**Here, we continue our understanding of drug targets, this time focusing on enzyme kinetics.**

Enzymes are biological macromolecules that act as catalysts in biochemical reactions. The metabolic processes and chemical reactions that sustain life often involve enzymes. Enzymes basically decrease the Gibbs free energy of activation (ΔG^{‡}), accelerating the rate of reaction. The majority of enzymes are proteins, and proteinaceous enzymes will be the focus of this section. Catalytic RNA molecules are also known and they are referred to as ribozymes. Just like other proteins under normal conditions, proteinaceous enzymes adopt certain three dimensional conformations that are crucial to their activity. Enzymes sometimes recruit other organic molecules or inorganic cofactors which may be essential to catalytic activity. An enzyme’s **substrate** is the molecule upon which enzymes act. In biochemistry, the suffix *-ase* is used to name the majority of enzymes.

As drug targets, enzymes are highly important. Enzyme inhibition has many applications. Many enzymes can be inhibited to treat a vast range of medical conditions. HMG-CoA reductase inhibitors as an example were described in a previous article. Other examples of drugs that act on enzymes include the antimalarial compound **pyrimethamine** (dihydrofolate reductase inhibitor), **clavulanic acid** (β-lactamase inhibitor), and the antineoplastic agent **lapatinib** (dual tyrosine kinase inhibitor).

#### Introduction to Michaelis-Menten Kinetics

Enzyme kinetics is the study of enzyme-catalysed chemical reactions. Enzymes are highly efficient catalytic systems. Generally, they can enhance reaction rates by a factor of 10^{5} – 10^{17}. Studying the kinetics of enzyme-catalysed reactions can be useful in determining an enzyme inhibitor’s properties. **Michaelis-Menten kinetics** is a widely known and useful model of enzyme kinetics. The model was named after Leonor Michaelis and Maud Menten, the individuals who described the mathematical model relating reaction rate and substrate concentration **[S]**. In chemistry, square brackets are often used to denote concentration (for example, [X] means ‘concentration of X’). Square brackets are also used when describing coordination complexes.

The site where substrate molecules binds to the enzyme and undergoes chemical reactions is referred to as the **active site** (sometimes referred to by some as the **orthosteric site**). Sites other than a protein’s active site are referred to as the **allosteric site**. Inhibitors can bind to the active site or to an allosteric site. Enzyme inhibitors that are able to form covalent bonds with the enzyme are referred to as **irreversible inhibitors**. Inhibitors that bind to active sites through intermolecular bonds are called **reversible inhibitors.** Reversible inhibitors are divided into:

- Competitive Inhibition
- Uncompetitive Inhibition
- Noncompetitive Inhibition

It is important to remember that different people may use different symbols or characters in the following equations. Don’t be bothered by this. It is more important to understand the meaning of the equations. If your understanding of basic algebra and chemical kinetics is a little rusty, it would be a good idea to go over those before proceeding.

The rate of enzyme-catalysed reactions is often referred to as enzyme velocity or simply velocity. Note that enzyme velocities are typically reported at time zero. µ mol min^{-1} is a commonly used unit of measurement for V_{0}. The complex formed between an enzyme and its substrate is referred to as an **ES complex** (enzyme-substrate complex). The Michaelis-Menten model of enzyme kinetics uses the concept of the formation of the ES complex. Under the model, the enzyme and the substrate form an ES complex. The formed ES complex can either form the product P and regenerate the enzyme, or dissociate back to E and S. The model is expressed in the equation shown below.

\[E + S \overset{k_1}{\underset{k_2}{\rightleftharpoons}} ES \stackrel{k_3}{\to} E + P\]

The formation of the product (P) from ES is assumed to be irreversible. The reaction rate constants are k_{1}, k_{2}, and k_{3}.

- k
_{1}: rate constant for the formation of ES - k
_{2}: rate constant for the dissociation of ES back to E + S - k
_{3}: rate constant for the formation of E + P

Based on studies of many enzymes, it was found that under conditions of relatively low [S], the rate increases almost linearly as [S] increases. In other words, the rate is directly proportional to [S] at relatively low [S]. At higher [S], the rate increases by smaller and smaller amounts as [S] increases until the reaction rate becomes almost constant.

