Everything about Michaelis-menten Kinetics totally explained
Michaelis-Menten kinetics describes the
kinetics of many
enzymes. It is named after
Leonor Michaelis and
Maud Menten. This kinetic model is relevant to situations where the concentration of enzyme is much lower than the concentration of substrate (for example where enzyme concentration is the limiting factor), and when the enzyme isn't
allosteric.
History
The modern relationship between substrate and enzyme concentration was proposed in
1903 by Victor Henri. A microscopic interpretation was thereafter proposed in
1913 by
Leonor Michaelis and
Maud Menten, following earlier work by
Archibald Vivian Hill. It postulated that enzyme (catalyst) and substrate (reactant) are in fast equilibrium with their complex, which then dissociates to yield product and free enzyme.
The current derivation, based on the quasi steady state approximation, namely that the concentrations of the intermediate complexes don't change, was proposed by Briggs and Haldane.
Determination of constants
To determine the maximum rate of an enzyme mediated reaction, a series of experiments is carried out where the
substrate concentration (
[S]) is increased until a constant initial rate of product formation is achieved. This is the
maximum velocity (
Vmax) of the enzyme under the conditions of the experiment. In this state, enzyme active sites are saturated with substrate.
Reaction rate/velocity V
The reaction rate
V is the number of reactions per second catalyzed per mole of the enzyme. The reaction rate increases with increasing substrate concentration [S],
asymptotically approaching the maximum rate
Vmax. There is therefore no clearly-defined substrate concentration at which the enzyme can be said to be saturated with substrate. A more appropriate measure to characterise an enzyme is the substrate concentration at which the reaction rate reaches half of its maximum value (
Vmax/2). This concentration can be shown to be equal to the Michaelis constant (
KM).
Michaelis constant 'Km'
For enzymatic reactions which exhibit simple Michaelis-Menten kinetics and in which product formation is the rate-limiting step (for example, when
k2 <<
k-1)
Km≈
k-1/
k1=
Kd, where
Kd is the
dissociation constant (
affinity for
substrate) of the
enzyme-
substrate (ES) complex. However, often
k2 >>
k-1, or
k2 and
k-1 are comparable, in which case nothing can be said about the enzyme affinity from the Michaelis constant alone.
The Michaelis constant can be defined as:
A plot of ([S]/v) versus ([S]) yields a slope = K
m/V
max and an intercept = 1/V
max.
The Hanes-Woolf regression was proposed in 1932 and 1957. The Hanes-Woolf regression has very little sensitivity to data error. It has some bias toward fitting the data in the middle and high concentration range.
There are two kinds of nonlinear least squares (NLLS) regression techniques that can be used to optimize the Michaelis-Menten equation. They differ only on how the goodness-of-fit is defined. In the v-NLLS regression method, the best goodness-of-fit is defined as the curve with the smallest
vertical error between the optimized curve and the data. In the n-NLLS regression method, the best goodness-of-fit is defined as the curve with the smallest
normal error between the optimized curve and the data. Using the vertical error is the most common form of NLLS regression criteria. Definitions based on the normal error are less common. The normal error is the error of the datum point to the nearest point on the optimized curve. It is called the normal error because the trajectory is normal (that is, perpendicular) to the curve.
It is a common misconception to think that NLLS regression methods are free of bias. However, it's important to note that the v-NLLS regression method is biased toward the data with low [S] values. This is because the Michaelis-Menten equation has a sharp rise at low concentration values, which results in a large vertical error if the regression doesn't optimize this region of the graph well. Conversely, the n-NLLS regression method doesn't have any significant bias toward any region of the saturation curve.
Whereas linear regressions are relatively easy to pursue with simple programs, such as excel or hand-held calculators, the nonlinear regressions are much more difficult to solve. The NLLS regressions are best pursued with any of various computer programs.
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