Enzymes and Metabolism

Enzyme structure, function, and regulation of metabolic pathways

⚡ Enzymes and Metabolism

Energy and Metabolism

Thermodynamics in Biology:

Free energy (ΔG): energy available to do work
Exergonic reactions: ΔG < 0 (release energy, spontaneous)
Endergonic reactions: ΔG > 0 (require energy input)

ATP (Adenosine Triphosphate):

Universal energy currency
Stores energy in phosphate bonds
ATP → ADP + P releases ~30.5 kJ/mol

Enzymes

What are enzymes?

Biological catalysts (usually proteins)
Speed up reactions without being consumed
Lower activation energy (Ea)
Do NOT change ΔG of reaction

Structure:

Active site: region where substrate binds
Substrate: reactant molecule
Specific shape determines which substrates bind

Mechanism:

Induced fit model:
- Enzyme changes shape when substrate binds
- Active site molds around substrate
- Forms enzyme-substrate complex
- Products released, enzyme returns to original shape
Enzyme + Substrate ⇌ ES complex → Enzyme + Product

Factors Affecting Enzyme Activity

1. Temperature

Optimal temperature maximizes activity
Too low: slow molecular movement
Too high: denaturation (lose shape)
Most human enzymes optimal at 37°C

2. pH

Each enzyme has optimal pH
Extreme pH denatures enzyme
Examples:
- Pepsin (stomach): pH 2
- Trypsin (intestine): pH 8

3. Substrate Concentration

Low [S]: activity increases with more substrate
High [S]: enzyme saturation (plateau)
Maximum velocity (Vmax) reached

4. Enzyme Concentration

More enzyme = more activity
Linear relationship (if substrate abundant)

Enzyme Regulation

Competitive Inhibition

Inhibitor competes with substrate for active site
Similar shape to substrate
Can be overcome by adding more substrate

Noncompetitive Inhibition

Inhibitor binds to allosteric site (not active site)
Changes enzyme shape → active site altered
Cannot be overcome by adding substrate

Allosteric Regulation

Regulatory molecule binds to allosteric site
Can be activator or inhibitor
Changes enzyme shape and activity

Feedback Inhibition

End product inhibits earlier enzyme in pathway
Prevents overproduction
Example: ATP inhibits glycolysis enzymes

Cofactors and Coenzymes

Cofactors: inorganic helpers (metal ions like Zn²⁺, Fe²⁺)
Coenzymes: organic helpers (vitamins like NAD⁺, FAD)
Required for enzyme function

Key Concepts

Enzymes lower activation energy but don't change ΔG
Active site binds substrate with high specificity
Induced fit: enzyme changes shape upon binding
Temperature and pH affect enzyme shape and activity
Competitive inhibitors block active site
Noncompetitive inhibitors change enzyme shape
Feedback inhibition regulates metabolic pathways

📚 Practice Problems

1Problem 1medium

❓ Question:

Explain enzyme kinetics: (a) describe how substrate concentration affects reaction rate and sketch a Michaelis-Menten curve, (b) define Km and Vmax, and (c) explain what it means if an enzyme has a low Km vs. high Km.

💡 Show Solution

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:

| Enzyme | Km | Affinity | Function | |--------|-----|---------|----------| | Hexokinase | 0.1 mM | High | Glucose uptake (all cells) | | Glucokinase | 10 mM | Low | Glucose sensing (liver) |

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

2Problem 2medium

❓ Question:

💡 Show Solution

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:

| Enzyme | Km | Affinity | Function | |--------|-----|---------|----------| | Hexokinase | 0.1 mM | High | Glucose uptake (all cells) | | Glucokinase | 10 mM | Low | Glucose sensing (liver) |

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

3Problem 3medium

❓ Question:

💡 Show Solution

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:

| Enzyme | Km | Affinity | Function | |--------|-----|---------|----------| | Hexokinase | 0.1 mM | High | Glucose uptake (all cells) | | Glucokinase | 10 mM | Low | Glucose sensing (liver) |

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

4Problem 4hard

❓ Question:

A researcher studies two inhibitors of enzyme X. Inhibitor A increases Km but doesn't change Vmax. Inhibitor B decreases Vmax but doesn't change Km. (a) Identify each type of inhibition, (b) explain the mechanism of each, and (c) sketch Lineweaver-Burk plots for both.

💡 Show Solution

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:

| Type | Active site? | Km | Vmax | Overcome with ↑[S]? | |------|--------------|-----|------|---------------------| | Competitive | Yes (competes) | ↑ | Same | Yes | | Non-competitive | No (allosteric) | Same | ↓ | No | | Uncompetitive | ES complex only | ↓ | ↓ | No |

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)

5Problem 5hard

❓ Question:

💡 Show Solution

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)

6Problem 6hard

❓ Question:

💡 Show Solution

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)

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