enzyme kinetics and associated reactor design: determination of the kinetic parameters of

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


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)


where rmax = k3CE0 and KM = f(rate constants)

- rS rmaxCS =

KM + CS rP =


S

P[E]

rmax is proportional to the initial concentration of the enzyme

KM is a constant


- rS rmaxCS =

KM + CS

Cs

rmax

rmax

2

KM

-rs

Catalyzed reactionCatalyzed reaction

uncatalyzed reaction



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


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


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)


=- rS

CS

rmax

KM

rmax

1+

CS (14)

CS

- rS

CS

1rmax

- KM

The Langmuir Plot


=- rS

CS

rmax

KM

rmax

1+

CS (14)

CS

- rS

CS

1rmax

- KM

The Langmuir Plot

Determine rmax more accurately than the other plots.


(15)

- rS

1

KM

rmax

- KM

The Lineweaver-Burk Plot

=- rS

1

rmax

KM

rmax

1+

CS

1

CS

1

1


(15)

- rS

1

KM

rmax

- KM


=- 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


(16)

- rS

KM

KM

The Eadie-Hofstee Plot

CS

-rS

rmax

=- rSrmax KM-

CS

- rS


(16)

- rS

KM

KM


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


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.


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



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



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


The Langmuir

Plot

The Lineweaver-

Burk Plot


rmax

KM

R2

Comparison of the results


The Langmuir

Plot

The Lineweaver-

Burk Plot


rmax 0.63 0.51 0.54

KM 2.93 1.78 1.89

R2 94.9% 84.6% 66.2%



The Langmuir

Plot

The Lineweaver-

Burk 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



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.


The effects of temperature on reaction rate


Temperature (deg C)

Rea

ctio

n r

ate

Optimal for most human enzymes

Optimal for some thermophillic bacterial enzymes


The effects of pH on reaction rate


pH

Rea

ctio

n r

ate

Optimal for pepsin (a stomach enzyme)

Optimal for trypsin (an intestinal enzyme)


Effect of shear


Complex enzyme kinetics

- learn about inhibited enzyme kinetics

- learn about allosteric enzymes and their kinetics


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.


Inhibited enzyme reactions

Inhibitors are also classified as competitive and non-competitive inhibitors.


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



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)



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)



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



- rS

1

- KM


rmax

1

CS

1

1 - KM, app

1 CI = 0 (no inhibitor)

CI > 0



In the presence of a competitive inhibitor, the maximal rate of the reaction (rmax) is unchanged, but the Michaelis constant (KM) is increased.


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



E + S ES E + Pk1

k2

k3

E + I EIk4

k5

EI + S EISk6

k7

ES + I ESIk8

k9



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



rP = rmax,appCS

KM + CS (23)

where

KM = k2 / k1 (6)

(5)rmax = k3CE0

rmax,app < rmax

rmax,app =(1 + CI / KI)

rmax(24)



- rS

1

- KM


rmax

1

CS

1

1

CI = 0 (no inhibitor)

CI > 0

rmax,app

1



In the presence of a non-competitive inhibitor, the maximal rate of the reaction (rmax) is lower but the Michaelis constant (KM) is unchanged.


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


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


Sigmoid kinetics

1 - θ

CSn

K + CSn

(26)


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


“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


Substrate inhibition

Cover it next time


Uncompetitive inhibition

Cover it next time


Allosteric enzyme


Cover next time in relation to competitive inhibition

enzyme kinetics and associated reactor design: determination of the kinetic parameters of

Documents