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Michaelis-Menten equation
The ratio of kcat to K m can be used to describe an enzyme's catalytic efficiency. We also note that: kcat K m =k 1 k 2 k − 1 k 2
k 1 is the on rate for binding. The efficiency of catalysis cannot be greater than the “efficiency” of collisions . k 2 / (k
- k 2 ) describes the fraction of all encounters between E and S that result in product formation. I.e. it is the efficiency of the conversion of ES to P. Finally we can say: kcat K m k coll k 1 k coll =efficiency of collisions
Catalytic efficiency
Why care about the kcat/K m ratio? We can use it to compare the rates at which an enzyme catalyzes a reaction with different substrates. Suppose you have two different substrates S 1 and S 2 present at concentrations [S 1 ] and [S 2 ] along with the enzyme. The rates of reactions will be given by: v 1 =kcat 1 [ES 1 ] and v 2 =kcat 2 [ES 2 ] Recall that [ES] can be expressed in terms of free [E] and [S]: [ES ]=
[E ][S]
K
m So then we can write the two rate equations as: v 1 = kcat (^1) K m [E ][S 1 ] and v 2 = kcat (^2) K m [E ][S 2 ]
Catalytic efficiency
kcat f
/K
mf
kcat r
/K
mr
=K
eq From this we can see that the catalytic efficiencies of the forward and reverse reactions are related by the equilibrium constant.
Determination of kcat and K
m Just as for the analysis of binding equilibria our objective is to measure reaction rates at numerous concentrations of substrate and a fixed enzyme concentration. Unless one is able to measure v at very low and very high [S], estimation of Vmax and K m from a direct plot is difficult. (Actual values for this data are Vmax=1 μM/s and K m = 7 μM
Determination of kcat and K
m We can use the Lineweaver-Burk plot (double reciprocal plot) to linearize the Michaelis-Menten equation: 1 v
K
m Vmax
[S ]
Vmax A plot of 1/v versus 1/[S] should give a straight line with a slope of K m /Vmax and a “y-intercept” of 1/Vmax. As in the case of fitting equilibrium binding data with the double reciprocal plot, errors are distorted by the presence of the reciprocals.
Determination of kcat and K
m The Lineweaver-Burk plot (double reciprocal plot): Linear least-squares fit gives: Vmax=0.88 +/- 0.05 μM/s and K m = 5.0 +/- 0.5 μM
Determination of kcat and K
m The Eadie-Hofstee plot: Linear least squares fit gives: Vmax = 0.94 +/- 0.05 μM/s and K m = 5.9 +/- 0.7 μM
Determination of kcat and K
m Now running the experiment properly with data in triplicate: Non-linear least squares fit gives: Vmax = 0.98 +/- 0.02 μM/s and K m = 6.7 +/- 0.4 μM
Determination of kcat and K
m So if the linearization methods may be not so accurate (depending on the errors in the data), why would we continue to use them? ● (^) Ease of plotting when you have no computer to do a non-linear fit. ● (^) Identification of kinetics that do not fit the Michaelis-Menten model. ● (^) Analysis of kinetics of enzyme inhibition.
Reversible Inhibition of Enzyme Activity
Consider the general scheme of inhibition:
E + S ES E + P
I I
EI ESI
K
m k cat K I
K
I Four possible types of inhibition can exist, depending on the values of K I and K I
Reversible Inhibition of Enzyme Activity
Substituting the expressions for [ES], [EI] and [ESI] into the last equation gives: v = k cat
[E ][S ]
K
m
[E ]
T
=[E ]
[E ][I ]
K
I
[E ][S ]
K
m
[E ][S ][I ]
K
m
K
I If we divide that into: we get: v [E ] T
k cat
[S]
K
m 1
[I ]
K
I
[S]
K
m
[S][I ]
K
m
K
I
Reversible Inhibition of Enzyme Activity
Which can be rearranged to: v = k cat
[E ]
T
[S ]
K
m
[I ]
K
I
[S ]
[S ][I ]
K
m
K
I
Dividing numerator and denominator by (1 + [I]/K I ) gives: v =
k cat
[I ]
K
I
[E ]
T
[S ]
K (^) m
[I ]
K
I
[I ]
K
I
[S ]
Types of Reversible Inhibition
● (^) Competitive ● (^) Non-competitive ● (^) Uncompetitive ● (^) Mixed
Competitive Inhibition
Km kcat
kcat app =kcat Km app =Km
[I ]
K
I
K
I
