Enzyme Kinetics:

(a) Effect of substrate concentration:

At low [S]:

• Few substrate molecules
• Many active sites available
• Increasing [S] sharply increases rate
• First-order kinetics (rate ∝ [S])

At intermediate [S]:

• Some active sites occupied
• Rate increases but less steeply
• Mixed-order kinetics

At high [S]:

• All active sites saturated
• Maximum rate achieved
• Further ↑[S] has no effect
• Zero-order kinetics (rate independent of [S])

Michaelis-Menten Curve:

Velocity (v)
    ^
Vmax|_ _ _ _ _ _ _ _ _ ___________
    |                  /
    |                 /
Vmax|_ _ _ _ _ _ _   /
 2  |             \ /
    |              X  ← Km
    |            / |
    |          /   |
    |        /     |
    |      /       |
    |____/____|____|_____________> [S]
              Km

(b) Definitions:

Vmax (Maximum velocity):

• Rate when enzyme is saturated with substrate
• All active sites occupied
• Cannot go faster (limited by enzyme concentration)
• Units: μmol/min, mM/s, etc.

Km (Michaelis constant):

• Substrate concentration at half-maximal velocity (Vmax/2)
• Measure of affinity:
  • Low Km = high affinity (reaches Vmax quickly)
  • High Km = low affinity (needs more substrate)
• Units: mM, μM, nM

Michaelis-Menten equation:

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

At [S] = Km: $v = \frac{V_{max} \cdot K_m}{K_m + K_m} = \frac{V_{max}}{2}$

(c) Interpretation of Km values:

Low Km (e.g., 0.01 mM):

• ✓ High affinity for substrate
• Enzyme binds substrate tightly
• Reaches Vmax at low [S]
• Efficient at low substrate concentrations
• Example: Hexokinase for glucose (Km ~ 0.1 mM)
  • Works well even when blood glucose is normal (5 mM)

High Km (e.g., 10 mM):

• ✗ Low affinity for substrate
• Enzyme binds substrate weakly
• Needs high [S] to reach Vmax
• Only efficient when substrate abundant
• Example: Glucokinase in liver (Km ~ 10 mM)
  • Only active when blood glucose is high (after meal)
  • Acts as "glucose sensor"

Comparison:

Physiological significance:

• Hexokinase: active even at low glucose → ensures cells get energy
• Glucokinase: only active at high glucose → liver stores excess as glycogen

$\boxed{\text{Low } K_m = \text{ high affinity; High } K_m = \text{ low affinity}}$

Lineweaver-Burk plot (double reciprocal):

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

Linearizes data:

• y-intercept = 1/Vmax
• x-intercept = -1/Km
• slope = Km/Vmax

Enzyme Inhibition Analysis:

(a) Identification:

Inhibitor A: ↑ Km, Vmax unchanged $\boxed{\text{Competitive inhibition}}$

Inhibitor B: ↓ Vmax, Km unchanged $\boxed{\text{Non-competitive inhibition}}$

(b) Mechanisms:

Competitive Inhibition (Inhibitor A):

Mechanism:

• Inhibitor structurally similar to substrate
• Competes for same active site
• Binds reversibly to free enzyme (E)
• E + I ⇌ EI (inactive)

Effect:

• Km increases (apparent affinity decreases)
  • Need more substrate to outcompete inhibitor
  • K_m(app) = Km(1 + [I]/Ki)
• Vmax unchanged
  • Can still reach max rate with enough substrate
  • High [S] outcompetes inhibitor

Example: Malonate inhibits succinate dehydrogenase

• Malonate similar to succinate
• Competes for active site in Krebs cycle

Equation: $v = \frac{V_{max}[S]}{K_m(1 + [I]/K_i) + [S]}$

Overcoming: ↑ substrate concentration

Non-competitive Inhibition (Inhibitor B):

Mechanism:

• Inhibitor binds to allosteric site (not active site)
• Can bind to E or ES complex
• E + I ⇌ EI, ES + I ⇌ ESI
• Changes enzyme conformation → reduces activity

Effect:

• Vmax decreases (fewer functional enzyme molecules)
  • V_max(app) = Vmax/(1 + [I]/Ki)
  • Essentially reduces [E]_total
• Km unchanged
  • Affinity for substrate not affected
  • Substrate still binds normally to unaffected enzymes

Example: Heavy metals (Pb²⁺, Hg²⁺) bind to sulfhydryl groups

• Distort protein shape
• Reduce activity

Equation: $v = \frac{V_{max}[S]}{(K_m + [S])(1 + [I]/K_i)}$

Overcoming: CANNOT overcome with ↑ [S]

(c) Lineweaver-Burk Plots:

Competitive Inhibition:

1/v  ^
     |    \  +Inhibitor (slope ↑)
     |     \
     |  \   \
     |   \   \
     | No inh.\
     |     \   \
     |______\___\_________> 1/[S]
     |       \   \
     |        \   \
             -1/Km(app)
              ↑         -1/Km
        (shifts left)   (no inhibitor)

Key features:

• Same y-intercept (1/Vmax unchanged)
• Different x-intercepts (Km changes)
• Different slopes (steeper with inhibitor)
• Lines converge on y-axis

Non-competitive Inhibition:

1/v  ^
     |
     |    +Inhibitor (higher y-int)
     |___________________
     |    \
     |     \ No inhibitor
     |      \____________
     |       \
     |________\___________> 1/[S]
              |
            -1/Km
        (same x-intercept)

Key features:

• Different y-intercepts (1/Vmax changes)
• Same x-intercept (-1/Km unchanged)
• Different slopes
• Lines converge on x-axis (if pure non-competitive)

Comparison Table:

Mixed inhibition (bonus):

• Both Km and Vmax change
• Can bind E or ES with different affinities
• Lines intersect above or below x-axis

$\boxed{\text{Competitive: } \uparrow K_m; \text{ Non-competitive: } \downarrow V_{max}}$

Clinical relevance:

• Statins: competitive inhibitors of HMG-CoA reductase (cholesterol synthesis)
• Aspirin: irreversible inhibitor of COX enzyme (anti-inflammatory)
• Methotrexate: competitive inhibitor of dihydrofolate reductase (cancer treatment)