Every cell consumes dissolved oxygen (DO), so total oxygen demand — the
oxygen uptake rate (OUR) — scales with the whole population. The reactor
replenishes DO by aeration + agitation, but that oxygen-transfer rate (OTR)
has a fixed ceiling set by the hardware. While demand stays under the
ceiling, DO holds near saturation and the culture grows happily. Once the
growing population pushes OUR past the OTR ceiling, DO can no longer keep
up and collapses toward zero — starved cells switch to anaerobic metabolism,
dumping toxic byproduct that stalls further growth.
OUR = q_O2 · N (demand grows with population N)
OTR = OTR_max · (DO_sat − DO)/DO_sat (transfer capped by OTR_max)
dDO/dt = OTR − OUR
dN/dt = μ_max · N · (DO/(DO+K)) · (1 − byproduct/tox) · (1 − N/N_cap)
dByproduct/dt = k_an · N · max(0, 1 − DO/DO_crit)
- Transfer-rate ceiling — raises or lowers OTR_max, i.e. how much aeration/agitation hardware the reactor has.
- Cell population — grows while DO is healthy; growth throttles as DO or byproduct worsen.
- Dissolved O₂ — healthy near 100% until demand exceeds the ceiling, then crashes.
- Byproduct — accumulates only once cells are oxygen-starved (anaerobic fallback), and above a toxic threshold halts growth entirely.