Michaelis-Menten Equation PDF

# The Michaelis-Menten equation ## Introduction The study of enzyme kinetics is the study of how quickly an enzyme catalyzes its reaction. Whereas enzyme activity quantifies the amount of substrate consumed or product produced per unit time, enzyme kinetics focuses on the rate of substrate consump...

# The Michaelis-Menten equation ## Introduction The study of enzyme kinetics is the study of how quickly an enzyme catalyzes its reaction. Whereas enzyme activity quantifies the amount of substrate consumed or product produced per unit time, enzyme kinetics focuses on the rate of substrate consumption or product production as a function of the amount of substrate and enzyme present. The kinetics of an enzyme are often described by the Michaelis-Menten equation. This equation relates the reaction velocity $V_o$ to the maximum possible reaction velocity $V_{max}$, the Michaelis constant $K_M$, and the substrate concentration $[S]$. $V_o = \frac{V_{max}[S]}{K_M + [S]}$ The rate of an enzyme-catalyzed reaction can change based on various factors, including substrate availability, enzyme concentration, and enzyme identity. Different isozymes (ie, different enzymes that catalyze the same reaction) may therefore have different kinetic parameters that allow them to perform different functions in various cells or tissues. A scenario in which two isozymes display different kinetics is graphed in Figure 5.1. ## 5.1.01 Assumptions of the Michaelis-Menten Equation The rate of a nonelementary reaction depends on the rate constants for each elementary step as well as the identification of the rate-determining step. Because enzyme-catalyzed reactions involve binding and unbinding in addition to the chemical reaction itself, the reaction scheme for an enzyme-catalyzed reaction can be simplified into three steps: enzyme-substrate binding, catalysis (ie, conversion of substrate to product), and enzyme-product unbinding (Figure 5.2). - Step 1: Substrate binding - Step 2: Substrate-product conversion (often the rate-determining step) - Step 3: Product unbinding In the reaction scheme shown in Figure 5.2, $E$ is the free enzyme, $S$ is the free substrate, $ES$ is the enzyme-substrate complex, $EP$ is the enzyme-product complex, and $P$ is the product. The forward rate constant for Step 1 is $k_1$, and the reverse rate constant is $k_{-1}$. Similarly, $k_2$, $k_{-2}$, $k_3$, and $k_{-3}$ are the forward and reverse rate constants for their respective steps. Step 2, in which bound substrate is converted to bound product, is typically assumed to be the slow, rate-determining step, whereas the binding and unbinding rates are assumed to occur much faster. In reality, enzymes often follow more complex mechanisms than the one presented in Figure 5.2. For example, some enzymes may include multiple catalytic steps or reaction intermediates. Nevertheless, the simplified model is still applicable to many of these enzymes given the appropriate experimental conditions. The more complex models are generally not tested on the exam, and enzymes are therefore assumed to follow Michaelis-Menten kinetics unless otherwise specified. Importantly, enzymes that exhibit cooperativity do not follow Michaelis-Menten kinetics. See Concept 3.1.04 for more information on cooperativity with general protein-ligand interactions and Concept 5.3.03 for more information on the effects of cooperativity on enzyme kinetics specifically. To derive the Michaelis-Menten equation, several assumptions must be made. These assumptions do not apply to physiological kinetics of an enzyme in vivo but apply only to experimental measurements of enzyme kinetics in vitro. The assumptions and the experimental conditions under which they are valid are irreversibility, the steady state approximation, and the free ligand approximation. - **Irreversibility** Michaelis-Menten kinetics assumes that the enzymatic reaction is irreversible that once product forms (Step 2), it does not revert to substrate. Similarly, the subsequent step of product unbinding is irreversible; free product does not bind again to the enzyme. Given these assumptions, the two sequential, irreversible steps can be combined into a single catalytic step, represented by the forward rate constant $k_{cat}$. No reverse catalytic rate constant applies because this step is assumed to be irreversible. - **Steady state approximation** The steady state approximation supposes that concentration of the enzyme-substrate complex $[ES]$ remains constant throughout the experiment (Figure 5.5). Experimentally, this assumption requires the initial substrate concentration to be large enough that it does not appreciably change during the measurement period. If $[S]$ were to appreciably deplete, then the $[E] + [S] = [ES]$ rapid equilibrium would also shift in accordance with Le Châtelier's principle, which would then cause $[ES]$ to change. In addition, the initial reaction velocity $V_o$ is not measured directly but is instead extrapolated from measurements at later timepoints. Doing this ensures that the data do not include pre-steady state measurements, which would be the reaction rates measured while $ES$ levels were building up but before steady state has been reached. - **Free ligand approximation** In the derivation of the Michaelis-Menten equation, the variable $[S]$ is understood to be the concentration of free (ie, unbound) substrate. However, in practice experimenters rarely measure the free substrate level. Instead, the free ligand (or for enzymes, free substrate) concentration is assumed to be approximately equal to the initial substrate concentration used to set up the experiment. For this assumption to be valid, the substrate concentration must be orders of magnitude greater than the total enzyme concentration (ie, $[S] >> [E_{tot}]$). Under these conditions, the substrate concentration $[S]$ does not appreciably decrease even if the enzyme is fully bound (Figure 5.7).