Exponential Fed-Batch Feed Rate and Growth Simulator
Design and inspect exponential nutrient feed profiles F(t) that maintain a constant sub-maximal specific growth rate (μset < μmax) after initial batch carbon depletion in the interactive E. coli bioreactor simulator. The workspace tracks how feed concentration Sf, maintenance coefficient ms, liquid volume accumulation V(t), and dissolved-oxygen ceilings interact across a 24-hour fed-batch horizon, and opens on the pump sheet: the feed rate to set for each interval and the volume fed so far.
Runs the shipped E. coli preset (7 L initial volume, 10 L total vessel working limit, 37 °C) through initial batch depletion followed by exponential glucose feeding.
Derive the open-loop exponential feeding law from substrate and maintenance balances
The goal of an exponential fed-batch phase is to hold the specific growth rate μ at a chosen constant target μset (typically 0.15 to 0.35 h⁻¹ in E. coli, well below μmax to prevent acetate overflow and postpone oxygen starvation) after the initial batch carbon source is exhausted at time t = t0. Under quasi-steady-state carbon limitation, the rate of substrate mass entering the vessel in the feed stream (F(t) × Sf) matches the total cellular consumption rate for new biomass synthesis plus non-growth maintenance: qS,set × X(t) V(t) = (μset / YX/S + ms) × X(t) V(t).
Because total biomass X(t) V(t) grows exponentially as X0 V0 exp(μset(t − t0)) whenever specific growth stays locked at μset, the volumetric feed flow rate F(t) must increase along the exact same exponential exponent exp(μset(t − t0)). Notice that even though total biomass X·V grows strictly exponentially, the biomass concentration X(t) in g L⁻¹ increases slightly sub-exponentially because the liquid volume V(t) in the denominator is simultaneously rising as feed enters the vessel.
Watch for open-loop feed runaway when oxygen limitation or volume caps intervene
An open-loop exponential feed equation is a feedforward schedule: the pump calculates F(t) from elapsed clock time (t − t0) rather than measuring actual real-time biomass or residual sugar. That assumption holds only so long as carbon remains the sole rate-limiting nutrient. In the pre-loaded 24-hour E. coli simulation (7 L initial volume in a 10 L vessel), rising cell density drives volumetric oxygen uptake (OUR) upward until dissolved oxygen C drops and the oxygen Monod term C/(Ko + C) throttles actual specific uptake below the feed target.
Once actual specific growth falls below μset, the pre-programmed exponential pump delivers more glucose per hour than the oxygen-limited cells can oxidize, causing residual substrate S(t) to spike upward while liquid volume V(t) approaches the 10 L vessel ceiling. Inspecting the coupled biomass, substrate, volume, and dissolved-oxygen curves in the workspace makes this physical transition between carbon limitation and oxygen/volume limitation immediately visible.
Equations and variables
Exponential feed volumetric flow rate for constant specific growth μset
F(t) = [ (μset / YX/S + ms) × X0 V0 / (Sf − S) ] × exp(μset × (t − t0))
- F(t)
- Volumetric feed rate at time t ≥ t0 (start of fed-batch phase) (L h⁻¹)
- μset
- Desired constant specific growth rate setpoint (chosen below μmax and oxygen ceiling) (h⁻¹)
- YX/S, ms
- True biomass-on-substrate yield (g g⁻¹) and specific maintenance coefficient (g g⁻¹ h⁻¹)
- X0 V0
- Total cell dry weight (g biomass) in the reactor at feed initiation time t0 (g)
- Sf, S
- Feed reservoir substrate concentration and quasi-steady residual reactor concentration (Sf ≫ S) (g L⁻¹)
Vessel liquid volume integration under exponential feeding
V(t) = V0 + ∫[t0..t] F(τ) dτ = V0 + (F(t0) / μset) × (exp(μset (t − t0)) − 1)
- V(t)
- Total liquid broth volume in the reactor at time t (bounded by vessel working capacity Vmax) (L)
- V0, F(t0)
- Liquid volume (L) and initial feed flow rate (L h⁻¹) at t = t0
Scope and limitations
- Open-loop exponential feeding assumes that total biomass X(t)·V(t) grows exactly at exp(μset(t − t0)); if dissolved oxygen drops or temperature shifts slow actual growth below μset, unconsumed substrate accumulates rapidly.
- Volumetric feed rate F(t) and liquid volume V(t) grow exponentially; any real run eventually hits either the pump/vessel maximum working volume Vmax or the volumetric oxygen transfer ceiling (OTRmax).
- Model trajectories are numerical integrations of the stated balance equations and composite parameters, not validated process forecasts for a commercial fermenter.
Sources and provenance
- Bioreactor model methods and independent verificationOriginal reference calculations; see source notices
Model outputs are simulations, not observations from a reactor.
- Lee 1996 high cell-density culture of Escherichia coliArticle citation; exponential fed-batch feeding equations and maintenance accounting
- Haverkorn van Rijsewijk et al. 2011 E. coli flux and growth measurementsCited parameter facts; existing preset attribution
Technical questions
Why must the exponential feed rate scale with total biomass X0·V0 rather than biomass concentration X0 alone?
The feed stream supplies grams of carbon per hour (F × Sf) to the entire vessel inventory. Total cellular carbon demand per hour is the specific uptake rate qS multiplied by the total grams of biomass in the reactor (X × V), not concentration per liter. Omitting V0 from the pre-exponential coefficient underfeeds any reactor larger than 1 L.
Why is residual substrate S usually neglected in the denominator (Sf − S ≈ Sf) during design?
In a concentrated glucose feed reservoir, Sf is typically 400 to 600 g L⁻¹, whereas the quasi-steady-state residual glucose concentration S in a carbon-limited fed-batch culture is held around 0.05 to 0.2 g L⁻¹. Because S is less than 0.05% of Sf, Sf − S is virtually equal to Sf while the culture remains carbon-limited.
What happens when an exponential feed run reaches the oxygen transfer limit or vessel capacity?
Because oxygen demand OUR scales exponentially with total biomass while maximum volumetric transfer OTRmax is bounded by agitation and aeration limits, dissolved oxygen eventually falls toward Ko. If the open-loop pump keeps increasing F(t) exponentially after oxygen limits growth, glucose accumulates in the broth and vessel volume hits the 10 L working limit.
Connected workflows
- Monod fed-batch growth simulation
Inspect the complete Monod fed-batch growth and maintenance balances.
- Oxygen transfer and uptake modelling
Calculate volumetric oxygen transfer rates (OTR) that cap exponential feeding.
- Viable-cell seeding volumes
Compare dynamic fed-batch growth against static multiwell cell seeding calculations.