CP504 – Lecture 8
Cellular kinetics and associated reactor design:
Modelling Cell Growth
 Approaches to modelling cell growth
 Unstructured segregated models
 Substrate inhibited models
 Product inhibited models
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Cell Growth Kinetics
The most commonly used model for μ is given by the Monod
model:
μ=
μm CS
(47)
KS + CS
where μmax and KS are known as the Monod kinetic parameters.
Monod Model is an over simplification of the complicated
mechanism of cell growth.
However, it adequately describes the kinetics when the
concentrations of inhibitors to cell growth are low.
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Cell Growth Kinetics
Let’s now take a look at the cell growth kinetics,
limitations of Monod model, and alternative models.
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Approaches to modelling cell growth:
Unstructured Models
Structured Models
(cell population is treated
as single component)
(cell population is treated
as a multi-component
system)
Nonsegregated
Models
Segregated Models
(cells are treated as
homogeneous)
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(cells are treated
heterogeneous)
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Approaches to modelling cell growth:
Unstructured
Nonsegregated
Models
(cell population is treated
as single component, and
cells are treated as
homogeneous)
Simple and applicable
to many situations.
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Structured
Segregated Models
(cell population is treated
as a multi-component
system, and cells are
treated heterogeneous)
Most realistic, but are
computationally
complex.
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Unstructured, nonsegregated models:
Monod model:
μ=
μm CS
KS + CS
Most commonly
used model for
cell growth
μ : specific (cell) growth rate
μm : maximum specific growth rate at saturating substrate
concentrations
CS : substrate concentration
KS : saturation constant (CS = KS when μ = μm / 2)
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Unstructured, nonsegregated models:
Monod model:
μ=
μm CS
KS + CS
Most commonly
used model for
cell growth
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μ (per h)
0.8
0.6
0.4
μm = 0.9 per h
Ks = 0.7 g/L
0.2
0
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0
5
10
Cs (g/L)
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Assumptions behind Monod model:
- One limiting substrate
- Semi-empirical relationship
- Single enzyme system with M-M kinetics being
responsible for the uptake of substrate
- Amount of enzyme is sufficiently low to be
growth limiting
- Cell growth is slow
- Cell population density is low
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Other unstructured, nonsegregated models
(assuming one limiting substrate):
Blackman equation:
μ = μm if CS ≥ 2KS
μ =
Tessier equation:
Moser equation:
Contois equation:
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μm CS
2 KS
if CS < 2KS
μ = μm [1 - exp(-KCS)]
μ =
μ =
μm CSn
KS + CSn
μm CS
KSX CX + CS
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Blackman equation:
μ = μm
μ=
μm CS
2 KS
if CS ≥ 2 KS
if CS < 2 KS
This often fits the data better
than the Monod model, but the
discontinuity can be a problem.
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μ (per h)
0.8
0.6
μm = 0.9 per h
Ks = 0.7 g/L
0.4
0.2
0
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0
5
Cs (g/L)
10
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Tessier equation:
μ = μm [1 - exp(-KCS)]
1
μ (per h)
0.8
0.6
μm = 0.9 per h
K = 0.7 g/L
0.4
0.2
0
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0
2
4
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Cs (g/L)
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Moser equation:
μ =
μm CSn
When n = 1, Moser equation describes
Monod model.
KS + CSn
1
μ (per h)
0.8
0.6
0.4
μm = 0.9 per h
Ks = 0.7 g/L
0.2
Monod
n = 0.25
n = 0.5
n = 0.75
0
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0
2
4
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Cs (g/L)
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Contois equation:
μ =
μm CS
Saturation constant (KSX CX ) is
proportional to cell concentration
KSX CX + CS
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Extended Monod model:
μ=
Extended Monod model includes
a CS,min term, which denotes the
minimal substrate concentration
needed for cell growth.
μm (CS – CS,min)
KS + CS – CS,min
1
μ (per h)
0.8
0.6
μm = 0.9 per h
Ks = 0.7 g/L
CS,min = 0.5 g/L
0.4
0.2
0
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0
5
Cs (g/L)
10
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Monod model for two limiting substrates:
μ = μm
CS1
CS2
KS1 + CS1
KS2 + CS2
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Monod model modified for rapidly-growing,
dense cultures:
Monod model is not suitable for rapidly-growing, dense cultures.
The following models are best suited for such situations:
μ=
μ=
μm CS
KS0 CS0 + CS
μm CS
KS1 + KS0 CS0 + CS
where CS0 is the initial substrate concentration and KS0 is
dimensionless.
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Monod model modified for substrate inhibition:
Monod model does not model substrate inhibition.
Substrate inhibition means increasing substrate concentration
beyond certain value reduces the cell growth rate.