The shape of the curve is hyperbolic. L. Michaelis and M. Menten arrived with an equation that describes the above relationship. The equation is named the **Michaelis-Menten equation **and is given by:

\[v = \frac{V_{max}[S]}{K_m + [S]}\]

**Michaelis-Menten Equation**

Note that in chemistry, the lowercase Greek letter ν (nu) is sometimes used to denote the rate of reaction. The derivation of the equation will not be shown here. As seen in the graph, ν increases as [S] increases, asymptotically approaching V_{max}. When L. Michaelis and M. Menten derived the equation, a new constant, **K _{M}** (Michaelis-Menten constant) was defined. K

_{M}is defined as the concentration of the substrate where the reaction rate is half of the maximum reaction velocity, V

_{max}. The unit of measurement of K

_{M}is M. The magnitude of KM varies widely depending on the enzyme and the substrate. K

_{M}is also equal to the sum of k

_{2}and k

_{3}over k

_{1}.

\[K_m = \frac{k_2 + k_3}{k_1}\]

The value of K_{M} can also give insight on substrate binding affinity for the enzyme. High K_{M} values usually indicate weak substrate affinity for the enzyme, whereas low K_{M} values usually indicate strong substance affinity for the enzyme.

#### Lineweaver-Burk Plot

Since ν asymptotically approaches V_{max}, it is extremely difficult to assess V_{max} from plots of [S] against ν. The parameters of the Michaelis-Menten Equation can be determined through linearised forms of the equation such as the Lineweaver-Burk equation and the Hanes-Woolf equation. The **Lineweaver-Burk plot** would be the plot of 1/[S] (x-axis) against 1/ν (y-axis). As you would recall from mathematics, the equation of a straight line is *y = mx + c *where *m* is the slope of the equation and *c* is the constant. The Lineweaver-Burk equation has this form.

\[y = mx + c\]

**Equation of a Straight Line**

\[\frac{1}{v} = \frac{K_m}{V_{max}} . \frac{1}{[S]} + \frac{1}{V_{max}}\]

**Lineweaver-Burk Equation**

Extracting the value of V_{max} is easier because it is simply the reciprocal of the constant of the equation. Once V_{max} is known, K_{M} can be determined from the slope of the line. The x-axis intercept is equal to -1/K_{m} and the y-axis intercept is equal to 1/V_{max}. Lineweaver-Burk plots are extremely useful. They can also provide information on the type of inhibition compounds give.

**Lineweaver-Burk Plot**

#### Lineweaver-Burk Equation Derivation

\[v = \frac{V_{max}[S]}{K_m + [S]}\]

**Michaelis-Menten Equation**

Rearranging the Michaelis-Menten equation gives (1)

\[\frac{V_{max}}{v} = \frac{K_m + [S]}{[S]}\hspace{5 mm}(1)\]

Dividing both sides of the equation with V_{max} gives (2)

\[\frac{1}{v} = \frac{K_m + [S]}{[S]V_{max}}\hspace{5 mm}(2)\]

Rearranging (2) gives (3)

\[\frac{1}{v} = \frac{K_m}{[S]V_{max}} + \frac{[S]}{[S]V_{max}}\hspace{5 mm}(3)\]

The [S] cancels to give (4)

\[\frac{1}{v} = \frac{K_m}{[S]V_{max}} + \frac{1}{V_{max}}\hspace{5 mm}(4)\]

Rearranging (4) gives the Lineweaver-Burk equation

\[\frac{1}{v} = \frac{K_m}{V_{max}} . \frac{1}{[S]} + \frac{1}{V_{max}}\]

**Lineweaver-Burk Equation**

#### Competitive Inhibitors

Reversible competitive inhibitors compete with substrate molecules for the enzyme’s active site. These inhibitors are typically structurally similar to the substrate. Enzymes may bind to the substrate or the inhibitor but not both simultaneously. At very high [S], more substrate can bind to the enzyme and they can outcompete the inhibitor, overcoming inhibition.

\[\begin{align*}&E + S{\rightleftharpoons}ES{\to}P \\&+ \\&I \\&{\downarrow} \\&EI\end{align*}\]

Statins are competitive inhibitors. ACE inhibitors such as perindopril and fosinopril are other examples of competitive inhibitors. ACE inhibitors are mainly employed as antihypertensives and can be used to treat congestive heart failure.

