- The steady state approximation states that the rate of change for unstable intermediates in low concentration is zero.
In the previous section, we examined two-step processes in which the second step was rate determining and utilized the equilibrium rates to derive an overall rate expression. In this section, we will discuss an alternate method that is commonly employed in both chemical reactions as well as enzyme kinetics. Given its prevalence in biological systems in Michaelis-Menton kinetics, we will use the classic enzyme model outlined below
Where E represents an enzyme, S represents the substrate, ES represents an enzyme-substrate complex and P is the product.
As with the example in Section 3.2, the first step is in equilibrium and the second step is rate determining. Based on this, the overall rate of this reaction is:
Just like before, it is very difficult to reliably measure [ES] over the course of the reaction, so we must find a way to express [ES] in terms of the reactants. To do this, we are going to make a steady state approximation, which approximates that the rate of change for unstable intermediates in low concentration is zero. In other words, the intermediate is formed at the same rate it is used up such that its concentration stays the same. The following graph helps visualize what this means.
This graph shows the relative concentration of the product, substrate, and enzyme-substrate complex [ES] throughout the reaction. You will note that after a short period of time, the concentration of ES does not change significantly (shown in yellow); as soon as an amount of ES is converted to product, it is rapidly replenished in the initial equilibrium step. Mathematically we can express this as:
when the reaction is in the steady state regime (shown in yellow).
We can now solve for the rate of ES formation and set the rate to 0. To do this, add all of the rate expressions involved in the formation of ES and subtract all of the rate expressions involved in the removal of ES.
Solving for [ES] gives:
Substituting this into the overall rate expression gives:
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