Enzyme Kinetics and Inhibition

Buffer Solutions

Identification of a Buffer: A solution containing a weak acid (e.g., $NaH2PO4$ ) and its conjugate base (e.g., $Na2HPO4$ ) in significant, often equal, concentrations and volumes is a buffer. The problem explicitly states equal concentrations and volumes of these two compounds.
pH Calculation for a Buffer: When a weak acid and its conjugate base are present in equal concentrations, the pH of the buffer solution is equal to the $pKa$ of the weak acid.
Relevant Dissociation: The two compounds given, $NaH2PO4$ and $Na2HPO4$ , are involved in the second dissociation step of phosphoric acid: ${H2PO4}^- \rightleftharpoons {HPO_4}^{2-} + H^+ $
pKa Values:
- First dissociation: $pKa_1 = 2.15$
- Second dissociation (relevant here): $pKa_2 = 6.8$
- Third dissociation: $pKa_3 = 12.38$
Concentrations of Other Species: At the pH of $6.8$ (which is equal to $pKa2$ ), the concentrations of other phosphate species (e.g., ${H3PO4}$ or ${PO4}^{3-}$ ) will be vanishingly small. This is because their respective $pKa$ values are far removed from $6.8$
- For ${H3PO4}$ ( $pKa_1 = 2.15$ ), the difference is about $5$ pH units, meaning its concentration would be $100,000$ times ( $10^5$ ) less.
- For ${PO4}^{3-}$ ( $pKa3 = 12.38$ ), the difference is also about $5$ pH units, meaning its concentration would likewise be $100,000$ times ( $10^5$ ) less.

Enzyme Kinetics: Basic Principles

Experimental Setup for Initial Rate Plots:
- Each point on an initial rate vs. substrate concentration plot is obtained by conducting a separate experiment.
- For each experiment, a specific substrate concentration, enzyme amount, temperature, and salt concentration are set.
- The reaction is started, and the initial rate ( $V_0$ ) is determined.
- Subsequent points are obtained by repeating the experiment, changing only the substrate concentration.
Interpreting the Plot (Hyperbolic Curve): This plot is a part of a rectangular hyperbola.
$V_{max}$ (Maximum Velocity):
- Read on the y-axis of the initial rate plot.
- It represents the maximum initial rate achieved when the enzyme is saturated with substrate.
$K_M$ (Michaelis Constant):
- This is the substrate concentration that results in an initial rate equal to half of $V{max}$ ( $V0 = 0.5 V_{max}$ ).
- Common Mistake: $KM$ is not equal to half $V{max}$ ( $KM \neq 0.5 V{max}$ ); it is the substrate concentration at which the rate is half $V{max}$ . A low $KM$ value indicates good interaction and binding between the enzyme and substrate.

Lineweaver-Burk Plot (Double Reciprocal Plot)

Purpose: To linearize the Michaelis-Menten kinetic data for easier determination of kinetic parameters, avoiding the guesswork involved in drawing asymptotes on the hyperbolic plot.
Construction:
- Take the reciprocal of all initial rates ( $1/V_0$ ) and all substrate concentrations ( $1/[S]$ ).
- Plot $1/V_0$ (y-axis) versus $1/[S]$ (x-axis).
Result: The plot yields a straight line.
- This follows the linear equation: $y = mx + b$
- Where: $y = 1/V_0$ and $x = 1/[S]$
Interpreting the Lineweaver-Burk Plot:
- y-intercept: The point where the line crosses the y-axis is equal to $1/V{max}$ . To find $V{max}$ , take the reciprocal of this value (e.g., if y-intercept is $0.01$ , $V_{max} = 1/0.01 = 100$ ).
- x-intercept: The point where the line crosses the x-axis (by interpolating back as $[S]$ cannot be negative) is equal to $-1/KM$ . To find $KM$ , take the reciprocal of the numerical value and ignore the negative sign (e.g., if x-intercept is $-0.1$ , $K_M = 1/0.1 = 10$ ).
- Slope (m): The slope of the line is equal to $KM/V{max}$ .

