
rmaxCS
KM + CS
From the data, we could calculate the substrate utilization
rate (-rs) as follows:
Prof. R. Shanthini
23 Sept 2011
We could rearrange
- rS =
rmaxCS
KM + CS
to get the following 3 linear forms:
CS
- rS
1
- rS
- rS
Prof. R. Shanthini
23 Sept 2011
=
KM
rmax
+
1
=
=
rmax
+
rmax -
1
rmax
CS
KM
1
rmax
CS
KM
- rS
CS
(14)
(15)
(16)
The Langmuir Plot
CS
- rS
=
KM
rmax
+
1
CS
(14)
rmax
CS
- rS
1
rmax
- KM
Prof. R. Shanthini
23 Sept 2011
CS
The Langmuir Plot
CS
- rS
=
KM
rmax
+
1
CS
(14)
rmax
CS
- rS
1
rmax
- KM
Prof. R. Shanthini
23 Sept 2011
Determine rmax
more accurately
than the other
plots.
CS
The Lineweaver-Burk Plot
1
- rS
1
=
rmax
+
KM
1
rmax
CS
(15)
1
- rS
KM
rmax
1
- KM
Prof. R. Shanthini
23 Sept 2011
1
CS
The Lineweaver-Burk Plot
1
- rS
1
=
1
- rS
1
- KM
Prof. R. Shanthini
23 Sept 2011
rmax
+
KM
1
rmax
CS
(15)
- Gives good estimates of rmax, but
not necessarily KM
KM
- Data points
at low substrate
rmax
concentrations influence the slope
and intercept more than data points
at high Cs
1
CS
The Eadie-Hofstee Plot
- rS
=
rmax -
KM
- rS
CS
(16)
- rS
KM
rmax
KM
Prof. R. Shanthini
23 Sept 2011
-rS
CS
The Eadie-Hofstee Plot
- rS
=
rmax -
KM
- rS
CS
(16)
- rS
- Can be subjected to large errors
since both coordinates contain (-rS)
KM
- Less bias on point at low Csrmax
than
with Lineweaver-Burk plot K
M
Prof. R. Shanthini
23 Sept 2011
-rS
CS
Data:
CS
-rS
(mmol/l)
-(mmol/l.min)
1
0.20
2
3
0.22
0.30
5
0.45
7
0.41
10
0.50
Prof. R. Shanthini
23 Sept 2011
Determine the M-M kinetic
parameters for all the
three methods discussed
in the previous slides.
The Langmuir Plot
25
CS/(-rS) min
20
15
10
y = 1.5866x + 4.6417
R2 = 0.9497
5
0
0
2
4
6
CS (mmol/l)
8
10
rmax = 1 / slope = 1 / 1.5866 = 0.63 mmol/l.min
Prof. R. Shanthini
23 Sept 2011
KM = rmax x intercept = 0.63 x 4.6417 = 2.93 mmol/l
The Lineweaver-Burk Plot
1/(-rS) l.min/mmol
6
5
4
3
2
y = 3.4575x + 1.945
R2 = 0.8463
1
0
0
0.2
0.4
0.6
1/CS l/mmol
0.8
1
rmax = 1 / intercept = 1 / 1.945 = 0.51 mmol/l.min
Prof. R. Shanthini
23 Sept 2011
KM = rmax x slope = 0.51 x 3.4575 = 1.78 mmol/l
The Eadie-Hofstee Plot
(-rS) mmol/l.min
0.6
y = -1.8923x + 0.5386
2
R = 0.6618
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
(-rS)/CS per min
0.2
rmax = intercept = 0.54 mmol/l.min
Prof. R. Shanthini
23 Sept 2011
KM = - slope = 1.89 mmol/l
0.25
Comparison of the results
The
Langmuir
Plot
rmax
KM
R2
Prof. R. Shanthini
23 Sept 2011
The
LineweaverBurk Plot
The EadieHofstee Plot
Comparison of the results
The
LineweaverBurk Plot
0.51
The EadieHofstee Plot
rmax
The
Langmuir
Plot
0.63
KM
2.93
1.78
1.89
R2
94.9%
84.6%
66.2%
Prof. R. Shanthini
23 Sept 2011
0.54
Comparison of the results
The
LineweaverBurk Plot
0.51
The EadieHofstee Plot
rmax
The
Langmuir
Plot
0.63
KM
2.93
1.78
1.89
R2
94.9%
84.6%
66.2%
Determine
rmax more
accurately than
the other plots
Gives good
estimates of
rmax, but not
necessarily KM
Can be
subjected to
large errors
Prof. R. Shanthini
23 Sept 2011
0.54
The effects of pH and temperature on reaction rate
Most enzymes function over a broad range of pHs and
temperatures.