#### Uncompetitive Inhibitors

Uncompetitive inhibitors are enzyme inhibitors that bind to the ES complex. Uncompetitive inhibitors are sometimes referred to as anti-competitive inhibitors. A Lineweaver-Burk plot of uncompetitive enzyme inhibition is shown below. Lithium, a known uncompetitive inhibitor, is used to treat conditions such as manic depression.

\[\begin{align*}E + S{\rightleftharpoons}E&S{\to}P \\+& \\I& \\{\downarrow}& \\E&SI\end{align*}\]

#### Noncompetitive Inhibitors

Noncompetitive inhibitors can bind to the enzyme whether or not a substrate has already been bound. Noncompetitive inhibition is a form of allosteric inhibition. It is important to mention that not all inhibitors that bind at allosteric sites are noncompetitive inhibitors. The binding of the inhibitor at an allosteric site causes an overall change of the three dimensional structure of the enzyme. The consequence of this conformational change is reduced catalytic activity.

\[\begin{align*}&E + S{\rightleftharpoons}ES{\to}P \\&+\hspace{14 mm}+ \\&I\hspace{17 mm}I \\&{\downarrow}\hspace{15 mm}{\downarrow} \\E&I+S{\rightleftharpoons}ESI\end{align*}\]

The antidepressant, tranylcypromine is an example of a nonselective noncompetitive, irreversible inhibitor of the monoamine oxidase enzyme (MAO). Tranylcypromine is unable to distinguish between the MAO-A and MAO-B isozymes of MAO. This drug is also used as an anxiolytic agent.

**Tranylcypromine**

#### Hanes-Woolf Plot

The **Hanes-Woolf plot** is another method that linearises the Michaelis-Menten equation, plotting [S] (x-axis) against [S]/ν (y-axis). Just like the Lineweaver-Burk equation, the Hanes-Woolf equation is of the form *y = mx + c*. The Hanes-Woolf plot is thought to be more accurate than Lineweaver-Burk for the determination of kinetic parameters. The plot and the equation are shown below.

**Hanes-Woolf Plot**

\[y = mx + c\]

**Equation of a Straight Line**

\[\frac{[S]}{v} = \frac{1}{V_{max}} . [S] + \frac{k_m}{V_{max}}\]

**Hanes-Woolf Equation**

#### Hanes-Woolf Equation Derivation

\[v = \frac{V_{max}[S]}{K_m + [S]}\]

**Michaelis-Menten Equation**

Inverting the Michaelis-Menten Equation gives (1)

\[\frac{1}{v} = \frac{k_m + [S]}{V_{max}[S]}\hspace{5 mm}(1)\]

Multiplying both sides of the equation with [S] gives (2)

\[\frac{[S]}{v} = \frac{[S](k_m + [S])}{V_{max}[S]}\]

\[\frac{[S]}{v} = \frac{k_m + [S]}{V_{max}}\hspace{5 mm}(2)\]

Rearranging (2) gives (3)

\[\frac{[S]}{v} = \frac{[S]}{V_{max}} + \frac{k_m}{V_{max}}\hspace{5 mm}(3)\]

Rewriting (3) gives the Hanes-Woolf equation

\[\frac{[S]}{v} = \frac{1}{V_{max}} . [S] + \frac{k_m}{V_{max}}\]

**Hanes-Woolf Equation**

Studying enzyme kinetics can prove beneficial to understanding how drugs work at enzymes. A combination of an understanding of a drug’s mode of action and structure-activity relationships, it may be possible to apply drug design methods to improve drug activity.

#### Summary of Equations

\[v = \frac{V_{max}[S]}{K_m + [S]}\] | \[K_m = \frac{k_2 + k_3}{k_1}\] |

Michaelis-Menten Equation |
Michaelis-Menten Constant |

\[\frac{1}{v} = \frac{K_m}{V_{max}} . \frac{1}{[S]} + \frac{1}{V_{max}}\] | \[\frac{[S]}{v} = \frac{1}{V_{max}} . [S] + \frac{k_m}{V_{max}}\] |

Lineweaver-Burk Equation |
Hanes-Woolf Equation |

#### Further Reading

Chemical kinetics is described in most introductory chemistry textbooks. Enzyme kinetics is described in far greater detail in many undergraduate level biochemistry text books. Many of these books also show the derivation of the Michaelis-Menten equation.

Fundamentals of Enzyme Kinetics 4^{th} Edition

*A. Cornish-Bowden*, © 2012 Wiley-VCH Verlag & Co. KGaA, Weinheim, Germany.

Enzyme Kinetics: Principles and Methods 2^{nd} Edition

*H. Bisswanger*, © 2008 Wiley-VCH Verlag & Co. KGaA, Weinheim, Germany.

Quantum Mechanical Methods for Enzyme Kinetics

*Annu. Rev. Phys. Chem.*, 2002,**53**, pp 467-505.