Fed-Batch Feed Rate Calculator for E. coli and Yeast

The growth phase starts at limiting-substrate depletion. Let $M=XV$ be total dry biomass. A constant target growth rate gives $dM/dt=\mu M$, so $M(t)=M_0e^{\mu t}$ with $M_0=X_0V_0$.

Choose the yield basis before calculating substrate demand. With a maintenance-free true growth yield, the Pirt relation is $q_S=\mu/Y_{X/S}^{G}+m_S$. With an observed yield measured at the selected growth rate, $q_S=\mu/Y_{X/S}^{obs}$ and the additional maintenance term is zero because maintenance is already included. With negligible residual substrate, $S_fF=q_SM$. Thus $F(t)=F_0e^{\mu t}$, $F_0=q_SM_0/S_f$, and cumulative feed is $V_f=F_0(e^{\mu t}-1)/\mu$. The liquid balance gives $V=V_0+V_f$ and $X=M/V$. This derivation uses the standard balances discussed by Pirt (1975), Korz et al. (1995) and the published Acatian worked example; it uses the initial biomass inventory rather than substituting the changing culture volume into the exponential prefactor.

Whole-reactor oxygen demand is $OUR=q_{O2}M$, converted from mmol/h to mol/h. Estimated total heat is $Q=H_{O2}OUR+P_{aux}$, converted from kJ/h to kW. The editable default $H_{O2}=460$ kJ/mol follows the respiratory oxycaloric approximation described in Altwasser et al. (2017), §2.5, Eq. 2. It is a positive heat-removal load here.

The oxygen and pump equality times are $\ln(OTR_{cap}/OUR_0)/\mu$ and $\ln(F_{max}/F_0)/\mu$. Cooling is converted to an equivalent oxygen capacity $(Q_{cap}-P_{aux})/H_{O2}$ before applying the same logarithmic relation. The liquid-fill equality is $\ln[1+\mu(V_{max}-V_0)/F_0]/\mu$. The first capacity equality is the minimum of these four times. The displayed endpoint is the earlier of that equality and the requested duration. Tied limits all apply; the label displays one of them.

The growth threshold is a separate physiological input constraint: the target must remain strictly below it. No post-limit growth, feed cap, or automatic lower-growth phase is implied. Checkpoint flows are instantaneous values of the continuous ramp, not piecewise-constant pump commands.

For the yeast substrate-demand check, the attached Vos et al. open-access article text gives the maintenance estimate on text-extraction page 6 and the approximate aerobic chemostat uptake on text-extraction page 13. These attachment page numbers are not printed journal page numbers.

The default physiology is one matched row of Van Hoek et al. (1998), Table 1: S. cerevisiae DS28911 on glucose, μ = 0.15 h⁻¹, observed yield 0.49 g/g and measured qO₂ = 3.9 mmol/(g dry biomass·h), at 30 °C, pH 5, and dissolved oxygen above 60% air saturation. This implies qS = 0.30612 g/(g·h), or 1.699 mmol glucose/(g·h), agreeing with the separately reported glucose uptake of 1.7 mmol/(g·h) after rounding. The reference ceiling is 0.28 h⁻¹, where the table first reports small ethanol formation. No separate maintenance coefficient is inferred from that single row.

• Reference biology: S. cerevisiae DS28911, glucose, aerobic mineral medium, 30 °C, pH 5, oxygen-nonlimiting (study dissolved oxygen >60% air saturation). The matched μ, observed yield and oxygen uptake come from Van Hoek et al. (1998), Table 1.
• Observed-yield mode includes maintenance in the measured yield; an extra maintenance term is not added. Pirt mode instead needs a maintenance-free true yield and a separately measured maintenance coefficient.
• Custom biological parameters must describe the same strain, substrate, growth rate, temperature, pH, oxygenation and induction state. The reference physiological values are fixed until Custom is selected.
• The editable 2 L starting volume, 10 g/L dry biomass, 500 g/L glucose feed and equipment capacities are engineering planning inputs, not a reproduction of the source experiment.
• One cell-free, limiting-substrate feed with negligible residual substrate after feed start; well-mixed growth and constant matched parameters within the phase.
• Feed is the only volume-changing stream; no broth removal, sampling, evaporation, base, antifoam or other additions.
• OTR and cooling inputs are usable minima across the phase and include the required dissolved-oxygen floor and operating margins.
• Nonmetabolic heat is constant. The respiratory heat-per-oxygen relation is approximate.

Use for preliminary planning of a single aerobic, substrate-limited biomass-growth phase in E. coli or yeast. A target below the entered overflow threshold is necessary but does not ensure absence of acetate, ethanol or local substrate excess. Lee (1996) discusses the broader high-cell-density cultivation constraints.

Observed-yield mode is valid only at the growth rate and conditions at which that yield was measured; it already contains maintenance. Choose Pirt mode only when a maintenance-free true yield and a matched maintenance coefficient are available. The yeast cross-check uses an approximate published glucose-uptake measurement from aerobic chemostats to check the Pirt substrate-demand submodel; it does not reproduce a retentostat or establish a fed-batch recipe.

Re-estimate yield, maintenance, oxygen uptake and heat yield when growth rate, induction, temperature, medium or product burden changes. Product-associated substrate demand, viability loss, secretion, inhibition, mixing gradients and dynamic kLa are outside this model. It does not estimate recombinant protein or GLP-1 precursor titre.

Equality means the entered available capacity is used up. A time beyond the requested phase is conditional extrapolation. At equality, review the operating strategy and design a new phase using appropriate uptake kinetics or feedback; continuing the old exponential biomass assumption is invalid.

The viewer's US mode converts supported quantities such as biomass mass and power. Laboratory units including L, L/h, g/L and mol/h retain their declared units in the automatic switch. Volume and concentration fields also offer ft³ and lb/ft³ through their individual unit controls.

The yeast reference is a respiratory chemostat-derived physiology basis, not a validated high-cell-density fed-batch recipe. Confirm that these uptake rates remain representative as dry biomass, broth properties and product burden change. The preset's 0.28 h⁻¹ overflow ceiling belongs to DS28911 under the cited conditions. Custom culture data require their own threshold. Carbon/redox closure checks should use the actual substrate, biomass composition and by-products; a generic biomass formula is only an approximate consistency screen. The published synthetic Acatian reference cases are retained explicitly as mathematical transfer checks.