The pH of the environment significantly affects the activity of enzymes, which in turn influences microbial growth rates. This model assumes that the maximum specific growth rate (\(\mu_{max}\)) is directly proportional to the activity of a single, key enzyme whose activity is pH-dependent.
The model considers three forms of the enzyme: a protonated inactive form (EH⁺), the active form (E), and a deprotonated inactive form (E⁻). The dissociation constants \(K_1\) (for EH⁺ ⇌ E + H⁺) and \(K_2\) (for E ⇌ E⁻ + H⁺) describe the transitions between these forms.
The resulting equation for \(\mu_{max}\) as a function of hydrogen ion concentration (\([H^+]\)) is:
μmax = (k⋅etot) / ( 1 + [H⁺]/K1 + K2/[H⁺] )
Where:
- μmax: Maximum specific growth rate at the given pH.
- k⋅etot: The theoretical maximum specific growth rate achievable if the enzyme were fully active (i.e., at the optimal pH). Often referred to as the pH-independent maximum rate constant. Units typically h⁻¹.
- [H⁺]: Hydrogen ion concentration, calculated from pH as \(10^{-pH}\). Units: mol/L.
- K1: First dissociation constant (units: mol/L). Represents the pH range where the enzyme starts losing activity due to protonation (low pH).
- K2: Second dissociation constant (units: mol/L). Represents the pH range where the enzyme starts losing activity due to deprotonation (high pH).
This model typically produces a bell-shaped curve when \(\mu_{max}\) is plotted against pH, with the optimum pH located between pK₁ and pK₂ (where pK = -log₁₀K).