However, they have an optimal pH and temperature for peak
activity.
In general, enzyme activities increase with increasing
temperatures; however, as temperatures get higher, enzymes
begin to denature.
Most enzymes are also sensitive to pH.
As with temperature, the optimal pH for an enzyme depends on
the environment in which it normally functions.
Prof. R. Shanthini
23 Sept 2011
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
The effects of temperature on reaction rate
Reaction rate
Optimal for most
human enzymes
Prof. R. Shanthini
23 Sept 2011
Optimal for some
thermophillic
bacterial
enzymes
Temperature (deg C)
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
The effects of pH on reaction rate
Optimal for trypsin
(an intestinal
enzyme)
Reaction rate
Optimal for pepsin
(a stomach
enzyme)
Prof. R. Shanthini
23 Sept 2011
pH
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
Effect of shear
Prof. R. Shanthini
23 Sept 2011
Complex enzyme kinetics
- learn about inhibited enzyme kinetics
- learn about allosteric enzymes and their kinetics
Prof. R. Shanthini
23 Sept 2011
Inhibited enzyme reactions
Inhibitors are substances that slow down the rate of enzyme
catalyzed reactions.
There are two distinct types of inhibitors:
- Irreversible inhibitors form a stable complex with enzymes
and reduce enzyme activity (e.g. lead and cadmium)
- Reversible inhibitors interact more loosely with enzymes
and can be displaced.
Prof. R. Shanthini
23 Sept 2011
Inhibited enzyme reactions
Inhibitors are also classified as competitive and non-competitive
inhibitors.
Prof. R. Shanthini
23 Sept 2011
Competitive inhibition
A competitive inhibitor has a chemical and structural similarity
to the substrate.
It competes with the substrate for the position at the
active site of the enzyme.
The rate of the reaction slows down because the active site
is occupied by the competitive inhibitor, making the active site
less accessible to the substrate.
Prof. R. Shanthini
23 Sept 2011
https://ibhumanbiochemistry.wikispaces.com/C.7.5
Competitive inhibition
Competitive inhibitors (denoted by I) compete with substrate to
occupy the active site of the enzyme.
k1
E+S
ES
k3
E+P
k2
k4
E+I
EI
k5
where
rP = k3 CES
CE0 = CE + CES + CEI
Prof. R. Shanthini
23 Sept 2011
(17)
(18)
Competitive inhibition
Assuming rapid equilibrium, we get
k1 CE CS = k2 CES
KM =
k2
k1
=
CE CS
CES
(19)
k4 CE CI = k5 CEI
KI =
Prof. R. Shanthini
23 Sept 2011
k5
k4
=
CE CI
CEI
(20)
Competitive inhibition
Combining (17) to (20), we get
rP =
where
k3CE0CS
=
KM (1 + CI / KI) + CS
rmax = k3CE0
KM,app = KM (1 + CI / KI)
KM = k2 / k1
Prof. R. Shanthini
23 Sept 2011
rmaxCS
KM,app + CS
(5)
(22)
(6)
KM,app > KM
(21)
Competitive inhibition
The Lineweaver-Burk Plot
1
CI > 0
- rS
CI = 0 (no inhibitor)
1
1
- KM
- KM, app
1
rmax
1
CS
Prof. R. Shanthini
23 Sept 2011
Competitive inhibition
In the presence of a competitive inhibitor, the maximal rate of
the reaction (rmax) is unchanged, but the Michaelis constant
(KM) is increased.
