explain that the maximum rate of reaction (Vmax) is used to derive the Michaelis–Menten constant (Km), which is used to compare the affinity of different enzymes for their substrates

Learning Objective

Explain how the maximum rate of reaction (V_max) is used to derive the Michaelis–Menten constant (K_m), and show how K_m can be used to compare the affinity of different enzymes for their substrates. In doing so, discuss quantitatively how substrate concentration, enzyme concentration, temperature, pH, inhibitors and immobilisation affect V_max and/or K_m.

1. Introduction to Enzyme Kinetics

• Enzymes are biological catalysts that lower the activation energy of a reaction.
• The rate of an enzyme‑catalysed reaction depends on:

  • the physical‑chemical environment (temperature, pH, etc.)
  • the interaction between enzyme and substrate (affinity, saturation)

• When every enzyme molecule is bound with substrate the reaction proceeds at its theoretical maximum, V_max. The relationship between V_max, substrate concentration ([S]) and the Michaelis–Menten constant (K_m) is described by the Michaelis–Menten equation (see Section 3).

2. Quantitative Effect of the Six Factors on V_max and K_m

2.1 Substrate concentration ([S])

• Increasing [S] increases the initial rate (v) because more enzyme‑substrate (ES) complexes form.
• When [S] ≫ K_m, the enzyme is saturated and v approaches V_max.
• Effect on kinetic parameters: no change in either V_max or K_m; they are intrinsic properties of the enzyme–substrate pair.

2.2 Enzyme concentration ([E]total)

• More enzyme molecules mean more active sites are available.
• Mathematically, V_max = k_cat [E]total, where k_cat (turnover number) is the number of substrate molecules converted per enzyme per second.
• Effect on kinetic parameters:

  • ↑ V_max (proportional to [E]total)
  • K_m unchanged (affinity is an intrinsic property).

2.3 Temperature

• Rising temperature increases kinetic energy → more frequent, energetic collisions → higher reaction rate.
• Up to the optimum temperature:

  • ↑ V_max (greater catalytic turnover)
  • K_m may fall slightly (enhanced binding) or remain unchanged, depending on the enzyme.

• Beyond the optimum, thermal denaturation distorts the active site:

  • ↓ V_max (loss of catalytic activity)
  • ↑ K_m (lower apparent affinity).

2.4 pH

• Each enzyme has an optimum pH at which ionisable side‑chains in the active site are correctly charged.
• Deviation from the optimum:

  • ↓ V_max (reduced catalytic efficiency)
  • ↑ K_m (weaker substrate binding) – the effect is usually more pronounced for acids or bases that alter the charge of key residues.

2.5 Inhibitors

*Uncompetitive inhibition is included in the A‑Level specification (Topic 3, sub‑topic 3.2).

2.6 Immobilised Enzymes

• Enzyme molecules are physically trapped (e.g., entrapment in calcium alginate beads, covalent attachment to a solid support).
• Typical kinetic consequences:

  • ↓ V_max – diffusion of substrate into the matrix limits the rate at which ES complexes can form.
  • ↑ K_m (apparent) – the effective substrate concentration at the active site is lower because of mass‑transfer resistance.

• Advantages (relevant for industry and practical work):

  • Easy separation of enzyme from product → reusable catalyst.
  • Often increased thermal and pH stability because the support matrix protects the protein.
  • Facilitates continuous‑flow processes.

2.7 Summary Table – Direction of Change for Each Factor

3. Michaelis–Menten Kinetics

The relationship between initial velocity (v) and substrate concentration ([S]) for many enzymes is:

• V_max – rate when every enzyme molecule is saturated with substrate.
• K_m – substrate concentration at which v = ½ V_max. It provides a quantitative measure of affinity: lower K_m → higher affinity.
• From the definition of V_max we have V_max = k_cat [E]total, where k_cat is the turnover number (s⁻¹).

4. Determining V_max and K_m from Experimental Data

4.1 Michaelis–Menten Plot (v vs [S])

1. Measure initial rates (v) for a series of substrate concentrations, keeping [E]total, temperature and pH constant.
2. Plot v (y‑axis) against [S] (x‑axis). The curve approaches an asymptote – read this as V_max.
3. The substrate concentration at which the curve reaches half‑maximal height is K_m.

4.2 Lineweaver–Burk (double‑reciprocal) Plot – Worked Example

Given the following data for an enzyme assay (enzyme concentration constant, temperature = 37 °C, pH = 7.0):

1. Calculate the reciprocals 1/[S] (µM⁻¹) and 1/v (min mg µmol⁻¹).
2. Plot 1/v (y‑axis) against 1/[S] (x‑axis). Fit a straight line (least‑squares).
3. From the linear equation 1/v = (K_m/V_max)·(1/[S]) + 1/V_max obtain:

   • y‑intercept = 1/V_max → V_max = 1 / (y‑intercept)
   • x‑intercept = –1/K_m → K_m = –1 / (x‑intercept)

Using the data above (rounded to two decimal places):

Linear regression gives the equation 1/v = 0.84·(1/[S]) + 0.20.

