prof. r. shanthini 23 sept 2011 enzyme kinetics and associated reactor design: determination of the...
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Prof. R. Shanthini 23 Sept 2011
Enzyme kinetics and associated reactor design:
Determination of the kinetic parameters of
enzyme-induced reactions
CP504 – Lecture 4
- 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
E + S ES E + Pk1
k2
k3
which is equivalent to
S
P[E]
S for substrate (reactant)
E for enzyme
ES for enzyme-substrate complex
P for product
Simple Enzyme Kinetics (in summary)
Prof. R. Shanthini 23 Sept 2011
where rmax = k3CE0 and KM = f(rate constants)
- rS rmaxCS =
KM + CS rP =
Simple Enzyme Kinetics (in summary)
S
P[E]
rmax is proportional to the initial concentration of the enzyme
KM is a constant
Prof. R. Shanthini 23 Sept 2011
- rS rmaxCS =
KM + CS
Cs
rmax
rmax
2
KM
-rs
Catalyzed reactionCatalyzed reaction
uncatalyzed reaction
Simple Enzyme Kinetics (in summary)
Prof. R. Shanthini 23 Sept 2011
How to determine the kinetic parameters rmax and KM ?
Carry out an enzyme catalysed experiment, and measure the substrate concentration (CS) with time.
From the data, we could calculate the substrate utilization rate (-rs) as follows:
t Cs - rs
0 50
10 45
15 41
rmaxCS =
KM + CS - rS
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.
From the data, we could calculate the substrate utilization rate (-rs) as follows:
t Cs - rs
0 50 (50-45)/10
10 45 (45-41)/5
15 41
rmaxCS =
KM + CS - rS
Prof. R. Shanthini 23 Sept 2011
rmaxCS =
KM + CS - rS
We could rearrange
to get the following 3 linear forms:
=- rS
CS
rmax
KM
rmax
1+ CS
=- rS
1
rmax
KM
rmax
1+
CS
1
=- rSrmax KM-
CS
- rS
(15)
(14)
(16)
Prof. R. Shanthini 23 Sept 2011
=- rS
CS
rmax
KM
rmax
1+
CS (14)
CS
- rS
CS
1rmax
- KM
The Langmuir Plot
Prof. R. Shanthini 23 Sept 2011
=- rS
CS
rmax
KM
rmax
1+
CS (14)
CS
- rS
CS
1rmax
- KM
The Langmuir Plot
Determine rmax more accurately than the other plots.
Prof. R. Shanthini 23 Sept 2011
(15)
- rS
1
KM
rmax
- KM
The Lineweaver-Burk Plot
=- rS
1
rmax
KM
rmax
1+
CS
1
CS
1
1
Prof. R. Shanthini 23 Sept 2011
(15)
- rS
1
KM
rmax
- KM
The Lineweaver-Burk Plot
=- rS
1
rmax
KM
rmax
1+
CS
1
CS
1
1
- Gives good estimates of rmax, but not necessarily KM
- Data points at low substrate concentrations influence the slope and intercept more than data points at high Cs
Prof. R. Shanthini 23 Sept 2011
(16)
- rS
KM
KM
The Eadie-Hofstee Plot
CS
-rS
rmax
=- rSrmax KM-
CS
- rS
Prof. R. Shanthini 23 Sept 2011
(16)
- rS
KM
KM
The Eadie-Hofstee Plot
CS
-rS
rmax
=- rSrmax KM-
CS
- rS
- Can be subjected to large errors since both coordinates contain (-rS)
- Less bias on point at low Cs than with Lineweaver-Burk plot
Prof. R. Shanthini 23 Sept 2011
CS
(mmol/l)
-rS
-(mmol/l.min)
1 0.20
2 0.22
3 0.30
5 0.45
7 0.41
10 0.50
Data:
Determine the M-M kinetic parameters for all the three methods discussed in the previous slides.
