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How to determine Km and Vmax
A simple chemical reaction with a single substrate shows a linear relationship between the rate of formation of product and the concentration of substrate, as shown below:
For an enzyme-catalysed reaction, there is usually a hyperbolic relationship between the rate of reaction and the concentration of substrate, as shown below:
(A) At low concentration of substrate, there is a steep increase in the rate of reaction with increasing substrate concentration. The catalytic site of the enzyme is empty, waiting for substrate to bind, for much of the time, and the rate at which product can be formed is limited by the concentration of substrate which is available.
(B) As the concentration of substrate increases, the enzyme becomes saturated with substrate. As soon as the catalytic site is empty, more substrate is available to bind and undergo reaction. The rate of formation of product now depends on the activity of the enzyme itself, and adding more substrate will not affect the rate of the reaction to any significant effect.
The rate of reaction when the enzyme
is saturated with substrate is the maximum rate of reaction, Vmax.
The relationship between rate of reaction and concentration of substrate depends on the affinity of the enzyme for its substrate. This is usually expressed as the Km (Michaelis constant) of the enzyme, an inverse measure of affinity.
For practical purposes, Km is the concentration of substrate which permits the enzyme to achieve half Vmax. An enzyme with a high Km has a low affinity for its substrate, and requires a greater concentration of substrate to achieve Vmax."
The Km of an enzyme, relative to the concentration of its substrate under normal conditions permits prediction of whether or not the rate of formation of product will be affected by the availability of substrate.
An enzyme with a low Km relative to the physiological concentration of substrate, as shown above, is normally saturated with substrate, and will act at a more or less constant rate, regardless of variations in the concentration of substrate within the physiological range.
An enzyme with a high Km relative to the physiological concentration of substrate, as shown above, is not normally saturated with substrate, and its activity will vary as the concentration of substrate varies, so that the rate of formation of product will depend on the availability of substrate.
If two enzymes, in different pathways, compete for the same substrate, then knowing the values of Km and Vmax for both enzymes permits prediction of the metabolic fate of the substrate and the relative amount that will flow through each pathway under various conditions.
In order to determine the amount of an enzyme present in a sample of tissue, it is obviously essential to ensure that the limiting factor is the activity of the enzyme itself, and not the amount of substrate available. This means that the concentration of substrate must be high enough to ensure that the enzyme is acting at Vmax. In practice, it is usual to use a concentration of substrate about 10 - 20-fold higher than the Km in order to determine the activity of an enzyme in a sample.
If an enzyme is to be used
to determine the concentration of substrate in a sample
(e.g. glucose oxidase is used to measure plasma glucose), then the substrate
must be the limiting factor, and the concentration of substrate must be below
Km, so that the rate of formation of product increases steeply with increasing
concentration of substrate, so providing a sensitive assay for the substrate."
Km and Vmax are determined by incubating
the enzyme with varying concentrations of substrate; the results can be plotted
as a graph of rate of reaction (v) against concentration of substrate ([S],
and will normally yield a hyperbolic curve, as shown in the graphs above.
The relationship is defined by the Michaelis-Menten equation:
= Vmax / (1 + (Km/[S]))
It is difficult to fit the best hyperbola through the experimental points, and difficult to determine Vmax with any precision by estimating the limit of the hyperbola at infinite substrate concentration. A number of ways of re-arranging the Michaelis-Menten equation have been devised to obtain linear relationships which permit more precise fitting to the experimental points, and estimation of the values of Km and Vmax. There are advantages and disadvantages associated with all three main methods of linearising the data.
The Lineweaver-Burk double reciprocal plot rearranges the Michaelis-Menten equation as:
1 / v = 1 / Vmax + Km / Vmax x 1 / [S]
plotting 1/v against 1/[S] give a straight line:
This is the most widely used method of linearising the data, and generally gives the best precision for estimates of Km and Vmax. However, it has the disadvantage of placing undue weight on the points obtained at low concentrations of substrate (the highest values of 1/[S] and 1/v). These are the points at which the precision of determining the rate of reaction is lowest, because the smallest amount of product has been formed.
The Eadie-Hofstee plot rearranges the Michaelis-Menten equation as:
v = Vmax - Km x v / [S]
plotting v against v / [S] gives a straight line:
This plot overcomes the problem of uneven spacing of points, and undue weight given to points at low concentrations of substrate. However, it has the disadvantage that v, which is a dependent variable, is used on both axes, and hence errors in measuring the rate of reaction are multiplied, resulting in lower precision of the estimates of Km and Vmax
The Hanes plot rearranges the Michaelis-Menten equation as:
[S] / v = Km / Vmax + [S] / Vmax
plotting [S] / v against [S] gives a straight line:
This plot overcomes the problem of uneven spacing of points, and undue weight given to points at low concentrations of substrate. However, it has the disadvantage that [S] is used on both axes, and hence pipetting errors, which lead to errors in the true concentration of substrate available, are multiplied, resulting in lower precision of the estimates of Km and Vmax.