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We recall the results from the previous section for the enzyme mechanism model that leads to the Michaelis-Menten (MM) equation of enzyme kinetics. Refer to the previous section as needed for the details.
In the above mechanism: E denotes the enzyme, S denotes the substrate, P denotes the products and ES denotes the enzyme-substrate complex. By use of the Steady-State Approximation (SSA) on the [ES], the intermediate, we arrive at the Michaelis-Menten (MM) equation listed below. As is commonly done, we write it in terms of initial rate of product formation. The use of the initial velocity (v_0) - usually defined as the measured velocity before more than about 10 percent of the substrate (S) has been converted to product (P) minimizes complicating factors such as the reversibility of the second step of the mechanism as well as inhibition of the enzyme by product and progressive inactivation of the enzyme.
Michaelis-Menten Equation:
In the above equation, [E]_T is the total enzyme concentration that is initially delivered to the system. K_M is termed the Michaelis constant and is defined as
As the [S] becomes very large, the velocity of the reaction will not increase indefinitely, but, for a fixed amount of [E]_T - will reach a limiting value termed V_{max}, the maximal velocity. Previously, you were asked to prove that at large [S], i.e. for [S] » K_M, a maximal velocity (V_{max}) is reached - at saturation. So,
Note: when [S] » K_M, i.e. v_0 approximately equals V_{max}, we have zero-order kinetics, i.e. the rate of reaction does not depend on [S], but rather on the rate of "catalytic conversion" and release of the product by the enzyme-substrate complex (ES). Substituting V_{max} for k_2[E]_T in our boxed expression above gives the "usual" form of the Michaelis-Menten (MM) equation:
This expression is the basic equation of enzyme kinetics. This equation was plotted previously and is reproduced below.
The above equation reveals several interesting features. First of all, the Michaelis constant (K_M) can be seen to be the [S] concentration for which the (initial) rate reaction reaches half (1/2) of the maximum, i.e.
Since K_M is directly proportional to k_2, this means that an enzyme with a small K_M value (see the table at the end of this handout for some representative enzyme-substrate systems) will achieve maximal catalytic activity at low [S]. Additional interesting information is revealed if we look at the limiting case of small [S], i.e. [S] » K_M. If the substrate concentration is so small such that [S] » K_M, then K_M + [S] in the denominator of the MM equation approximately equals K_M. If we approximate the denominator of the MM equation merely by K_M, the MM equation becomes
Refer to the plot of v_0 versus [S] and convince yourself that the rate of reaction is first-order (linear) at small [S] and zero-order at large [S].
Note that k_2 is sometimes referred to as the "catalytic constant" since - according to the proposed mechanism above - it is the rate constant for the enzyme reacting with the substrate (i.e. as a catalyst) to produce the product. Therefore, another useful parameter in Michaelis-Menten enzyme kinetics is the turnover number of an enzyme. It is a measure of the number of reaction processes, i.e. number of ES complexes that are converted to product, per enzyme molecule per unit time. Since k_2 is the rate constant that determines this conversion rate, it is easy to see that
See the table below for K_M and k_{cat} for some representative enzyme-substrate systems.
| Enzyme (E) | Substrate (S) | K_M (in M) | k_{cat} (in s^{-1}) | k_{cat}/K_M (in M^{-1}s^{-1}) |
|---|---|---|---|---|
| Acetylycholine esterase | Acetylcholine | 9.5 x 10^{-5} | 1.4 x 10^4 | 1.5 x 10^6 |
| Carbonic Anhydrase | CO_2 | 0.012 | 1.0 x 10^6 | 8.3 x 10^7 |
| Carbonic Anhydrase | HCO_3- | 0.026 | 4.0 x 10^5 | 1.5 x 10^7 |
| Catalase | H_2O_2 | 0.025 | 1.0 x 10^7 | 4.0 x 10^8 |
| Fumerase | Fumerate | 5.0 x 10^{-6} | 800 | 1.6 x 10^8 |
| Fumerase | Malate | 2.5 x 10^{-5} | 900 | 3.6 x 10^7 |
| Urease | Urea | 0.025 | 1.0 x 10^4 | 4.0 x 10^5 |
Analysis of Enzyme Kinetic Data According to Michaelis-Menten Model
There are several available methods for determining the parameters from the MM equation - such as K_M and V_{max}. In practice, it is very difficult to accurately determine V_{max} from a direct plot of v_0 versus [S] (why?). A better method for determining the values of V_{max} and K_M was formulated by Hans Lineweaver and Dean Burk and is termed the Lineweaver-Burk or double reciprocal plot. Specifically, it is a plot of 1/v_0 versus 1/[S]. As we will see below, such a plot - according to the Michaelis-Menten model - should yield a line with a slope of K_M/V_{max} and a 1/v_0 ("y") intercept of 1/V_{max}.
In order to understand the above remarks, we recall the MM equation - written in terms of V_{max} and K_M is given by
Taking the reciprocal of both sides of the above equation yields
Simplifying the above equation gives the Lineweaver-Burk relationship:
Thus, if 1/v_0 is assigned as the dependent variable ("y") and 1/[S] is assigned as the independent variable ("x"), it is easy to see that the coefficient of 1/[S] ("x") is K_M/V_{max} (slope) and the 1/v_0 ("y") intercept is 1/V_{max}. Also, the extrapolated 1/[S] ("x") intercept is (-1/K_M). A disadvantage of the Lineweaver-Burk plot is that most experimental measurements involve relatively high [S] and are thus crowded toward the left side of the plot (see an example plot here). In addition, for small values of [S], small errors in v_0 will lead to large errors in 1/v_0 and thus to large errors in K_M and in V_{max}. If (when) you take a course in biochemistry, you will learn of more sophisticated plots that can give more accurate values of the Michaelis-Menten parameters mentioned above.
There is also a relevant diagram, a plot of competitive inhibitor behavior, and a final Lineweaver-Burk plot showing the effect of a competitive inhibitor.
