| Balances with fed-batch terms |  | Kinetics |
---|---|---|---|
Biophase | \( \frac{{\boldsymbol{d}\boldsymbol{X}}_{\boldsymbol{v}}}{\boldsymbol{d}\boldsymbol{t}}=\left(\boldsymbol{\mu} -{\boldsymbol{\mu}}_{\boldsymbol{d}}\right)\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( \frac{{\boldsymbol{dX}}_{\boldsymbol{t}}}{\boldsymbol{dt}}=\boldsymbol{\mu} \bullet {\boldsymbol{X}}_{\boldsymbol{v}}-{\boldsymbol{K}}_{\boldsymbol{Lys}}\bullet \left({\boldsymbol{X}}_{\boldsymbol{t}}-{\boldsymbol{X}}_{\boldsymbol{v}}\right) \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{X}}_{\boldsymbol{t}} \) | Cell growth & death | \( \boldsymbol{\mu} ={\boldsymbol{\mu}}_{\boldsymbol{max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Glc}}+{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Gln}}}{{\boldsymbol{c}}_{\boldsymbol{Gln}}+{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Gln}}} \) \( {\boldsymbol{\mu}}_{\boldsymbol{d}}={\boldsymbol{\mu}}_{\boldsymbol{d},\boldsymbol{\min}}+{\boldsymbol{\mu}}_{\boldsymbol{d},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}}{{\boldsymbol{K}}_{\boldsymbol{S},\boldsymbol{Glc}}+{\boldsymbol{c}}_{\boldsymbol{Glc}}} \) |
Liquid phase | \( \frac{{\boldsymbol{dc}}_{\boldsymbol{Glc}}}{\boldsymbol{dt}}=-{\boldsymbol{q}}_{\boldsymbol{Glc}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( +\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Glc},\boldsymbol{F}}-\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Glc}} \) \( \frac{{\boldsymbol{dc}}_{\boldsymbol{Gln}}}{\boldsymbol{dt}}=-{\boldsymbol{q}}_{\boldsymbol{Gln}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( +\frac{{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Gln},\boldsymbol{F}}-\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Gln}} \) \( \frac{{\boldsymbol{dc}}_{\boldsymbol{Lac}}}{\boldsymbol{dt}}={\boldsymbol{q}}_{\boldsymbol{Lac}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Lac}} \) \( \frac{{\boldsymbol{dc}}_{\boldsymbol{Amm}}}{\boldsymbol{dt}}={\boldsymbol{q}}_{\boldsymbol{Amm}}\bullet {\boldsymbol{X}}_{\boldsymbol{v}} \) \( -\frac{{\boldsymbol{F}}_{\boldsymbol{Glc}}+{\boldsymbol{F}}_{\boldsymbol{Gln}}}{\boldsymbol{V}}\bullet {\boldsymbol{c}}_{\boldsymbol{Amm}} \) | Substrate uptake & metabolite production | \( {\boldsymbol{q}}_{\boldsymbol{Glc}}={\boldsymbol{q}}_{\boldsymbol{Glc},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Glc}}+{\boldsymbol{k}}_{\boldsymbol{Glc}}}\bullet \left(\frac{\boldsymbol{\mu}}{\boldsymbol{\mu} +{\boldsymbol{\mu}}_{\boldsymbol{max}}}+\mathbf{0.5}\right) \) \( {\boldsymbol{q}}_{\boldsymbol{Gln}}={\boldsymbol{q}}_{\boldsymbol{Gln},\boldsymbol{\max}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Gln}}}{{\boldsymbol{c}}_{\boldsymbol{Gln}}+{\boldsymbol{k}}_{\boldsymbol{Gln}}} \) \( {\boldsymbol{q}}_{\boldsymbol{Lac}}={\boldsymbol{Y}}_{\boldsymbol{Lac}/\boldsymbol{Glc}}\bullet \frac{{\boldsymbol{c}}_{\boldsymbol{Glc}}}{{\boldsymbol{c}}_{\boldsymbol{Lac}}}\bullet {\boldsymbol{q}}_{\boldsymbol{Glc}}-{\boldsymbol{q}}_{\boldsymbol{Lac},\boldsymbol{uptake}} \) if c Glc  ≥ 0.5 mM : q Lac,uptake  = 0 if c Glc  < 0.5 mM : q Lac,uptake  = q Lac,uptake,max q Amm  = YAmm/Gln ∙ q Gln |