Enzyme Kinetics – BIOC 192 Lecture 9

Learning Objectives

• Compare enzymes using $K<em>M$, $V</em>{max}$, $k<em>{cat}$, $k</em>{cat}/K_M$
• Measure these via Michaelis–Menten approach
• Recognise biological range & implications of kinetic parameters

Progress Curve & Initial Velocity

• Plot [product] vs. time → initial slope = $V_0$ (use t≈0)
• Always work under conditions where [S] ≫ [E] so [S] is ~constant during initial period

Effect of Enzyme Concentration

• With saturating substrate, $V \propto [E]$ (directly proportional)

Effect of Substrate Concentration

• At low [S]: first-order (rate increases linearly)
• At high [S]: active sites saturated → zero-order (rate independent of [S])
• Resulting $V$ vs $[S]$ curve is a rectangular hyperbola

Michaelis–Menten Equation

• $V = \frac{V<em>{max}[S]}{K</em>M + [S]}$
• Simplifying assumptions:
  • Product does not revert to substrate
  • Steady-state: $\frac{d[ES]}{dt}=0$
  • Initial rate: [S] remains ~constant, [S] ≫ [E]
• Mechanism: $E + S \overset{k<em>1}{\underset{k</em>{-1}}{\rightleftharpoons}} ES \xrightarrow{k_2} E + P$

Key Kinetic Parameters

• $V_{max}$ – maximum velocity when $[S]\to\infty$
• $K<em>M$ – substrate concentration at $V = V</em>{max}/2$
  • Low $K<em>M$ ⇒ high affinity, high $K</em>M$ ⇒ low affinity
  • Approx. dissociation constant when $k<em>2 \ll k</em>{-1}$
• $k<em>{cat} = \frac{V</em>{max}}{[E]_T}$ (turnover number; s⁻¹)
• Catalytic efficiency: $\frac{k<em>{cat}}{K</em>M}$ (best overall measure)

Determining $K<em>M$ and $V</em>{max}$

• Direct reading from hyperbola is imprecise → use double-reciprocal (Lineweaver–Burk) plot:
  • Plot $1/V$ vs $1/[S]$
  • Intercepts: y = $1/V<em>{max}$, x = $-1/K</em>M$
  • Slope = $K<em>M/V</em>{max}$

Physiological & Comparative Insights

• In cells, [S] often < $K_M$ → enzyme rate sensitive to changes in [S]
• For multi-substrate enzymes, lower $K<em>M$ indicates preferred substrate (e.g., hexokinase: $K</em>M$ glucose 0.05 mM vs fructose 1.5 mM)

Typical Values

• Fast enzymes: Catalase $k_{cat} \approx 4\times10^7\;s^{-1}$, Carbonic anhydrase $10^6\;s^{-1}$
• Range across biology spans $10^{-2}$–$10^7\;s^{-1}$; efficiencies vary widely

Summary Points

• Measure $V<em>0$ across [S] → fit to Michaelis–Menten → obtain $V</em>{max}, K_M$
• Knowing $[E]<em>T$ allows $k</em>{cat}$; efficiency = $k<em>{cat}/K</em>M$
• High $k<em>{cat}$ & low $K</em>M$ give the most efficient enzymes

Quick Self-Check

1. What does $V_0$ represent?
2. Relationship between $V$ and $[S]$ at low [S]?
3. With excess substrate, $V$ is ____ to [E].
4. Which two parameters come from $V$ vs [S] curve?
5. Why does $V$ plateau at high [S]?
6. Equation describing hyperbola?
7. Axes of Lineweaver–Burk plot?
8. Meaning of y-intercept? x-intercept?
9. Significance of $K<em>M$ and of high $K</em>M$?
10. Define $k_{cat}$ and catalytic efficiency.