Additional Enzyme Kinetic Parameters

$k_{cat}$ (Turnover Number):
- Represents how many times a single enzyme molecule performs its catalytic function (i.e., converts substrate to product) within a given amount of time.
- Units: Reciprocal time (e.g., $s^{-1}$ ).
- Interpretation: A $k_{cat}$ of $10,000\ s^{-1}$ means one enzyme molecule can produce $10,000$ product molecules in one second.
- Determination Conditions: $k_{cat}$ is determined when the enzyme is saturated with substrate (i.e., at high substrate concentrations where all enzyme molecules are occupied).
- Significance: Higher $k_{cat}$ indicates higher intrinsic catalytic ability of the enzyme.
Catalytic Efficiency (Specificity Constant):
- Formula: $k{cat}/KM$
- Significance: This parameter reflects the overall effectiveness of the enzyme in converting substrate to product at low substrate concentrations, which are more typical in living systems than saturating conditions.
- Interpretation: A high $k{cat}/KM$ value indicates an enzyme that works very well under physiological (low substrate) conditions.
Diffusion Limit: This is the maximum rate at which an enzyme and substrate can diffuse together in an aqueous solution, typically around $10^8$ or $10^9$ $M^{-1}s^{-1}$ .

Enzyme Inhibition

General Concept: Inhibitors are compounds that decrease enzyme activity. They can be reversible or irreversible.
Product Inhibition: The product of an enzymatic reaction can act as an inhibitor. Since the product also interacts with the enzyme at the active site (often structurally similar to the substrate), a high concentration of product can bind to the enzyme, preventing the substrate from binding and thus decreasing enzyme activity.
Competitive Inhibition:
- Mechanism: A competitive inhibitor (often structurally similar to the substrate) binds reversibly to the enzyme's active site, competing directly with the natural substrate.
- Reversibility: Competitive inhibition is reversible. The inhibitor binds via weak interactions and can unbind. If the normal substrate is present at a sufficiently high concentration, it can outcompete the inhibitor for the active site.
- Effect on Kinetic Parameters:
 - $V_{max}$ : Stays the same (because at very high substrate concentrations, the substrate can overcome the inhibition and still reach the maximal rate).
 - Apparent $KM$ : Increases (a higher substrate concentration is required to reach half $V{max}$ because of the competition).
- Lineweaver-Burk Plot Changes:
 - y-intercept ( $1/V_{max}$ ): Remains unchanged.
 - x-intercept ( $-1/KM$ ): Moves closer to the origin (as $KM$ increases, $-1/K_M$ becomes a smaller negative number).
Irreversible Inhibition ("Suicide Substrates"):
- Mechanism: These inhibitors also bind to the active site but then form a covalent bond with the enzyme, permanently inactivating it.
- Reversibility: Not reversible.
Uncompetitive Inhibition:
- Mechanism: The inhibitor binds only to the enzyme-substrate (ES) complex, not to the free enzyme or the active site directly.
- Effect on Kinetic Parameters:
  - $V_{max}$ : Decreases.
  - $K_M$ : Stays the same (as stated in the lecture).
- Lineweaver-Burk Plot Changes: This was not explicitly detailed, but generally, both the x and y intercepts change, with parallel lines appearing.
Mixed Inhibition: Briefly mentioned as another type of inhibition.
Practical Role of Inhibitors: Inhibitors are often used as tools by biochemists in the laboratory to study enzyme activity and mechanisms, rather than primarily as in vivo regulators in the immune system (though some exist).

Enzyme Substrate Specificity and Reaction Mechanisms

Multiple Substrates/Reactions: While the discussed kinetic models typically assume a single substrate, enzymes can interact with more than one substrate or catalyze reactions involving multiple substrates.
Kinetic Parameters for Different Substrates: If an enzyme can act on multiple substrates (e.g., similar molecules), the kinetic parameters ( $KM, V{max}$ ) will likely differ for each, with the intended or 'designed' substrate usually showing a lower $KM$ (tighter binding) and potentially higher $V{max}$ (more efficient catalysis).
Kinetics and Mechanisms: Kinetics is a valuable tool for deducing the step-by-step mechanism of an enzymatic reaction.
Spectrometry in Reaction Progress: Spectrophotometers are commonly used to monitor the progress of reactions by tracking changes in absorbance over time, which can then be converted to concentration over time data.

Interpreting Enzymatic Mechanisms (Lysozyme Example)

Mechanism Interpretation: Students are expected to interpret given enzymatic mechanisms, not memorize them.
Lysozyme Example:
- Substrate: The enzyme acts on two sugar molecules (monosaccharides) linked by a glycosidic bond, which are components of a bacterial cell wall.
- Enzyme Active Site: The active site contains a glutamic acid residue, among other parts that stabilize the substrate.
- Reaction Type: The enzyme catalyzes the hydrolysis of the glycosidic bond. This is identified by the breaking of a covalent bond using water ( $H_2O$ ).
- Biological Significance: Lysozyme hydrolyzes the bonds in bacterial cell walls, thus killing the bacteria. It is present in human tears, providing an immune defense mechanism by preventing bacteria from infecting the eyes.