Week 10 - Enzymology

Lecture 1: Enzymology 10

1. Catalytic Efficiency & the Diffusion Limit

Core idea: Enzymes cannot exceed the rate at which molecules collide in solution.

• Catalytic perfection occurs when
  kcat/KM ≈ 10⁸–10⁹ M⁻¹ s⁻¹
  → reaction occurs immediately upon E–S collision.
  (Transcript: “the limit is about 10⁹… as soon as the substrate and enzyme collide…”)

• Most enzymes are NOT perfect.
  Bar‑Even et al. survey:

  • Average kcat ≈ 10 s⁻¹

  • Average kcat/KM ≈ 10⁵ M⁻¹ s⁻¹
    (Transcript: “average enzyme was pretty moderate… 10 turnovers per second… kcat/KM about 10⁵”)

• Evolutionary logic:

  • High‑flux pathways (e.g., glycolysis) → fast enzymes.

  • Secondary metabolism → slower enzymes.

  • “Maximal rates may not evolve where weaker selection pressures exist.”

2. Michaelis–Menten Kinetics Refresher

Rectangular hyperbola

• Plot of Vo vs [S] gives hyperbolic curve.

• Vo ∝ [E] at all [S]
  (Transcript: “velocity is directly proportional to enzyme concentration at all substrate concentrations.”)

Why we need linear transformations

• Cannot experimentally reach infinite [S] → cannot directly observe Vmax.

• Use Lineweaver–Burk (double reciprocal):

  

  
  From the plot:

• Y‑intercept = 1/Vmax

• X‑intercept = –1/KM

• Slope = KM/Vmax

• Other transformations exist (Eadie–Hofstee), but only LB is examinable.

3. Why Data Quality Depends on Assumptions

The Michaelis–Menten derivation requires:

Assumption 1 — Initial rates

• Must measure before:

  • substrate depletion occurs

  • product accumulates enough to drive the reverse reaction
    (Transcript: “we want to ignore the reverse reaction… ensure substrate is not being depleted.”)

Assumption 2 — [S]free ≈ [S]added

• Only true at early time points.

Assumption 3 — [E] ≪ [S]

• Ensures ES formation does not significantly deplete substrate.

4. Designing a Valid Enzyme Assay

Goal: Ensure product formation is linear with:

• time, and

• enzyme concentration
  (Transcript: “rates should be proportional to both enzyme concentration and time.”)

Preliminary experiments required

Use the lowest [S] planned for the assay because it is most vulnerable to depletion.

A. Linearity with time

• Plot [P] vs time at several [E].

• Choose a time window where curves are straight.

B. Linearity with enzyme concentration

• Plot [P] vs [E] at several time points.

• Choose an [E] range where product formation is proportional to [E].

Why not use extremely low [E] and short times?

• Harder to detect product (poor sensitivity).

• Longer times + higher [E] (within linearity) → better signal.

5. What Happens When Assumptions Are Violated

Case study from transcript

• Student obtained a sigmoidal‑looking Vo vs [S] curve.

• Problem: underestimation of rates at low [S] due to substrate depletion.

• After re‑establishing linear conditions → curve became rectangular hyperbola.

Key lesson:
Sigmoidal artefacts often arise from bad assay design, not real cooperativity.

6. Comparing Enzymes Using KM, kcat, kcat/KM

Example: CYP1A1, CYP1A2, CYP2A13 metabolising theophylline

(Table from slides + transcript discussion)

Which has highest affinity?

• Lowest KM → CYP1A2
  (Transcript: “KM should be small… 1A2 has the lowest KM.”)

Which contributes most in vivo?

Depends on:

1. Drug concentration

   • At low [S] → high‑affinity enzyme dominates (CYP1A2).

   • At high [S] → enzyme with highest Vmax dominates (CYP1A1).

2. Relative enzyme abundance

   • If CYP2A13 is 10× more abundant, it becomes dominant despite lower catalytic efficiency.

Take‑home: Contribution = f([S], kcat, KM, [E]).

7. Enzyme Inhibition — Competitive Inhibition

Mechanism

• Inhibitor resembles substrate and binds active site.

• Competes with S for E.

• Can be overcome by ↑[S].

Kinetic effects

• Vmax unchanged

• KM increases by factor α
  
  

  

Lineweaver–Burk pattern

• Lines intersect at Y‑axis.

• Slope increases with [I].

• Y‑intercept (1/Vmax) unchanged.

Example: Methotrexate

• Competitive inhibitor of dihydrofolate reductase.

• Binds same site as folate.

8. Binding Interactions & Inhibition Types

• Enzymes bind substrates via weak interactions (H‑bonds, ionic, van der Waals).

• Inhibitors can bind:

  • active site (competitive)

  • ES complex (uncompetitive)

  • allosteric site (noncompetitive)

• Irreversible inhibitors form covalent bonds → enzyme must be resynthesised.

Exam‑Ready Summary

• kcat/KM defines catalytic efficiency; diffusion limit ≈ 10⁹.

• Most enzymes are moderately efficient due to evolutionary constraints.

• Vo ∝ [E] at all [S].

• Use Lineweaver–Burk to obtain KM and Vmax.

• Valid kinetics requires initial rates, no substrate depletion, linearity with time and [E].

• Artefacts (e.g., false sigmoids) arise from violating assumptions.

