The Michaelis-Menten Equation This equation cannot be explicitly integrated, however, without...
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The Michaelis-Menten Equation
This equation cannot be explicitly integrated, however, without simplifying assumptions, two possibilities are:
1. Assumption of equilibrium. Leonor Michaelis and Maud Menten, building on the work of Victor Henri, assumed that k-1 » k2, so that the first step of the reaction reaches equilibrium.
Ks is the dissociation constant of the first step in the enzymatic reaction
v= Vmax = k2 ETET = E + ES
ES = E * S / Ks
v/Vmax = k2 ES / k2 ET = ES/ ET
v/Vmax = ES/ ET
v/Vmax = ES/ E + ES ¿How to know ES?
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The Michaelis-Menten Equation
1. Assumption of steady-state. Figure illustrates the progress curves of the various participants in reaction
under the physiologically common conditions that substrate is in great excess over Enzyme ([S] » [E]).
ES maintains a steady state and [ES] can be treated as having a constant value:
The so called steady state assumption, a more general condition than that of equilibrium, was first proposed in 1925 by G. E. Briggs and B. S. Haldane
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The Michaelis constant, KM , is defined as
The Michaelis-Menten Equation
Solving for [ES]: ES = E * S /(k-1+k2)/k1
Therefore: ES = E * S / KM
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The Michaelis-Menten Equation
The expression of the initial velocity (v0) of the reaction, the velocityat t=0, thereby becomes
The maximal velocity of a reaction, Vmax occurs at high substrate concentrations when the enzyme is saturated, that is, S>> Km, and ET is entirely in the ES form v= Vmax when
This expression, the Michaelis-Menten equation, is the basic equation of enzyme kinetic.
v/Vmax = ES/ (E + ES)
v/Vmax = (E*S)/Km/ (E + (E*S)/Km )
v/Vmax = S/ Km / (1 + S/Km)
v/Vmax = S / Km + S
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Significance of the Michaelis Constant
The Michaelis constant, KM, has a simple operational definition. At the substrate concentration at which [S] = KM, this equation
yields v0 = Vmax/2 so that
KM is the substrate concentration at which the reaction velocity is half maximal
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Vmax= 10 M/seg Km=10 x10-5 MSi en el ensayo se usaron 5mg/L de preparación enzimática, entonces:v= Vmax = k2 ET k2= 10/5 = 2 moles/mg seg
¿Qué predicciones podemos hacer a partir de esta información?
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- dS/dt = vi = So dX/dt
Al iniciar: t = 0, S = So
A cualquier tiempo:T = t S = S X = (So-S)/So
Graficar por ejemplo :
(So – S)/t vs (1/t) ln (So/S)
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O -
C
Time (sec)N
itrop
hen
ol
Two-Stage Catalysis of Chymotrypsin
O
CH3–C–O– –NO2
Nitrophenol acetate
OC
O
CH3–C HO– –NO2
+ H2O
O-HC
CH3COOH
Kinetics of reaction
Two-phasereaction
Acylation
Deacylation (slow step)
Adapted from Dressler & Potter (1991) Discovering Enzymes, p.169
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DESVIACIONESA M&M
dE/dt = -k1 E*S + k-1 ES + k3 ES’
dES/dt = k1 E*S – ES (k-1 + k2)
dES’/dt = k2 ES – k3 ES’
v = dP1/dt = k2 ES
v = dP2/dt = k3 ES’
ET= E + ES + ES’
From the steady state assumption::
dE/dt = 0 -k1 E*S + k-1 ES + k3 ES’
dES/dt = 0 ES = k1 E*S / (k-1 + k2)
dES’/dt = 0 ES’ = k2 ES / k3
dP2/dt = v = k3 ES’ v = k3 k2 ES / k3
v = k2 k1 E*S / (k-1 + k2)
Vmax = k3 ET Vmax = k3 (E + ES + ES’) Vmax = k3 (E + k1 E*S / (k-1 + k2) + k2 ES / k3 Vmax = k3 (E + k1 E*S / (k-1 + k2) + k2 k1 E*S / (k-1 + k2) / k3 ) Vmax = k3 E + k1 k3 E*S / (k-1 + k2) + k2 k1 E*S / (k-1 + k2)
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v / Vmax = k2 k1 E*S / (k-1 + k2) k3 E + k1 k3 E*S / (k-1 + k2) + k2 k1 E*S / (k-1 + k2)
v / Vmax = k2 k1 S k3 (k-1 + k2) + k1 k3 S + k2 k1 S
Vmax = k3 ET
v / Vmax = k2 k1 S k3 (k-1 + k2) + k1 S (k2 + k3)
v / Vmax = k2 S / (k2 + k3) k3 (k-1 + k2) / (k2 + k3) + k1 S
v = k2 k3 ET S / (k2 + k3)
k3 (k-1 + k2) / (k2 + k3) + k1 S
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K-2+
Vmax f = k2 k3 ET k-2 + k3 +k2
Vmax r = k-1 k-2 ET k-1 + k2 +k-2
Ks = k-1 k-2 + k-1 k3 + k2 k3 k1( k2 + k-2 + k3)
Kp = k-1 k-2 + k-1 k3 + k2 k3 k-3( k-1 + k2 + k-2)
Keq = Vmax f * Kp Vmax r * Ks
v = Vmax f Kp S – Vmax r Ks P
KsKp + KpS + KsP
v = Vmax f S – P / Keq
Ks + S + (Ks/Kp) P
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