Harper's Illustrated Biochemistry, Volume 1 - Murray R. 1993

Structure and Functions of Proteins and Enzymes
Enzymes: Kinetics
Michaelis–Menten Equation

Graphical Determination of the Michaelis constant K_M

The Substrate Concentration at which the velocity is equal to half of the maximum is denoted by K_M and is called the Michaelis constant. It can be determined from the plot of v versus [S] (Fig. 8.14). Note that K_M has the dimensions of molar concentration.

When [S] approaches K_M, v becomes highly sensitive to changes in [S]; in this region, the enzyme operates at half-maximal velocity. Many Enzymes are characterized by K_M values that roughly correspond to the physiological concentrations of their substrates.

The Michaelis–Menten equation

Class="center">

describes The behavior of many enzymes as substrate concentration changes. Using this equation, the dependence of the initial velocity of an enzymatic reaction on [S] and K_M can be illustrated by the following specific Examples.

Fig. 8.15. Diagram illustrating enzyme-substrate binding at low (A) and high (C) substrate concentrations, as well as at a substrate concentration equal to K_M (B). States A, B, and C correspond to points A, B, and C on the curve shown in Fig. 8.14.

1. [S] is much less than K_M (point A in Figs. 8.14 and 8.15). In this case, the term [S] in the denominator can be neglected, and the denominator is practically equal to K_M. The ratio of the two constants, V_mаx and K_M, can be replaced by a new constant k. Thus, we have:

(≃ means "approximately equal to").

In other words, when the substrate concentration is significantly lower than the concentration at which the reaction velocity is half-maximal (i.e., significantly less than K_M), the initial velocity v is proportional to the substrate concentration [S].

2. [S] is much greater than K_M (point C in Figs. 8.14 and 8.15). In this case, the term K_M in the denominator can be omitted, i.e.

This means that at a substrate concentration [S] far exceeding K_M, the initial velocity v is equal to the maximum velocity V_max.

3. [S] = K_M (point B in Figs. 8.14 and 8.15).

This means that at a substrate concentration equal to K_M, the initial reaction velocity v is half of the maximum. This also suggests a method for estimating K_M: experimentally determining the substrate concentration at which the initial velocity is equal to half of the maximum.

For many enzymes, determining V_max (and hence K_M) directly from the plot of v versus [S] is difficult. To overcome this, the reciprocal of the Michaelis–Menten equation is used, i.e.

and its right-hand side is expressed as the sum of two terms:

Hence, upon simplification, we obtain

Thus, we obtain the equation of a straight line

y = ax + b,

where y = 1/v and x = 1/[S].

If we plot y (i.e., 1/v) against x (i.e., 1/[S]), the intercept on the y-axis, b, will be equal to 1/V_mах, and the slope, a, will be K_M/V_max. The intercept on the x-axis (in the negative region) can be found by Setting y to zero. Thus

The double-reciprocal plot of the Michaelis—Menten equation is known as a Lineweaver—Burk plot (Fig. 8.16). From it, K_M can be determined either from the slope and the y-intercept or from the x-intercept in the negative region. Since [S] has the dimensions of molar concentration, K_M is also expressed in moles per liter. The velocity v can be expressed in any units, since K_M is independent of [Enz.]. The reciprocal plot allows K_M to be determined using a relatively small number of data points, which is why it is frequently used to find K_M.

When using the Lineweaver—Burk plot in practice to estimate K_M, one often finds that almost all data points cluster in the region of low substrate concentrations. This happens when measurements are taken at equal intervals of [S]. To avoid this, measurements should be carried out at values of [S] corresponding to equal intervals on the reciprocal scale.

An alternative approach to the experimental estimation of K_M and V_mах was proposed by Eadie and Hofstee. The Michaelis—Menten equation can be rearranged to the form

Fig. 8.16. Lineweaver—Burk double-reciprocal plot (1/v versus 1/[S]), used for the graphical Determination of k_M and V_max.

To find K_M and V_max, a plot of v/[S] (y-axis) versus v (x-axis) is constructed. In this case, the y-intercept equals V_mах/K_M, and the x-intercept equals V_max. The slope is equal to —1/K_M.

Although the Lineweaver—Burk and Eadie—Hofstee equations are quite convenient in certain cases, rigorous determination of K_M and V_mах requires appropriate statistical analysis.

Estimates of K_M are of great practical value. At substrate concentrations 100 times greater than K_M, the enzyme operates at virtually maximum velocity; therefore, the maximum velocity (V_max) reflects The amount of active enzyme present. This crucial fact is utilized to estimate Enzyme Concentration in a preparation. The value of K_M serves as a guide for how much substrate should be added to determine V_max. Reciprocal plots are also widely used in evaluating the action of inhibitors.

The relationship between K_M and K_d, the dissociation constant of the enzyme-substrate complex,

The affinity of an enzyme for its substrate is inversely proportional to the dissociation constant K_d of the Enz—S complex:

In other words, the lower the tendency of the enzyme-substrate complex to dissociate, the higher the affinity of the enzyme for the substrate.

The value of K_M for a given enzyme-substrate pair can serve as a rough measure of K_d. However, this is valid only if the assumption made in deriving the Michaelis—Menten equation holds true. This assumption states that the first step of the enzymatic reaction

is rapid and that equilibrium is constantly maintained at this stage. In other words, The rate of dissociation of Enz—S into Enz + S is much higher,

It follows from the Michaelis—Menten equation that the value of [S] at which

equals

But if

k_-1 » k₂,

then

k₂ + k ≃ k_-1

and

Under these conditions, 1/K_M = 1/K_M and equals the affinity of the enzyme for the substrate. If k₂ + k_-1≉ k_-1, then 1/K_M will yield an underestimated value of the affinity.