• Enzyme contribution depends on [S], kcat, KM, and [E].

• Competitive inhibition: ↑KM, Vmax unchanged, LB lines intersect at Y‑axis.

Lecture 2: Enzymology 11

1. Uncompetitive Inhibition

Core idea: Inhibitor binds only to ES, at an allosteric site, forming ESI.

Mechanism

• I binds after substrate is bound → ES → ESI

• I does not resemble substrate

• Binding site ≠ active site (allosteric)

Kinetic consequences

• Vmax decreases

• KM decreases (enzyme appears to have ↑ affinity)

• Both decrease by same factor α′ = 1 + [I]/KI′

• Cannot be overcome by ↑[S]

Lineweaver–Burk pattern

• Parallel lines (distinctive hallmark)

Example

• Topoisomerase poisons (e.g., camptothecin)

  • Intercalate into DNA only when DNA is bound in the active site

  • Bind enzyme–DNA complex, blocking catalytic cycle

2. Non‑Competitive Inhibition

Core idea: Inhibitor binds to both E and ES, at an allosteric site.

Mechanism

• I binds to E → EI

• I binds to ES → ESI

• Two dissociation constants: KI (E), KI′ (ES)

Kinetic consequences

• Vmax decreases (always)

• KM may change or may not

  • Pure non‑competitive: KM unchanged (KI = KI′)

  • Mixed non‑competitive: KM ↑ or ↓ depending on relative KI vs KI′

Lineweaver–Burk patterns

• General: Lines converge behind the Y‑axis

• Pure: Converge on X‑axis

• Mixed:

  • KI < KI′ → lines meet above X‑axis

  • KI > KI′ → lines meet below X‑axis

Example

• Alanine inhibition of pyruvate kinase

  • Binds at “amino acid site” on both E and ES

  • Site is far from active site (domain interface)

3. Why Lineweaver–Burk Plots Matter

• Distinguish inhibitor type by pattern of lines:

  • Competitive: intersect on Y‑axis

  • Uncompetitive: parallel

  • Non‑competitive: converge behind Y‑axis

  • Pure non‑competitive: converge on X‑axis

4. Regulation of Enzyme Activity in Cells

Cells regulate flux through pathways using:

A. Amount of enzyme

• Transcriptional, post‑transcriptional, post‑translational

• Slow, costly

B. Amount of substrate

• Limited to substrate availability

• Not sufficient for pathway‑level coordination

C. Structural changes to enzyme

• Covalent modification

• Regulatory proteins

• Allosteric regulation (major focus)

5. Allosteric Regulation

Key features

• Effector binds allosteric site

• Causes conformational change → alters activity

• Can inhibit or activate

• Effects can be transmitted over long distances in multi‑domain proteins

Functional consequences

• Change in:

  • Active site geometry

  • Affinity for substrate

  • Exposure of new binding sites

  • Domain orientation

6. Regulation of Metabolic Pathways

Why regulate?

• Maintain balanced flux

• Prevent wasteful accumulation of end products

Regulatory enzymes typically:

• Catalyse first committed step

• Catalyse irreversible reactions (large –ΔG′)

Feedback inhibition

• End product inhibits first enzyme in pathway

• Classic mechanism for controlling flux

7. Case Study: Phosphofructokinase (PFK)

PFK is a central regulatory enzyme in glycolysis.

Structural basis

• Tetramer with T (tense, inactive) and R (relaxed, active) states

• Active sites and allosteric sites lie at domain interfaces

Cooperativity

• Binding of F6P shifts equilibrium → R state

• R state binds F6P better → positive cooperativity

• Produces sigmoidal substrate saturation curve

Homo- vs heterotropic effects

• Homotropic: F6P enhances its own binding

• Heterotropic: ATP, AMP, ADP, citrate, F2,6‑BP modulate activity

ATP as dual regulator

• ATP = substrate + allosteric inhibitor

• High ATP → binds allosteric site → shifts to T state → ↓ affinity for F6P

• Low ATP → R state favoured → ↑ glycolysis

8. Cooperativity & the Hill Equation

Used for enzymes with sigmoidal kinetics.

[ V_0 = \frac{V_{\max}[S]^n}{K_{0.5} + [S]^n} ]

• n > 1: positive cooperativity

• n < 1: negative cooperativity

• n = 1: no cooperativity (M–M kinetics)

• KM replaced by K0.5 (substrate conc. giving half‑maximal velocity)

9. Structural Basis of Allostery

• Allosteric sites often at interfaces between domains or subunits

• Small conformational shifts → large functional effects

• Example: PFK, pyruvate kinase, ATCase

Ultra‑Condensed Take‑Home Summary

• Uncompetitive: binds ES only → ↓KM, ↓Vmax → parallel LB lines

• Non‑competitive: binds E + ES → ↓Vmax → KM unchanged (pure) or changed (mixed)

• Pure: lines meet on X‑axis

• Mixed: lines meet above/below X‑axis

• Allosteric regulation: effector binds non‑active site → conformational change

• Cooperativity: substrate binding alters affinity of other sites → sigmoidal kinetics

• PFK: classic allosteric enzyme; regulated by ATP, AMP, citrate, F2,6‑BP

• Regulatory enzymes: first committed step, irreversible, control pathway flux