BIOE 312
Experiment 2
Construction and Simulation of a Simple Kinetic Growth Model for Bacteria
Contents
1.
Mathematic modelling of cell-based bioprocess ................................................................. 1
1.1.
Motivations for modelling ........................................................................................... 1
1.2.
Approaches to modelling ............................................................................................. 2
1.3.
Contruction of dynamic models .................................................................................. 4
1.4.
Common specific rate equations.................................................................................. 5
1.4.1.
(Cell)-specific growth rate.................................................................................... 5
1.4.2.
Specific substrate consumption rate ..................................................................... 7
1.4.1.
Specific metabolite and product production rate .................................................. 7
1.5.
2.
Limitation and inhibition effects in E.coli cultivations ............................................... 7
Experiment .......................................................................................................................... 8
2.1.
Overview of the experiment ........................................................................................ 8
2.2.
Exercise 1 – Construction of a simple model for E.coli growth from batch culture
data
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2.3.
Exercise 2 – Simulation of the model in batch mode with varying initial values. .... 10
1. Mathematic modelling of cell-based bioprocess
1.1. Motivations for modelling
A mathematical model of a process can be defined as a set of mathematical equations
describing the main phenomena that have an influence on the growth, death and metabolic
activities of cells (Engasser, Goergen, & Marc, 1998). Bailey (1998) gives some reasons why
one would want to construct a model of a bioprocess: to organize disparate information into a
coherent whole, to think logically about what components and interactions are important in a
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complex system, to discover new strategies, to make important corrections in the conventional
wisdom and to understand the essential qualitative features.
A mathematical description of a physical phenomenon can not replace or fully imitate the
reality itself. It is important to recognize that a model can serve as a “tool” for engineers and
every tool is designed for a purpose before it is actually made. Therefore a model must be
constructed only after its purpose is defined. It is then used only for that purpose and must be
supported with experimental evidence of its validity.
1.2. Approaches to modelling
Bioprocesses are inherently complex and this complexity has implications on modelling.
Figure 1 gives an overview of this complexity (Bailey & Ollis, 1986). As shown on this
diagram, cell population interactcs with its environment (growth medium) by consuming
nutrients and excreting products and producing heat energy. As cells grow in the medium,
they change its rheological properties and composition. As this composition changes, cells
react accordingly by adapting their metabolism. Due to the heterogeneous environmental
conditions, not all cells are in the same state of metabolic activity. Some cells die, while some
keep growing. Some cells may even experience genetic alterations.
This heterogeneity and complexity is undesired and often kept at a minimum level by
employing process control strategies on the most critical environmental factors. pH,
temperature and oxygen concentration can be kept at optimum levels since appropriate on-line
measuring equipment is widely available and suitable actions can be taken (e.g. base addition
for pH control). However, some components still can not be measured or even estimated on-
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Figure 1 - Summary of some important parameters, phenomena and interactions between environment and cell
population
line, such as nutrient concentrations.
Much of the focus in modeling of bioprocesses has been therefore the description of
interactions of the biological system (cells) with the substrate and metabolites affecting cell
growth and viability. Depending on the level of understanding required, various approaches
are available to the modelling the metabolism of cells.
A common classification of the approaches to the modelling of cell population kinetics is
given in Figure 2 (Bailey & Ollis, 1986). Accordingly, modelling approaches can be classified
in two dimensions as unstructured vs. structured and segregated vs. unsegregated models.
Segregated approach take the cell-to-cell heterogeneity into account whereas the unsegregated
refers to a description of cell population with an average cell approximation. In the other
dimension of classification, structured models include a description of intracellular metabolic
pathways in attempt to describe the dynamics of cell-environment interaction. On the
contrary, unstructured models assume a balanced growth of cells, i.e. intracellular pathways
are disregarded and cells are assumed to follow a fixed metabolic pattern. Obviously, a
structured and segregated approach would be the ideal case in describing population
dynamics. Construction of such models are however difficult as they require a non-trivial
mathematical structure of the system to be built up (segregated part) and formulation of rate
equations for intracellular reactions (structured part). Even unstructured and segregated