Prof. R. Shanthini 23 Sept 2011
The Langmuir Plot
y = 1.5866x + 4.6417
R2 = 0.94970
5
10
15
20
25
0 2 4 6 8 10CS (mmol/l)
CS/(
-rS)
min
rmax = 1 / slope = 1 / 1.5866 = 0.63 mmol/l.min
KM = rmax x intercept = 0.63 x 4.6417 = 2.93 mmol/l
Prof. R. Shanthini 23 Sept 2011
The Lineweaver-Burk Plot
y = 3.4575x + 1.945
R2 = 0.84630
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 11/CS l/mmol
1/(
-rS)
l.min
/mm
ol
rmax = 1 / intercept = 1 / 1.945 = 0.51 mmol/l.min
KM = rmax x slope = 0.51 x 3.4575 = 1.78 mmol/l
Prof. R. Shanthini 23 Sept 2011
The Eadie-Hofstee Plot
y = -1.8923x + 0.5386
R2 = 0.6618
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.05 0.1 0.15 0.2 0.25(-rS)/CS per min
(-r S
) m
mol
/l.m
in
rmax = intercept = 0.54 mmol/l.min
KM = - slope = 1.89 mmol/l
Prof. R. Shanthini 23 Sept 2011
The Langmuir
Plot
The Lineweaver-
Burk Plot
The Eadie-Hofstee Plot
rmax
KM
R2
Comparison of the results
Prof. R. Shanthini 23 Sept 2011
The Langmuir
Plot
The Lineweaver-
Burk Plot
The Eadie-Hofstee Plot
rmax 0.63 0.51 0.54
KM 2.93 1.78 1.89
R2 94.9% 84.6% 66.2%
Comparison of the results
Prof. R. Shanthini 23 Sept 2011
The Langmuir
Plot
The Lineweaver-
Burk Plot
The Eadie-Hofstee Plot
rmax 0.63 0.51 0.54
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
Comparison of the results
Prof. R. Shanthini 23 Sept 2011
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
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
The effects of temperature on reaction rate
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
Temperature (deg C)
Rea
ctio
n r
ate
Optimal for most human enzymes
Optimal for some thermophillic bacterial enzymes
Prof. R. Shanthini 23 Sept 2011
The effects of pH on reaction rate
https://wikispaces.psu.edu/display/230/Enzyme+Kinetics+and+Catalysis
pH
Rea
ctio
n r
ate
Optimal for pepsin (a stomach enzyme)
Optimal for trypsin (an intestinal enzyme)
Prof. R. Shanthini 23 Sept 2011
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.
https://ibhumanbiochemistry.wikispaces.com/C.7.5
Prof. R. Shanthini 23 Sept 2011
Competitive inhibition
Competitive inhibitors (denoted by I) compete with substrate to occupy the active site of the enzyme.
E + S ES E + Pk1
k2
k3
E + I EIk4
k5
rP = k3 CES (17)
CE0 = CE + CES + CEI
where
(18)
Prof. R. Shanthini 23 Sept 2011
Competitive inhibition
Assuming rapid equilibrium, we get
k1 CE CS = k2 CES
k4 CE CI = k5 CEI
k2
k1 KM =
CE CS
CES =
k5
k4 KI =
CE CI
CEI =
(19)
(20)
Prof. R. Shanthini 23 Sept 2011
Competitive inhibition
Combining (17) to (20), we get
k3CE0CSrP =
rmaxCS =
KM,app + CS (21)
KM (1 + CI / KI) + CS
where
KM,app = KM (1 + CI / KI) (22)
KM = k2 / k1 (6)
(5)rmax = k3CE0
KM,app > KM
Prof. R. Shanthini 23 Sept 2011
Competitive inhibition
- rS
1
- KM
The Lineweaver-Burk Plot
rmax
1
CS
1
1 - KM, app
1 CI = 0 (no inhibitor)
CI > 0
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.
https://ibhumanbiochemistry.wikispaces.com/C.7.5
Prof. R. Shanthini 23 Sept 2011
Non-competitive inhibition
E + S ES E + Pk1
k2
k3
E + I EIk4
k5
EI + S EISk6
k7
ES + I ESIk8
k9
Prof. R. Shanthini 23 Sept 2011
Non-competitive inhibition
k2
k1 = KM =
We could drive the rate equation (given on the next page) assuming the following:
k7
k6 = KIM
k5
k4 = KI =
k9
k8 = KMI
Prof. R. Shanthini 23 Sept 2011
Non-competitive inhibition
rP = rmax,appCS
KM + CS (23)
where
KM = k2 / k1 (6)
(5)rmax = k3CE0
rmax,app < rmax
rmax,app =(1 + CI / KI)
rmax(24)
Prof. R. Shanthini 23 Sept 2011
Non-competitive inhibition
- rS
1
- KM
The Lineweaver-Burk Plot
rmax
1
CS
1
1
CI = 0 (no inhibitor)
CI > 0
rmax,app
1
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 = rmaxCS
n
K + CSn
(25)
n = 1 gives Michaelis-Menten kinetics
n > 1 gives positive cooperativity
n < 1 gives negative cooperativity
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics
The Hill equation
Hill coefficientHill constant
Prof. R. Shanthini 23 Sept 2011
Sigmoid/Hill kinetics
Examples of the “S-shaped” sigmoidal/Hill curve, which is different from the hyberbolic curve of M-M kinetics.
n = 2n = 4
n = 6
Prof. R. Shanthini 23 Sept 2011
Sigmoid kinetics
1 - θ
CSn
K + CSn
(26)
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics
For an alternative formulation of Hill equation, we could rewrite (25) in a linear form as follows:
θln = n ln(CS) – ln (K)
rmax θ = =
rP
Prof. R. Shanthini 23 Sept 2011
“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?
CS
(mg/l)
-rS
(mg/l.h)
10 5
20 7.5
30 10
50 12.5
60 13.7
80 15
90 15
110 21.5
130 9.5
140 7.5
150 5.7
Prof. R. Shanthini 23 Sept 2011
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
http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Sigmoid_Kinetics
Cover next time in relation to competitive inhibition