Soit l’équation enzymatique,
EGlu His + S
A
Kü2
K2s
EGlu~His S
AKa\
Ks
EGlu His'^ + S V... EGlu His'^ S
/
Kai
/
EGlu His'^ + S
Ka\
V
K,s
EGlu His^ S
h
—*■ EGlu~ His^ + P
Figure 28. Modèle cinétique de la formation du complexe barnase-substrat.
Glu and Glu ~ représentent respectivement la forme neutre (protonée) et négativement char
gée (déprotonée) du Glu 73. His and His représentent respectivement la forme neutre (sim
plement protonée) et positivement chargée (doublement protonée) de l’His 102. EGlu His S
représente le complexe enzyme-substrat et EGluHis l’enzyme libre. Ka
\2sont les constantes
d’acidité des différents états de protonation de l’enzyme libre et Ka\2 sont celles du complexe,
Ki2s sont les constantes de dissociation des complexes enzyme-substrat etA^ est la constante
de vitesse de la réaction. Le cas où l’His 102 est simplement protonée et le Glu 73 protoné
n’est pas pris en compte car d’une part, il n’existe qu’en concentration marginale
(pKaciu C pKanis) ot d’autre part, il conduit à des équations plus complexes dont les paramè
tres ne peuvent être dérivés de manière univoque à partir des données expérimentales disponi
bles.
On peut écrire.
[EGlu-His^][H^] _ [EGlu-His][H^]
_ [EGlu-His^
Ka\ = S][H^
[EGlu His* 5] et Kar-, =
[EGlu-His 5][W+]
[EGlu-His* S]
K, = [EGlu-His^][S]
[EGlu-His+ S] Ku = [EGlu His*][S][EGlu His* S] et Ko, = [EGlu-His][S]
[EGlu-His 5]
(2)
(3)
La concentration totale d’enzyme [E]o est égale à,
[£]o = [EGlu-His 5] + [EGlu'His] + [EGlu'His*] +
[EGlu-His^ S] + [EGlu His^] + [EGlu His^ 5] (4)
Les constantes de dissociation peuvent être introduites dans l’Eq. (4),
[EGlu His 5] = [EGlu-His][S] _ [S][EGlu-His H^] K
ü2Ko.
mais aussi, [EGlu His 5] =
K2s[H-]
[EGlu-His^ 5] Ka^^
Un
[EGlu-His] = K
ü2[EGlu-His-^] _ Ka
2Ks [EGlu-His*^ 5]
[H-]
mais aussi, [EGlu His^] = K, [EGlu-His^ 5]
[S]
[EGlu His^] = [EGlu-His*][H*] _ K, [EGlu-His* 5][i7+]
Ka, Ka, [S]
^EG,u His‘ S] =
/Ct, Ka, K,,
[EGlu His* S] = [EGlu- His* S][H*]
Kah
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Sachant que,
Ka\ K^, =Ka, K, et Ka\ Ko, = Kuo K, (12)
On peut remplacer les termes de l’Eq. (4) par leur correspondants dans les
Eqs. (6), (7), (8), (9) et (11). On obtient,
_ [EGlu-His* 5]
[E]o !k(i + + !h.]
[51 ^ Ka^ ^ [;, + ]j
+ i+iïy +ai Kab (13)
Léquation de la vitesse de la réaction est,
V =
[£o] [5] k, X---!---- ^
I «J I J 2 f„ + | Kab_________ i"^i
. , [W~"l , ^2 [5] + K, X[«^1
(14)
En comparant l’Eq. (14) à l’équation de la vitesse de réaction d’un catalyse Michælienne ’clas-
sique’ où il n’existe qu’une seule forme de l’enzyme et du complexe enzyme—substrat,
V —
[E
q] [5] k,,
[5] + (15)
On obtient les expressions suivantes pour caractériser le mécanisme de la Figure 28,
kcal — k2 X ... ,
1 + ^ +1«Kab [//+] (16)
Ku = K,
X1
I[tf + 1
I1+1^ + ^
1«"1
(17)
!SsBL = hL X 1
Em Ks h ^ ^ K^i ^ [H+] X(18)
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