models are rarely observed in literature, unless they are constructed for a special purpose. As
a result, unstructured and unsegregated models have been the common type of models used in
bioprocessing as they include fewer parameters than structured models and therefore easier to
construct (Augusto, Barral, & Piccoli, 2007; Engasser et al., 1998). Nevertheless, simplified
growth phenomena and disregarded intracellular pathways require careful use of these models
since their extrapolative capability outside the region of their original validity is low.
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1.3. Contruction of dynamic models
Based on the steps found in Augusto et al. (2007) and Engasser et al. (1998), work-flow in
the construction of dynamic models can be formalized as follows:
Purpose of the model. Models essentially contain simplifications of the system they intend to
describe. On extremely complex systems such as a single living cell or a population of cells,
these simplifications can not be made without guidance. This guidance is provided by the
purpose of the model. It must be clearly desribed and well elaborated.
Structure of the model. After the purpose is set, decisions can be made on the necessary
amount of detail to include in the model. Structure of the model is then determined relating
the selected components to each other.
Formulation of the model.
Mass balance equations. These are easily constructed by applying principles of mass
conservation on each of the model components. Mass balacens are mostly determined by
physical characteristics of the system, i.e. whether it is operated in batch, fed-batch or
continuous mode (or their variations) and whether the vessel has special characteristics such
as multiple compartments or selective product removal systems.
Rate equations. These give the “dynamic” characteristics to the kinetic model. As
Figure 2 - An overview of approaches to the modelling of cell population kinetics
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components interact with each other via reaction mechanisms, mechanistic description of
these interactions via rate equations are vital in fulfillin the purpose of the model. Formulation
of rate equations is the most important step in the construction of a model.
Parameter fitting. A model consisting merely of formulas can only describe the kinetic
behaviour of the process qualitatively in a formalized way. In order to make it amenable to
computation analytically or via numerical analysis techniques, its parameters must be
determined. Accurate estimation of parameters are important for they form the quantitative
basis of the kinetic model. A careful design of experiments for parameter estimation based on
the kinetic model is advantageous in collection information rich data.
Validation of the final model. In order to demonstrate that the mechanisms behing the model
are correct and valid, its output (a simulated process) must be compared with an independent
set of measurement data from a real process. Depending on the success of validation, a need
to modify the model in order to meet the requirements may arise or it can be concluded that a
satisfactory description of the system is obtained.
1.4. Common specific rate equations
Due to the limited understanding of the metabolism and the unstructured approach used in
model-building, a wide variety of equations for cell-specific rates were proposed in the
literature for different strains and cultivation conditions. The following summarizes only
those ones that are necessary to initiate a modeling study and in no way includes all formulas
encountered in the literature.
1.4.1. (Cell)-specific growth rate
Quantitative description of cell growth is a primary goal in modelling studies since
cells are the primary components of a system influencing other components via consumption
or production. Cell growth is formulated via the cell-specific growth rate. Specific growth rate
is usually denoted by the Greek letter µ.
The most commonl used structure is a single-substrate limited growth equation which
was originally proposed by Jacques Monod to describe bacterial growth (Monod, 1949)
µ = µ𝑚𝑎𝑥
𝐶𝑆
𝐶𝑆 + 𝐾𝑆
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where µ𝑚𝑎𝑥 is the maximum specific growth rate under balanced growth conditions (1/time),
𝐶𝑆 is the limiting substrate concentration (mass/volume) and 𝐾𝑆 is the substrate limitation
constant for growth (mass/volume) or more clearly the substrate concentration at which the
growth rate is at its half of the maximum.
Monod’s equation describes a saturation-type substrate-limited growth pattern: cells
grow at a maximum rate until the concentration of the substrate starts to decrease to critical
levels. The growth rate drops to zero when substrate concentration is zero. Depending on the
value of 𝐾𝑆 , growth rate can be quickly or slowly “saturated”.
In cases where there is excessive toxic metabolite build-up such as lactic acid, ethanol
or acetic acid, growth can be decelerated or even cease due to the negative effect of these
components. For these cases, an “inhibition” structure can be incorporated into the above
equation:
µ = µ𝑚𝑎𝑥
𝐶𝑆
𝐾𝐼
𝐶𝑆 + 𝐾𝑆 𝐶𝐼 + 𝐾𝐼
where 𝐾𝐼 is called the half-inhibition constant for the inhibiting compound and 𝐶𝐼 is the
concentration of the inhibiting compound.
In some cases the inhibition by the toxic metabolite is more apparent than what would
be when the above equation is used. In many such cases, growth rate linearly decreases with
increasing inhibitor concentrations and stops when a critical inhibitor concentration is
