Governing Equations

Lac Operon Regulation Model:

Repressor Binding:
  [R·O] = K_R × [R] / (K_d + [R]) × [O]
  Active repressor = [R] when no lactose/IPTG
  Inactive repressor = [R·allolactose] or [R·IPTG]

cAMP-CAP Activation:
  [CAP·cAMP] = K_C × [CAP] × cAMP / (K_d + cAMP)
  cAMP ∝ 1 / [glucose]  (inverse relationship)
  CAP active only when [glucose] is low

Transcription Rate:
  d[mRNA]/dt = k_tx × (1 - [R·O]) × [CAP·cAMP] / (K_m + [CAP·cAMP]) - k_deg × [mRNA]

Translation:
  d[Protein]/dt = k_tl × [mRNA] - k_prot_deg × [Protein]

β-galactosidase Activity:
  Activity = k_cat × [Protein] × [substrate] / (K_m + [substrate])

Logic:
  - No glucose + lactose/IPTG → MAX expression (CAP on, repressor off)
  - Glucose present → LOW expression (CAP off, catabolite repression)
  - No lactose/IPTG → ZERO expression (repressor on, blocks promoter)
  - IPTG = gratuitous inducer (binds repressor, not metabolized)