Michaelis-Menten Kinetics
Enzyme Kinetics
Enzymatic reaction
k1 k2
E+S k-1
ES E+P
Rate expression for product formation
v = dP/dt = k2(ES)
d(ES)/dt = k1(E)(S)-k-1(ES)-k2(ES)
Conservation of enzyme
(E) = (E0) – (ES)
Two Methods to Proceed
• Rapid equilibrium assumption: define
equilibrium coefficient
K’m = k-1/k1 = [E][S]/[ES]
• Quasi-steady state assumption
[ES] = k1[E][S]/(k-1+k2)
• Both methods yield the same final equation
Michaelis- Menten Kinetics
Michaelis-Menten Kinetics
• When v= 1/2 Vmax, [S]= Km so Km is
sometimes called the half-saturation
constant and sometimes the Michaelis
constant
Vmax [S ] k 2 [E 0 ][S ]
v
K m [S ] K m [S ]
Michaelis-Menten Kinetics
Vmax [S ] k 2 [E 0 ][S ]
v
K m [S ] K m [S ]
• units on k2 are amount product per amount
of enzyme per unit time (also called the
“turnover number”). Units on E0 are
amount of enzyme (moles, grams, units,
etc.) per unit volume
• Km has the same units as [S] (mole/liter,
etc.)
Experimentally Determining Rate
Parameters for Michaelis-Menten
Kinetics
Lineweaver-Burk
Eadie-Hofstee
Hanes- Woolf
Batch Kinetics
Determining Parameters
• Rearrange the equation into a linear form.
• Plot the data.
• What kind of data would we have for an
experiment examining enzyme kinetics?
• Describe an experiment.
• The intercept and slope are related to the
parameter values.
Enzyme Kinetics Experiment
Place enzyme and substrate (reactants) in a
constant temperature, well stirred vessel.
Measure disappearance of reactant or
formation of product with time.
Why constant temperature?
Why well stirred?
What about the medium? Buffer?
Lineweaver-Burk
(double reciprocal plot)
– Rewrite Michaelis-Menten rate expression
1 Km 1 1
v Vmax [S ] Vmax
– Plot 1/v versus 1/[S]. Slope is Km/Vmax,
intercept is 1/Vmax
Graphical Solution
intercepts
1/ V
1 Km 1 1 Slope = Km/ Vmax
v Vmax [S] Vmax
1/ Vmax
-1/ Km 1/ [S]
Example: Lineweaver-Burk
-5 -5
[S] x 10 M V, M/min x 10
1.0 1.17
1.5 1.50
2.0 1.75
2.5 1.94
3.0 2.10
3.5 2.23
4.0 2.33
4.5 2.42
5.0 2.50
Resulting Plot
slope = Km/ Vmax= 0.5686
y intercept = 1/ Vmax= 2.8687
Michaelis-Menten Kinetics
Vmax [S ] k 2 [E 0 ][S ]
v
K m [S ] K m [S ]
Vmax = 1/2.8687 x 10-4 = 3.49 x 10-5 M/min
Km= 0.5686 x Vm = 1.98 x 10-5 M
Other Methods
• Eadie-Hofstee plot
v
v Vmax Km
[S ]
• Hanes- Woolf
[S ] K m 1
[S ]
v Vmax Vmax
Comparison of Methods
• Lineweaver-Burk: supposedly gives good
estimate for Vmax, error is not symmetric
about data points, low [S] values get more
weight
• Eadie-Hofstee: less bias at low [S]
• Hanes-Woolf: more accurate for Vmax.
• When trying to fit whole cell data – I don’t
have much luck with any of them!
Batch Kinetics
d[S ] Vmax [S ]
v
dt K m [S ]
integrate
[S 0 ]
Vmaxt [S 0 ] [S ] K m ln
[S ]
rearrange
[S 0 ] [S ] K m [S 0 ]
ln Vmax
t t [S ]
Inhibited Enzyme Kinetics
• Competitive Inhibition
• Noncompetitive Inhibition
• Uncompetitive Inhibition
• Substrate Inhibition
Effects of Temperature and pH
Experiments: Initial rate at
different substrate concentrations
E S1= 20 E S2=10 E S3=6.7 E S4=5 E S5=4
Measure S for a short time period. Calculate v from:
v = [S(time 0) – S(time 1)]/delta time
Time (min) S (g/L)
Experiment 0 20
Using S1 0.5 19.43
v= (20-19.3)g/L]/0.5 min = 1.14 g/L/min
Time (min) S (g/L )
Experiment 0 10
Using S2 0.5 9.565
v= (10-9.565)g/L]/0.5 min = 0.87 g/L/min
Experimental Data
S (mmol/L) v (mmol/L/min)
20 1.14
10 0.87
6.7 0.70
5.0 0.59
4.0 0.50
Problems with this method?