1
μ (per h)
0.8
0.6
0.4
0.2
0
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0
5
Cs (g/L)
10
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Monod model modified for cell growth with
noncompetitive substrate inhibition:
μ=
=
μm
(1 + KS/CS)(1 + CS/KI )
μm CS
KS + CS + CS2/KI + KS CS/KI
If KI >> KS then
μ=
μm CS
KS + CS + CS2/KI
where KI is the substrate inhibition constant.
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Monod model modified for cell growth with
competitive substrate inhibition:
μ=
μm CS
KS(1 + CS/KI) + CS
where KI is the substrate inhibition constant.
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Monod model modified for cell growth with product
inhibition:
Monod model does not model product inhibition (where
increasing product concentration beyond certain value reduces
the cell growth rate)
For competitive product inhibition:
μm CS
μ=
KS(1 + Cp/Kp) + CS
For non-competitive product inhibition:
μm
μ=
(1 + KS/CS)(1 + Cp/Kp )
where Cp is the product concentration and Kp is a product
inhibition constant.
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Monod model modified for cell growth with product
inhibition:
Ethanol fermentation from glucose by yeasts is an example of
non-competitive product inhibition. Ethanol is an inhibitor at
concentrations above nearly 5% (v/v). Rate expressions
specifically for ethanol inhibition are the following:
μ=
μ=
μm CS
(KS + CS)
μm CS
(KS + CS)
(1 + Cp/Cpm)
exp(-Cp/Kp)
where Cpm is the product concentration at which
growth stops.
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Monod model modified for cell growth with toxic
compound inhibition:
For competitive toxic compound inhibition:
μ=
μm CS
KS(1 + CI/KI) + CS
For non-competitive toxic compound inhibition:
μ=
μm
(1 + KS/CS)(1 + CI/KI )
where CI is the product concentration and KI is a constant to
be determined.
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Monod model extended to include cell death
kinetics:
μ=
μm CS
KS + CS
- kd
where kd is the specific death rate (per time).
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Beyond this slide, modifications will be made.
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Other unstructured, nonsegregated models
(assuming one limiting substrate):
Luedeking-Piret model:
rP =  rX + β CX
Used for lactic acid formation by Lactobacillus debruickii
where production of lactic acid was found to occur semiindependently of cell growth.
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Modelling μ under specific conditions:
There are models used under specific conditions. We will learn
them as the situation arises.
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Limitations of unstructured non-segregated
models:
• No attempt to utilize or recognize knowledge about
cellular metabolism and regulation
• Show no lag phase
• Give no insight to the variables that influence growth
• Assume a black box
• Assume dynamic response of a cell is dominated by an
internal process with a time delay on the order of the
response time
• Most processes are assumed to be too fast or too slow
to influence the observed response.
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Filamentous Organisms:
• Types of Organisms
– Moulds and fungi
– bacteria or yeast entrapped in a spherical gel particle
– formation of microbial pettlets in suspension
• Their growth does not necessarily increase the number of
cells, but increase them in length, and hence there will be
changes in physical properties like density of the cell mass
and viscosity of the broth
dR
 k  const
• Model - no mass transfer limitations
dt
where R is the radius of the cell floc or pellet or mold colony
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Filamentous Organisms:
• The product formation may be growth associated, which
means rate of product formation is proportional to the cell
growth rate (i.e., product is formed as a result of the primary
metabolic function of the cell)
r P =  rX
It happens mostly during the exponential growth phase
Examples:
- production of alcohol by the anaerobic fermentation of
glucose by yeast
- production of gluconic acid from glucose by Gluconobactor
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Filamentous Organisms:
• The product formation may be non-growth associated,
which means rate of product formation is proportional to the
cell concentration rather than cell growth rate (i.e., product is
formed as a result of the secondary metabolism)
rP = β CX
It happens at the end of the exponential growth phase or only
after entering into the stationary phase
Examples:
- production of antibiotics in batch fermentations
- production of vitamins in batch fermentations
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Other unstructured, nonsegregated models
(assuming one limiting substrate):
Luedeking-Piret model:
rP =  rX + β CX
Used for lactic acid formation by Lactobacillus debruickii
where production of lactic acid was found to occur semiindependently of cell growth.
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Filamentous Organisms:
Then the growth of the biomass (M) can be written as
dM
2 dR
  4R
 k p 4R 2 
dt
dt
or
dM
 M 2 / 3
dt
where
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  k p (36)
1/ 3
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Filamentous Organisms:
• Integrating the equation:
 1/ 3 t   t 
M   M0     
3 3

3
3
• M0 is usually very small then M  t
• Model is supported by experimental data.
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Chemically Structured Models :
• Improvement over nonstructured, nonsegregated models
• Need less fudge factors, inhibitors, substrate inhibition,
high concentration different rates etc.
• Model the kinetic interactions amoung cellular
subcomponents
• Try to use Intrinsic variables - concentration per unit cell
mass- Not extrinsic variables - concentration per reactor
volume
• More predictive
• Incorporate our knowledge of cell biology
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