Free Online Fed-Batch Feeding Calculator | Exponential Feed Trajectory

Q: What is the Crabtree effect and how does fed-batch prevent it?

The Crabtree effect describes aerobic fermentation where cells produce ethanol or acetate even in the presence of excess dissolved oxygen when glucose uptake exceeds respiratory capacity. Controlled feeding limits glucose availability, forcing 100% aerobic respiration.

Q: How do I select the optimal set-point growth rate (μ_set)?

$\mu_{\text{set}}$ is typically chosen at 60%–75% of maximum specific growth rate (e.g. $\mu_{\text{set}} = 0.15 ext{–}0.25\,\text{hr}^{-1}$ for *E. coli*) to optimize specific productivity while preventing oxygen limitation.

Q: What is DO-stat feeding?

When cells deplete available substrate, metabolic oxygen demand drops instantly, causing dissolved oxygen (DO) to spike. DO-stat automated controllers detect this spike and trigger the feed pump to administer the next bolus.

FEED FORMULATION & TARGETS RECIPE

Starting Broth Volume V₀ (L) 3.00 L

Biomass at Feed Start X₀ (g/L) 15.0 g/L

Feed Stock Substrate S_in (g/L) 500 g/L

Target Growth Rate μ_set (h⁻¹) 0.150 h⁻¹

Kept below μ_crit to suppress overflow acetate/ethanol

Yield Coeff Y_X/S (g DCW / g) 0.450 g/g

Feed Duration (hours) 12.0 h

SUBSTRATE FEED TRAJECTORY & BROTH VOLUME PROFILE F(t) • V(t)

Pump Setpoint Schedule (Sampled Timepoints)

Time (h)	Feed Rate (mL/h)	Broth Vol (L)	Biomass (g/L)	Glucose Fed (g)

UPSTREAM HANDOFF ➔ STEP 7
Calculated feed rate trajectory F(t) seamlessly drives continuous substrate delivery in Bioreactor 3D Simulator & Digital Twin (Step 7).

PROCESS BOUNDS & MASS BALANCE BALANCE

Specific Glucose Uptake (qs)

0.333 g/g/h

qs = μ_set / Yx/s (Below Crabtree limit)

Total Glucose Consumed

505.0 g

Yielded ~227 g new dry cell biomass

Volume Expansion Ratio

+33.7% Increase

Final liquid volume: 4.01 L

Feeding Strategy Status

Optimal Exponential Profile

Maintains strictly constant specific growth rate. Avoids catabolite repression and glucose accumulation.

📚 Fed-Batch Fermentation Feeding Trajectory & Overflow Control Guide Upstream Bioprocess • Step 6 of 7

Theoretical Principles & Engineering Fundamentals

Fed-batch fermentation is the gold-standard production mode for therapeutic proteins, recombinant enzymes, and industrial biochemicals. By feeding concentrated carbon substrate incrementally, high cell densities ($>100 ext{ g/L CDW}$) are achieved without accumulating toxic fermentative byproducts (acetate in E. coli, ethanol in yeast).

An exponential feeding trajectory maintains a constant specific growth rate ($\mu_{ ext{set}} < \mu_{ ext{crit}}$) below the overflow threshold:

$$F(t) = rac{\mu_{ ext{set}}}{Y_{X/S} imes S_{ ext{feed}}} imes X_0 imes V_0 imes e^{\mu_{ ext{set}} imes t}$$

Governing Equations & Mathematical Formulations

Exponential Feed Rate Trajectory F(t) = \frac{\mu_{\text{set}}}{Y_{X/S} \times S_{\text{feed}}} X_0 V_0 e^{\mu_{\text{set}} t}

Maintains constant growth rate $\mu_{\text{set}}$ below the threshold of overflow metabolism.

Culture Volume Expansion V(t) = V_0 + \int_0^t F(\tau) d\tau

Tracks cumulative broth volume as concentrated substrate is metered into the vessel.

Biomass Yield on Substrate Y_{X/S} = \frac{\Delta X}{\Delta S}

Biomass produced per gram of carbon substrate consumed (typically 0.45–0.50 g/g for glucose).

Industrial Benchmark Data & Parameter Reference

Feeding Regime	Growth Rate Behavior	Byproduct Risk	Industrial Use Case
Exponential Feed	Constant (μ = μ_set)	Low (Controlled below μ_crit)	Recombinant protein expression
Linear Feed	Decaying (μ decreases with t)	Very Low	Secondary metabolite synthesis
DO-Stat Feeding	Feedback-driven (DO spikes trigger pump)	Zero Overflow	Automated high-density fermentation
pH-Stat Feeding	Feedback-driven (pH shifts trigger pump)	Low	Acetate/lactate consumption cycles

Frequently Asked Questions (Bioprocess Engineering FAQ)

What is the Crabtree effect and how does fed-batch prevent it?

The Crabtree effect describes aerobic fermentation where cells produce ethanol or acetate even in the presence of excess dissolved oxygen when glucose uptake exceeds respiratory capacity. Controlled feeding limits glucose availability, forcing 100% aerobic respiration.

How do I select the optimal set-point growth rate (μ_set)?