# Structural Biochemistry/Enzyme/Michaelis and Menten Equation

V$_0$ = V$_{max}$ ([S]/([S] + K$_M$))
The Michaelis-Menten equation arises from the general equation for an enzymatic reaction: E + S ↔ ES ↔ E + P, where E is the enzyme, S is the substrate, ES is the enzyme-substrate complex, and P is the product. Thus, the enzyme combines with the substrate in order to form the ES complex, which in turn converts to product while preserving the enzyme. The rate of the forward reaction from E + S to ES may be termed k$_1$, and the reverse reaction as k$_{-1}$. Likewise, for the reaction from the ES complex to E and P, the forward reaction rate is k$_2$, and the reverse is k$_{-2}$. Therefore, the ES complex may dissolve back into the enzyme and substrate, or move forward to form product.
At initial reaction time, when t ≈ 0, little product formation occurs, therefore the backward reaction rate of k$_{-2}$ may be neglected. The new reaction becomes:
E + S ↔ ES → E + P
Assuming steady state, the following rate equations may be written as:
Rate of formation of ES = k$_1$[E][S]
Rate of breakdown of ES = (k$_{-1}$ + k$_2$) [ES]
and set equal to each other (Note that the brackets represent concentrations). Therefore:
k$_1$[E][S] = (k$_{-1}$ + k$_2$) [ES]
Rearranging terms,
[E][S]/[ES] = (k$_{-1}$ + k$_2$)/k$_1$
The fraction [E][S]/[ES] has been coined K$_m$, or the Michaelis constant.
According to Michaelis-Menten's kinetics equations, at low concentrations of substrate, [S], the concentration is almost negligible in the denominator as K$_M$ >> [S], so the equation is essentially
V$_0$ = V$_{max}$ [S]/K$_M$
which resembles a first order reaction.
At High substrate concentrations, [S] >> K$_M$, and thus the term [S]/([S] + K$_M$) becomes essentially one and the initial velocity approached V$_{max}$, which resembles zero order reaction.
The Michaelis-Menten equation is:
In this equation:
V$_0$ is the initial velocity of the reaction.
V$_{max}$ is the maximal rate of the reaction.
[Substrate] is the concentration of the substrate.
Lineweaver-Burk Plot -
K$_m$ is the Michaelis-Menten constant which shows the concentration of the substrate when the reaction velocity is equal to one half of the maximal velocity for the reaction. It can also be thought of as a measure of how well a substrate complexes with a given enzyme, otherwise known as its binding affinity. An equation with a low K$_m$ value indicates a large binding affinity, as the reaction will approach V$_{max}$ more rapidly. An equation with a high K$_m$ indicates that the enzyme does not bind as efficiently with the substrate, and V$_{max}$ will only be reached if the substrate concentration is high enough to saturate the enzyme.
As the concentration of substrates increases at constant enzyme concentration, the active sites on the protein will be occupied as the reaction is proceeding. When all the active sites have been occupied, the reaction is complete, which means that the enzyme is at its maximum capacity and increasing the concentration of substrate will not increase the rate of turnover. Here is an analogy which helps to understand this concept easier.
Vmax is equal to the product of the catalyst rate constant (kcat) and the concentration of the enzyme. The Michaelis-Menten equation can then be rewritten as V= Kcat [Enzyme] [S] / (Km + [S]). Kcat is equal to K2, and it measures the number of substrate molecules "turned over" by enzyme per second. The unit of Kcat is in 1/sec. The reciprocal of Kcat is then the time required by an enzyme to "turn over" a substrate molecule. The higher the Kcat is, the more substrates get turned over in one second.
Km is the concentration of substrates when the reaction reaches half of Vmax. A small Km indicates high affinity since it means the reaction can reach half of Vmax in a small number of substrate concentration. This small Km will approach Vmax more quickly than high Km value.
When Kcat/ Km, it gives us a measure of enzyme efficiency with a unit of 1/(Molarity*second)= L/ (mol*s). The enzyme efficiency can be increased as Kcat has high turnover and a small number of Km.
Taking the reciprocal of both side of the Michaelis-Menten equation gives:
To determined the values of K$_M$ and Vmax. The double-reciprocal of Michaelis-Menten equation could be used.
A graph of the double-reciprocal equation is also called a Lineweaver-Burk, 1/Vo vs 1/[S]. The y-intercept is 1/Vmax; the x-intercept is -1/KM; and the slope is KM/Vmax. Lineweaver-Burk graphs are particularly useful for analyzing how enzyme kinematics change in the presence of inhibitors, competitive, non-competitive, or a mixture of the two.
There are four reversible inhibitors: competitive, uncompetitive, non-competitive and mixed inhibitors. They can be plotted on double reciprocal plot. Competitive inhibitors are molecules that look like substrates and they bind to active site and slow down the reactions. Therefore, competitive inhibitors increase Km value (decrease affinity, less chance the substrates can go to active site), and Vmax stays the same. On double reciprocal plot, competitive inhibitor shifts the x-axis (1/[s]) to the right towards zero compared to the slope with no inhibitor present. Uncompetitive inhibitors can bind close to the active site but don't occupy the active site. As a result, uncompetitive inhibitors lower Km (increase affinity) and lower Vmax. On double reciprocal plot, x-axis (1/[s]) is shifted to the left and up on the y-axis (1/V) compared to the slope with no inhibitor. Non-competitive inhibitors are not bind to the active site but somewhere on that enzyme which changes its activity. It has the same Km but lower Vmax to those with no inhibitors. On the double reciprocal plot, the slope goes higher on y-axis (1/V) than the one with no inhibitor. Km value is numerically equal to the substrate concentration at which the half of the enzyme molecules are associated with substrate. km value is an index of the affinity of enzyme for its particular substrate.Non competitive inhibition has no effect to the value of Km.