Biokinetics

Model fermentation growth rate and substrate consumption

Inputs

g/L
1/hr
OD600

Model

HourCellsSubstrate
ctrl+v to paste data
g/L

Calculation

The relationship between cell mass, substrate, and fermentation volume is determined through differential equations. The rate of change for each component depends on the fermentation phase:

Batch Phase The initial growup of the cells at constant volume.

dXdt=r⋅X=μmax⋅SKs+S⋅X\frac{dX}{dt} = r⋅X = \frac{\mu_{max} ⋅ S}{K_{s}+S} ⋅ X
dSdt=−r⋅X/Ya=−μmax⋅SKs+S/Ya⋅X\frac{dS}{dt} = -r⋅X/Y_{a} = -\frac{\mu_{max} ⋅ S}{K_{s}+S}/Y_{a} ⋅ X
dVdt=0\frac{dV}{dt} = 0

Fed Batch Cell growth at increasing volume.

dXdt=r⋅X−FV⋅X=μmax⋅SKs+S⋅X−FV⋅X\frac{dX}{dt} = r⋅X - \frac{F}{V}⋅X = \frac{\mu_{max} ⋅ S}{K_{s}+S} ⋅ X - \frac{F}{V}⋅X
dSdt=F(t)∗(Sf−S)V−rX/Ya=F(t)∗(Sf−S)V−μmax⋅SKs+S/Ya⋅X\frac{dS}{dt} = \frac{F(t)*(S_f - S)}{V}-rX/Y_{a} = \frac{F(t)*(S_f - S)}{V}-\frac{\mu_{max} ⋅ S}{K_{s}+S}/Y_{a} ⋅ X
dVdt=F(t)\frac{dV}{dt} = F(t)

Where

Yxs=Yxsmax⋅μmaxμmax−Yxsmax⋅msY_{xs} = \frac{Y_{xs max}⋅\mu_{max}}{\mu_{max}-Y_{xs max}⋅ms}
F0=V0⋅XbSf⋅μmax⋅zYxsmax+msF_{0} = \frac{V_{0}⋅X{b}}{S_{f}} ⋅ \frac{\mu_{max}⋅z}{Y_{xs max}}+ms
F(t)=F0eμmax⋅z⋅tF(t) = F_{0} e^{\mu_{max}⋅z⋅t}
Ya=μmax⋅Yxsmaxμmax−Yxsmax⋅msY_{a} = \frac{\mu_{max}⋅Y_{xs max}}{\mu_{max}-Y_{xs max}⋅ms}

Definitions

X=X =
Dry cell concentartion
S=S =
Substrate concentration
V=V =
Cummulative fermentation volume
V0=V_{0} =
Intial fermentation volume
μmax=\mu_{max} =
Maximum specific growth rate
Ks=K_{s} =
Half-velocity constant
Sf=S_{f} =
Feed substrate concentration
Xb=X_{b} =
Batch phase final cell concentration
ms=ms =
Cell maintenance consumption rate
z=z =
Specific growth rate scaling factor
Yxsmax=Y_{xs max} =
Max biomass/substrate yield
F(t)=F(t) =
Feed rate

Created with help from Phil Vo