• y‑intercept = 0.20 → V_max = 1 / 0.20 = 5.0 µmol min⁻¹ mg⁻¹
• x‑intercept = –0.84⁻¹ = –1.19 → K_m = 1 / 1.19 ≈ 0.84 µM

Interpretation: The enzyme reaches half‑maximal velocity at a very low substrate concentration (0.84 µM), indicating high affinity.

Limitations of the Lineweaver–Burk Plot

• Both axes are reciprocals; any experimental error at low [S] (large 1/[S] values) is magnified, giving disproportionate weight to the least reliable points.
• Modern practice prefers non‑linear regression of the Michaelis–Menten equation, but the LB plot remains a required skill for the Cambridge A‑Level exam.

4.3 Eadie–Hofstee Plot (optional)

Plots v (y‑axis) against v/[S] (x‑axis). The slope equals –K_m and the y‑intercept equals V_max. It reduces the weighting problem of the LB plot because the axes are not reciprocals.

5. Inhibition – Kinetic Signatures

• Competitive inhibition: inhibitor competes with substrate for the active site.

  • V_max unchanged.
  • Apparent K_m increases (requires more substrate to reach ½ V_max).
  • On a LB plot the lines intersect on the y‑axis.

• Non‑competitive inhibition: inhibitor binds a separate site, reducing the number of functional enzymes.

  • V_max decreases.
  • K_m unchanged.
  • LB lines intersect on the x‑axis.

• Uncompetitive inhibition (A‑Level): inhibitor binds only to the ES complex.

  • Both V_max and K_m decrease proportionally.
  • LB plots give a set of parallel lines (same slope, higher y‑intercept).

6. Immobilised Enzymes – Practical Implications

• Typical experiment: compare free catalase with alginate‑entrapped catalase.

  1. Prepare equal enzyme activities (e.g., same protein concentration) in free and immobilised forms.
  2. Measure initial rates at a range of H₂O₂ concentrations.
  3. Plot Michaelis–Menten and Lineweaver–Burk curves for each preparation.

• Interpretation:

  • A right‑shift of the curve (higher apparent K_m) indicates diffusion limitation.
  • A lower asymptote (lower V_max) shows that the maximum catalytic turnover is reduced, often because not all immobilised enzyme molecules are accessible.

7. Practical Investigation – Mapping to the Syllabus

The following activity satisfies the Cambridge requirement to plan an experiment, collect data, analyse and evaluate (Topic 3, sub‑topic 3.2):

1. Planning

   • Define the research question: “How does immobilisation affect the kinetic parameters of catalase?”
   • Identify variables:

     • Independent: enzyme form (free vs. immobilised), substrate concentration.
     • Dependent: initial rate (Δ[O₂] / Δt).
     • Controlled: temperature (25 °C), pH (7.0), enzyme concentration (standardised by protein assay).

   • Choose appropriate substrate concentrations (e.g., 0.5–10 mM H₂O₂) to cover the region below and above the expected K_m.

2. Data collection

   • Measure the volume of O₂ evolved in a fixed time interval for each substrate concentration.
   • Repeat each measurement at least three times to obtain an average and standard deviation.

3. Data analysis

   • Convert O₂ volume to µmol min⁻¹ mg⁻¹ (using gas‑law conversion).
   • Construct Michaelis–Menten and Lineweaver–Burk plots for both enzyme forms.
   • Calculate V_max and K_m from the LB plots (or by non‑linear regression if software is available).

4. Evaluation checklist

   • Sources of random error – timing of gas collection, temperature fluctuations.
   • Systematic error – inaccurate calibration of the gas syringe, incomplete mixing of substrate.
   • Specific to immobilisation – diffusion limitation within the bead, possible enzyme leakage.
   • Repeatability – have you performed replicates and obtained consistent values?
   • Improvement suggestions – use a stirring bath to minimise diffusion gradients, test different bead sizes.

8. Using K_m to Compare Enzyme Affinity

Because K_m is the substrate concentration required to reach half‑maximal velocity, it provides a direct, quantitative way to compare how tightly different enzymes bind the same substrate, or how one enzyme behaves with different substrates.

9. Summary (Key Points for Revision)

• V_max = k_cat [E]total – the theoretical maximum rate when the enzyme is fully saturated.
• K_m = [S] at which v = ½ V_max. Lower K_m → higher substrate affinity.
• Quantitative effects of the six factors:

  • Substrate concentration – no change in kinetic parameters.
  • Enzyme concentration – ↑ V_max, K_m unchanged.
  • Temperature – ↑ V_max up to optimum; beyond optimum ↓ V_max and ↑ K_m.
  • pH – optimum gives maximal V_max and minimal K_m; deviations cause ↓ V_max and ↑ K_m.
  • Inhibitors – competitive (↑ K_m), non‑competitive (↓ V_max), uncompetitive (↓ both).
  • Immobilisation – typically ↓ V_max and ↑ apparent K_m because of diffusion limits.

• Lineweaver–Burk plots allow extraction of V_max and K_m, but the double‑reciprocal transformation amplifies errors at low [S]; students should comment on this limitation.
• Uncompetitive inhibition, though optional in some curricula, is part of the A‑Level specification and must be recognised.
• Practical skills: design experiments, collect reliable data, construct kinetic plots, calculate V_max and K_m, and critically evaluate sources of error.
• Comparing K_m values provides a quantitative method for judging which enzyme has the stronger affinity for a given substrate.