Prof. R. Shanthini
23 Sept 2011
Non-competitive inhibition
Non-competitive inhibitor binds to the enzyme, but not on
the active site.
It therefore does not compete with the substrate.
However, non-competitive inhibitor causes the enzyme’s
active site to change shape and as a result, the substrate
can no longer bind to it, decreasing the rate of the reaction.
Prof. R. Shanthini
23 Sept 2011
https://ibhumanbiochemistry.wikispaces.com/C.7.5
Non-competitive inhibition
E+S
k1
ES
k2
E+I
k4
EI
k5
EI + S
k6
EIS
k7
ES + I
k8
k9
Prof. R. Shanthini
23 Sept 2011
ESI
k3
E+P
Non-competitive inhibition
We could drive the rate equation (given on the next page)
assuming the following:
k2
k1
k5
k4
Prof. R. Shanthini
23 Sept 2011
= KM =
= KI =
k7
k6
k9
k8
= KIM
= KMI
Non-competitive inhibition
rP =
rmax,appCS
(23)
KM + CS
where
rmax,app =
rmax
(1 + CI / KI)
(24)
rmax = k3CE0
(5)
KM = k2 / k1
(6)
Prof. R. Shanthini
23 Sept 2011
rmax,app < rmax
Non-competitive inhibition
The Lineweaver-Burk Plot
1
CI > 0
- rS
1
rmax,app
1
- KM
CI = 0 (no inhibitor)
1
rmax
1
CS
Prof. R. Shanthini
23 Sept 2011
Non-competitive inhibition
In the presence of a non-competitive inhibitor, the maximal
rate of the reaction (rmax) is lower but the Michaelis constant
(KM) is unchanged.
Prof. R. Shanthini
23 Sept 2011
Sigmoid/Hill kinetics
A particular class of enzymes exhibit kinetic properties that
cannot be studied using the Michaelis-Menten equation.
The rate equation of these unique enzymes is characterized
by Sigmoid/Hill kinetics as follows:
rP =
Hill constant
rmaxCSn
K + CS n
(25) The Hill
equation
Hill coefficient
n = 1 gives Michaelis-Menten kinetics
n > 1 gives positive cooperativity
n < 1 gives negative cooperativity
Prof. R. Shanthini
23 Sept 2011
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics
Sigmoid/Hill kinetics
Examples of the “S-shaped” sigmoidal/Hill curve, which is
different from the hyberbolic curve of M-M kinetics.
n=6
n=4
n=2
Prof. R. Shanthini
23 Sept 2011
Sigmoid kinetics
For an alternative formulation of Hill equation, we could
rewrite (25) in a linear form as follows:
θ =
ln
Prof. R. Shanthini
23 Sept 2011
θ
1-θ
rP
rmax
=
CSn
K + CS n
= n ln(CS) – ln (K)
(26)
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics
“Food for Thought”
Problem 3.13 from Shuler & Kargi:
The following substrate reaction rate (-rS)
data were obtained from enzymatic
oxidation of phenol by phenol oxidase at
different phenol concentrations (CS). By
plotting (-rS) versus (CS) curve, or
otherwise, determine the type of inhibition
described by the data provided?
Prof. R. Shanthini
23 Sept 2011
CS
(mg/l)
10
-rS
(mg/l.h)
5
20
30
7.5
10
50
60
80
12.5
13.7
15
90
110
130
15
21.5
9.5
140
150
7.5
5.7
Substrate inhibition
Cover it next time
Prof. R. Shanthini
23 Sept 2011
Uncompetitive inhibition
Cover it next time
Prof. R. Shanthini
23 Sept 2011
Allosteric enzyme
Cover next time in relation to competitive inhibition
Prof. R. Shanthini
23 Sept 2011
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics