Enzyme Kinetics and Inhibition Notes

Enzyme Kinetics and Inhibition

Learning Objectives

• Identify the properties of enzymes, their classification, regulation, and kinetics using equations, models, and graphs.

What is an Enzyme?

• Biological catalysts.
• Usually proteins, sometimes RNA (ribozymes).
• Essential for metabolism and DNA replication.
• Speed up reactions without being consumed.

Factors Influencing Reaction Rate

• Increasing temperature: Up to 42°C (onset of protein denaturation).
• Increasing concentration of reactants.
• Adding a catalyst.
• pH:
  • Pepsin in the stomach (acidic).
  • Lipase in the pancreas (basic).

Definition of Terms

• Substrate: Specific molecule that an enzyme acts on.
• Active Site: Special region on the enzyme where the substrate binds.
• Activation Energy ($\Delta G^{\ddagger}$): Minimum amount of energy needed for a chemical reaction to begin.
• Specificity: Enzyme’s ability to choose a specific substrate.

Enzymes and Reaction Speed

• Enzymes speed up chemical reactions by lowering the activation energy.
• Enzymes make reactions happen fast enough at body temperature and pH.
• Most biological reactions are too slow without enzymes.

Enzyme Classes and Reactions

1. Oxidoreductases: Oxidation-reduction reactions.
2. Transferases: Transfer of functional groups.
3. Hydrolases: Hydrolysis reactions.
4. Lyases: Group elimination to form double bonds.
5. Isomerases: Isomerization reactions.
6. Ligases: Bond formation coupled with ATP hydrolysis.
7. Translocases: Solute transport through membranes.

Example Reaction

• Alanine + α-Ketoglutarate catalyzed by alanine aminotransferase forms Pyruvate + Glutamate.

Catalytic Mechanisms

• Idealized Transfer Reaction: $(AB) + (C) \rightarrow (A) + (BC)$
• Involves a progress of reaction coordinate and free energy changes.

Definition of Terms (cont.)

• Reaction Coordinate: Progress of the reaction.
• Transition State: Point of highest free energy; midway between reactants and products.
• Enzyme-Substrate Complex: Temporary molecule formed when the enzyme binds the substrate.
• Activation Energy ($\Delta G^{\ddagger}$): Energy-requiring step or energy barrier. Height of $\Delta G^{\ddagger}$ determines the rate of a reaction; higher $\Delta G^{\ddagger}$ = less likely rxn will occur = slower.

Catalytic Mechanisms in Nature

• Involves reactants, products, and the reaction coordinate, showing the free energy change during the reaction.

Catalyst and Activation Energy

• Catalyst (enzyme) provides a reaction pathway with a lower $\Delta G^{\ddagger}$.
• Uncatalyzed reaction has a higher $\Delta G^{\ddagger}_{uncat}$.
• Catalyzed reaction has a lower $\Delta G^{\ddagger}_{cat}$.

Enzyme Catalysts and Free Energy Change

• Enzyme catalysts do not alter $\Delta G$ of a reaction.
• It merely provides a pathway from reactants to products through a transition state.
• Enzymes lower the height of $\Delta G^{\ddagger}$ by lowering the energy of the transition state.

Lock-and-Key Model

• The enzyme and substrate fit together like a key to a lock.
• Fails to explain the dynamic changes that accompany catalysis.
• Less accepted model.

Induced Fit Model

• Binding of substrate induces conformational changes in the active site of the enzyme.
• Explains dynamic changes.
• More accepted model.

Enzyme Kinetics

• Kinetics = how fast enzymes work.
• Helps understand enzymatic reactions and regulation.

Reaction Velocity

• Progress of reaction can be expressed as velocity (v).
• Rate of disappearance of substrate (S) or rate of appearance of the product (P).

Enzyme Kinetics and Substrate Concentration

• When enzyme concentration is held constant, the reaction velocity varies with [S] in a nonlinear (hyperbolic) fashion.
• Suggests that an enzyme physically combines with its substrate to form an ES complex.

Enzyme Kinetics: Reaction Scheme

• $E + S \rightleftharpoons ES \rightarrow E + P$

Enzyme Kinetics: Saturation

• As substrate concentration increases, the reaction velocity reaches a saturation point.

Enzyme Kinetics: Unimolecular Reactions

• Unimolecular (first-order) reaction: involves a single reactant.
• $k$ = rate constant in s$-1$
• $v$ is directly proportional to [A].

Enzyme Kinetics: Bimolecular Reactions

• Bimolecular (second-order) reaction: involves 2 reactants.
• $k$ = rate constant in M$-1$ s$-1$
• $v$ is directly proportional to [A] x [B].

Problem: Determining Reaction Velocity

• Determine the velocity of the reaction $X + Y \rightarrow Z$ when the sample contains 3 μM X and 5 μM Y and $k$ for the reaction is 400 M$-1$ s$-1$.
• Is the reaction uni- or bimolecular?
• Express the answer in M/s.

Solution

• The reaction is bimolecular.
• $v = 6 \times 10^{-9}$ M/s

Michaelis-Menten Equation

• Rate equation for an enzyme-catalyzed reaction.
• E binds its S (in an ES complex) before converting it to product.
• Overall reaction has first- & second-order reactions.
• Each with a characteristic rate constant.

Michaelis-Menten Equation: Historical Context

• Derived by Leonor Michaelis and Maude Menten.
• Mathematical description of the hyperbolic curve.

Michaelis-Menten Equation: Terms

• $v_0$ = initial velocity (velocity at the start of the reaction at time 0).
• $V_{max}$ = maximum reaction velocity; proportional to [E].
• $K_m$ = Michaelis constant.

Michaelis-Menten Equation: Formula

• $v<em>0 = \frac{V</em>{max}[S]}{K_m + [S]}$

Michaelis-Menten Equation: Problem

• An enzyme-catalyzed reaction has a $K<em>M$ of 1 mM and a $V</em>{max}$ of 5 nM⋅s$-1$. What is the reaction velocity when the substrate concentration is 0.25 mM?

Michaelis-Menten Equation: Answer

• Answer: $1 \times 10^{-9}$ M/s or 1 nM/s

Michaelis Constant ($K_m$)

• [S] at which $v<em>0$ is half-maximal (i.e., $v</em>0 = \frac{V<em>{max}}{2}$ when $[S] = K</em>m$).
• Indicates how efficiently an enzyme selects its substrate and converts it to product.
• Lower $K_m$ = enzyme is more effective at low [S].
• Higher $K_M$ = the less effective the enzyme is.
• Inversely proportional to the affinity of the enzyme for its substrate.

Michaelis-Menten Equation: Kinetics

• First-order kinetics: Rate of reaction is dependent on [S].
• Zero-order kinetics: Rate of reaction is independent of [S].

Lineweaver-Burk Plot

• In practice, hyperbolic curves are prone to misinterpretation; difficult to estimate the upper limit of the curve ($V_{max}$).
• Linear transformation = Lineweaver–Burk plot.

Lineweaver-Burk Plot: Equation

• Reciprocal of the Michaelis-Menten equation:
• $\frac{1}{v<em>0} = (\frac{K</em>M}{V<em>{max}}) \frac{1}{[S]} + \frac{1}{V</em>{max}}$

Lineweaver-Burk Plot: Slope-Intercept Form

• Recall: Slope-intercept form: $y = mx + b$
  • $x$ is the independent variable.
  • $m$ is the slope of the line.
  • $y$ is the dependent variable.
  • $b$ is the y-intercept of the line.

Lineweaver-Burk Plot: Equation (cont.)

• $\frac{1}{v<em>0} = (\frac{K</em>M}{V<em>{max}}) \frac{1}{[S]} + \frac{1}{V</em>{max}}$
• $y = mx + b$

Lineweaver-Burk Plot: Interpretation

• Plot of 1/v versus 1/[S] gives a straight line whose slope is $K<em>m /V</em>{max}$ and whose intercept on the 1/v axis is $1/V_{max}$.

Lineweaver-Burk Plot: Graph

• The extrapolated intercept on the 1/[S] axis is $-1/K_m$.
• $Slope = \frac{K<em>M}{V</em>{max}}$

Lineweaver-Burk Plot: Velocity Problem

• The velocity of an enzyme-catalyzed reaction was measured at several substrate concentrations. Calculate $K<em>M$ and $V</em>{max}$ for the reaction.
• Solution: Calculate the reciprocals of the substrate concentration and velocity, and then make a plot of 1/vo versus 1/[S] (a Lineweaver-Burk plot).

Lineweaver-Burk Plot: Calculations

• Recall: $\frac{1}{v<em>0} = (\frac{K</em>M}{V<em>{max}}) \frac{1}{[S]} + \frac{1}{V</em>{max}}$
• $Slope = \frac{K<em>M}{V</em>{max}}$

Lineweaver-Burk Plot: Solution

• The intercept on the 1/[S] axis (which is equal to $-1/K_M$) is -1.33 μM$-1$. Therefore,
• $K_M = -\frac{1}{-1.33 \mu M^{-1}} = 0.75 \mu M$
• The intercept on the 1/v axis (which is equal to $1/V_{max}$) is 0.33 mM$-1$⋅s. Therefore,
• $V_{max} = \frac{1}{0.33 mM^{-1} \cdot s} = 3.0 mM \cdot s^{-1}$

Enzyme Inhibition

• Enzyme inhibitor: Any substance that can diminish the velocity of an enzyme-catalyzed reaction.
• Many naturally occurring substances inhibit the activity of essential enzymes.
• Examples: antibiotics, pesticides, and other poisons.

Enzyme Inhibition: Reversible

• The inhibitor binds temporarily to the enzyme (e.g., to the active site or elsewhere).
• The enzyme can regain activity if the inhibitor is removed.
• Examples: Ibuprofen, mefenamic acid.

Enzyme Inhibition: Irreversible

• Permanently disables the enzyme molecule.
• The only way to restore function is to make new enzyme.
• Example: Aspirin.

Competitive Inhibition

• Inhibitor binds to the active site, competing with substrate.
• Increases $K_m$ due to low affinity.
• No change in $V_{max}$ due to unchanged [E].
• May be reversed by increasing [S]; outcompetes the inhibitor.

Competitive Inhibition: Alpha Factor

• $\alpha$ = factor that makes $K_m$ appear larger.
• Recall: High $K_m$, lower affinity.

Uncompetitive Inhibition

• Inhibitor binds only to the ES complex which is rare.
• Decreases $K_m$. Inhibitor stabilizes the ES complex. Enzyme