02_-_Enzyme_Kinetics - Quantitative Physiology I Molecular...

Quantitative Physiology I / Molecular and Cellular Systems, BMEN E4001x
Notes 02 - Enzyme kinetics
Chapter 1 of Keener & Sneyd
Life is based on regulation of processes (see B&B Chapt. 58)
Glycolysis is a key step in carbohydrate metabolism. This is the breakdown of glucose into
pyruvate, which goes into the citric acid cycle, yielding two energy stores, ATP and NADH.
Here are the first few steps, for which enzyme regulation plays a key role.
This process takes some ATP, converting it to the lower energy ADP, but ATP is replenished
later in metabolism. In this scheme, consider phosphofructokinase (PFK1)
• adds a second phosphate group (from ATP) to fructose, leading later to splitting of this
molecule
• ATP is a substrate for this enzyme
• ATP is also an inhibitor of PFK1, leading to feedback regulation of this enzyme and
metabolism
Enzyme Kinetics
In an enzymatic relation, something happens after the binding. So, our equations need to include
this conversion. However, measured kinetics show a slightly different kinetics than would be
alluded to using our bimolecular model.
Note that for the initial A+B->C reaction, the initial rate reaction, assuming [C](t=0)=0 is:
1 1
k ] B ][ A [
dt
] C [ d
abk
dt
dc
= ⇒ =
In contrast, enzymatic systems often show saturation.
A simple mechanism to express simple enzyme kinetics, Michaelis Menten (1913):
E P C E S 2
1
1
k
k
k
+  → 
   ←
→ 
+
-
This system allows regeneration of E.
It is also recognized that P+E-> C can conceptually happen, but it is assumed here
that P gets removed immediately.
G l u c o s e
F r u c t o s e 6 - P
F r u c t o s e 1 , 6 - P
P y r u v a t e
P F K 1

0
2
2 1 1
2 1 1
1 1
e c e
ck
dt
dp
ck ck sek
dt
dc
ck ck sek
dt
de
ck sek
dt
ds
= +
=
- - + =
+ + - =
+ - =
-
-
-
Equilibrium Approximation
In this original form, Michaelis and Menten assumed fast equilibrium between the S,E and C.
Thus:
k
1
se=k
-1
c
and by using e+c=e
0
, with a similar chain of algebra as for biomolecular binding, they got:
1
1
s
s
0
s
0
k
k
K ;
] S [ K
] S [
] E [ ] C [ ;
s K
s
e c
-
=
+
=
+
=
Quite reasonable, when it is recognized that this is the same form as biomolecular binding. Put
another way, this is bimolecular binding leading to a slow evolution of C into P + E. As the
next step is
] [
] [
2
C k
dt
P d
=
we can say that the rate of reaction is proportional to [C], or
1
1
S
S
0 2
k
k
K ;
] S [ K
] S [
] E [ k V
-
=
+
=

Quasi-steady-state approximation