Rate is not measured at a constant substrate
concentration – substrate decreasing. Must have
sensitive assay for substrate to measure initial rates.
20
18
16 regression
14 S/v = 0.6S + 5.6
12
S/v (min)
10
8
6 experimental data
regression
4
2
0
0 5 10 15 20 25
S (g/L)
Allosteric Enzyme Kinetics
In an enzyme with more than one substrate
binding site, binding of one substrate
molecule affects the binding of another.
n
d[S ] V max[S]
v n
dt K m [S ]n
n>1, cooperation; n0
I=0
Vmax is unchanged
1/Vmax
-1/Km -1/Km,app 1/[S]
Practice deriving kinetic
expressions
Derive competitive inhibition equation (3.22
in your text)? Write down all assumptions.
Noncompetitive Inhibition
Inhibitor binds to the enzyme, but not at the active
site. However, the enzyme affinity for substrate is
reduced.
E S ES P [E ][S ] [EI ][S ]
Km
[ES ] [ESI ]
I I
[E ][I ] [ES ][I ]
KI
EI S ESI [EI ] [ESI ]
[E 0 ] [E ] [ES ] [EI ] [ESI ]
v k 2 [ES]
Noncompetitive Inhibition
Cofactors and Coenzymes
Holoenzymes- three parts
• Apoenzyme- Protein portion
• Cofactor- inorganic ion (ex: metal ions),
improve the fit of enzyme with substrate
• Coenzyme- nonprotein organic molecule
(ex: NAD- nicotinamide adenine
dinucleotide), many synthesized from
vitamins (why vitamins are essential)
Noncompetitive Inhibition
Rate is given by:
Vmax Vmax,app
v
[I ] K m
K m
1 1 1
K I
[S ]
[S ]
Question: What is the magnitude of Vmax,app
relative to Vmax, and what will be the effect of
v? How can you moderate the effects of this
type of inhibition.
Noncompetitive Inhibition
1/v I>0
I=0
1/Vmax,app
1/Vmax
-1/Km 1/[S]
Km is unchanged
Uncompetitive Inhibition
Inhibitor binds only to ES complex, and not
to E alone.
E S ES P [E ][S ]
Km
[ES ]
I
[E ][I ]
KI
[EI ]
ESI
[E 0 ] [E ] [ES ] [ESI ]
v k 2 [ES]
Uncompetitive Inhibition
Rate is given by:
Vmax
[S ]
[I ]
1
K I Vmax,app [S ]
v
[S ] K m ,app [S ]
Km
[I ]
1
K I
What is the magnitude of Vmax,app relative to Vmax?
What is the magnitude of Km,app relative to Km?
Uncompetitive Inhibition
1/v
I>0
I=0
1/Vmax,app
1/Vmax
-1/Km,app -1/Km 1/[S]
Substrate Inhibition
No substrate inhibition
v
Substrate inhibition
Vmax [S ]
v
[S]2
K m [S ]
KSi
[S ]max. rate K mK Si
S
Enzyme Deactivation
• Enzymes are denatured by
– Temperature
– pH
– Radiation
– Irreversible binding by inhibitors
• Temperature can both increase
(thermal activation) and decrease
(thermal denaturation) rate
Temperature effects
At moderate temperatures, higher
temperatures give higher rates.
E a
v k 2 [E ], where k 2 Ae RT
At higher temperatures, rate starts to
decrease as enzyme denatures faster
d[E ] E d
k d t
k d [E ], or [E] [E 0 ]e , where kd Ad e d RT
dt
Temperature Effects
Effect on rate is a combination of the two effects
E a
k d t
v Ae RT
[E 0 ]e
Activation energy 10
kcal/mol
Deactivation energy 100
kcal/mol