Cellular kinetics and associated reactor design:

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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
µm CS
(47)
µ
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
Lets 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 (cell population is treated
as single component)
Structured Models (cell population is treated as
a multi-component system)
Nonsegregated Models (cells are treated as
homogeneous)
Segregated Models (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)
Structured Segregated Models (cell population is
treated as a multi-component system, and cells
are treated heterogeneous)
Most realistic, but are computationally complex.
Simple and applicable to many situations.
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Unstructured, nonsegregated models
Monod model
Most commonly used model for cell growth
µm CS
µ
KS CS
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
Most commonly used model for cell growth
µm CS
µ
KS CS
µ (per h)
µm 0.9 per h Ks 0.7 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
µm CS
if CS lt 2KS
µ
2 KS
Tessier equation
µ µm 1 - exp(-KCS)
µm CSn
Moser equation
µ
KS CSn
µm CS
Contois equation
µ
KSX CX CS
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Blackman equation
µ µm
if CS 2 KS
This often fits the data better than the Monod
model, but the discontinuity can be a problem.
µm CS
if CS lt 2 KS
µ
2 KS
µ (per h)
µm 0.9 per h Ks 0.7 g/L
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Tessier equation
µ µm 1 - exp(-KCS)
µ (per h)
µm 0.9 per h K 0.7 g/L
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Moser equation
When n 1, Moser equation describes Monod model.
µm CSn
µ
KS CSn
µ (per h)
µm 0.9 per h Ks 0.7 g/L
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Contois equation
Saturation constant (KSX CX ) is proportional to
cell concentration
µm CS
µ
KSX CX CS
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Extended Monod model
Extended Monod model includes a CS,min term,
which denotes the minimal substrate concentration
needed for cell growth.
µm (CS CS,min)
µ
KS CS CS,min
µ (per h)
µm 0.9 per h Ks 0.7 g/L CS,min 0.5 g/L
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Monod model for two limiting substrates
CS1
CS2
µm
µ
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
µm CS
µ
KS0 CS0 CS
µm CS
µ
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.
µ (per h)
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Monod model modified for cell growth with
noncompetitive substrate inhibition
µm
µ
(1 KS/CS)(1 CS/KI )
µm CS

KS CS CS2/KI KS CS/KI
µm CS
µ
If KI gtgt KS then
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
µm CS
µ
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
µm CS
(1 Cp/Cpm)
µ
(KS CS)
µm CS
exp(-Cp/Kp)
µ
(KS CS)
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
µm CS
µ
KS(1 CI/KI) CS
For non-competitive toxic compound inhibition
µm
µ
(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
µm CS
µ
- kd
KS CS
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 ß CX
Used for lactic acid formation by Lactobacillus
debruickii where production of lactic acid was
found to occur semi-independently 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
• Model - no mass transfer limitations
• 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)

rP ? 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 ß 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 ß CX
Used for lactic acid formation by Lactobacillus
debruickii where production of lactic acid was
found to occur semi-independently of cell growth.
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Filamentous Organisms
Then the growth of the biomass (M) can be written
as
where
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Filamentous Organisms

• Integrating the equation
• M0 is usually very small then
• 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