reached. This is described with the following equation:
µ = µ𝑚𝑎𝑥
𝐶𝑆
𝐶𝐼
(1 −
)
𝐶𝑆 + 𝐾𝑆
𝐶𝐼,𝑐𝑟𝑖𝑡
where 𝐶𝐼,𝑐𝑟𝑖𝑡 the critical inhibitor concentration above which no cell growth occurs.
It is also known that above certain concentrations, the limiting substrate may produce
toxic effects on cells. In this case, the inhibition of substrate is also incorporated into the
growth-rate equation:
µ = µ𝑚𝑎𝑥
𝐶𝑆
𝐾𝐼,𝑆
𝐶𝐼
(1 −
)
𝐶𝑆 + 𝐾𝑆 𝐶𝑆 + 𝐾𝐼,𝑆
𝐶𝐼,𝑐𝑟𝑖𝑡
where 𝐾𝐼,𝑆 is the half-inhibition constant for substrate.
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1.4.2. Specific substrate consumption rate
For substrate uptake, a maintenance model suggested by Pirt (1965) found widespread
use:
𝑞𝑆 =
1
𝑌𝑋⁄
𝜇 + 𝑚𝑆
𝑆
where 𝑞𝑆 is the cell-specific rate of substrate consumption (mass/cell*time); 𝑌𝑋⁄ is the
𝑆
maximum global substrate-to-cells yield factor (cell/mass) and 𝑚𝑆 is the specific substrate
uptake rate for maintenance (mass/time).
Alternatively, the specific substrate uptake rate can be modelled as a function of
substrate concentration:
𝑞𝑆 = 𝑞𝑆,𝑚𝑎𝑥
𝐶𝑆
𝐶𝑆 + 𝐾𝑆
1.4.1. Specific metabolite and product production rate
In a similar approach to Pirt (1965), specific metabolite production rate can also be
modelled by relating it linearly to specific growth rate (Luedeking & Piret, 1959):
𝑞𝑝 = 𝑌𝑃⁄ 𝜇 + 𝑚𝑝
𝑋
where 𝑞𝑝 is the cell-specific rate of product formation (mass/cell*time); 𝑌𝑃⁄ is the yield of
𝑋
product on formed cells and 𝑚𝑝 is the specific product formation rate from maintenance
(mass/time).
1.5. Limitation and inhibition effects in E.coli cultivations
It is widely known that growth of E.coli is limited by the availability of substrate in the
medium. Most growth media contain either glucose or glycerin as substrates and both of these
are growth-limiting (Dubach & Maerkl, 1992; Sezonov, Joseleau-Petit, & D’Ari, 2007).
These substrates show toxic effects as well, when supplied at high concentrations.
The most important unwanted side-product in E.coli cultivations is acetic acid, for this
compound inhibits both the growth and product formation in glucose or glycerin supplied
cultivations.
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2. Experiment
2.1. Overview of the experiment
In this experiment, you will construct and simulate an unstructured kinetic growth model
for E.coli using some of the data from Experiment 1 and new data that will be given for this
experiment.
First, you will formulate mass balances for batch and fed-batch cultivation modes. These
will take the form of an Ordinary Differential Equations (ODE) system as unsteady-state
mass/concentration balances.
Then, you will start with the formulation of kinetic phenomena by assuming certain
phenomena and choosing appropriate cell-specific rate equations. Parameters in these
equations will be fitted to the data.
After parameters are obtained, you will enter your growth model into a MATLAB-based
simulation tool and simulate batch and fed-batch processes with simple feeding strategies
(You will receive hands-on instructions on how to use the tool from your instructor on the day
of the experiment). The output from each simulation will be evaluated and assessed in terms
of their efficiency for supporting the growth of bacteria.
2.2. Exercise 1 – Construction of a simple model for E.coli growth from batch culture
data
A team of researchers investigate the possibility of using a strain of E.coli for recombinant
protein production. In order to determine the growth requirements of the bacteria, they test
several growth conditions in shake flask experiments. Based on the information found in the
literature, they assume glucose as a potential growth limiter & inhibitor and acetic acid as a
potential growth inhibitor.
To test the nature of the effects of these compounds, they conduct a series of batch cultivation
experiments in controlled environment in a series of bioreactors. For each experiment, they
vary the initial glucose and acetic acid concentrations and record specific growth rates in the
exponential phase.
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Glucose limitation
Initial
Glucose inhibitor
µ (1/h)
Initial
Glucose (g/l)
Acetic acid inhibition
µ (1/h)
Initial acetic
Glucose (g/l)
µ (1/h)
acid (g/l)
0.01
0.20
6
0.43
0
0.50
0.04
0.39
50
0.42
0.5
0.46
0.08
0.38
200
0.30
1
0.45
0.15
0.41
400
0.19
1.5
0.35
0.4
0.53
600
0.15
2.5
0.22
0.8
0.49
3
0.20
1.2
0.55
3.5
0.15
4
0.07
5
0.03
In addition to this, they record the specific consumption and production rates for glucose and
acetic acid that correspond to different specific growth rates:
qglc (g/g dcw/h)

qac (g/g dcw/h)
µ (1/h)
0.200
0.056
0.01
0.224
0.049
0.02
0.269
0.055
0.04
0.397
0.070
0.1
0.841
0.114
0.3
1.067
0.126
0.4
1.286
0.150
0.5
Transfer the values to an excel worksheet and plot all concentrations and specific rates
against the specific growth rate.

Examine the plots. What is the effect of glucose and acetic acid on the growth of cells?

Choose a formulation for cell specific growth rate and determine its parameters via a
least-squares fitting tool (e.g. the Solver Add-in from Analysis ToolPak in Excel).

What is the relationship between specific rates of consumption and production and
specific growth rate? Choose a formulation for relating these specific rates to specific
growth rate. Determine their parameters as in the previous step.
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
On a paper, formulate a mass balance for a hypothetical batch bioreactor on glucose,
acetic acid and biomass.

Insert your expressions for specific rates into the mass balance equation system.
2.3. Exercise 2 – Simulation of the model in batch mode with varying initial values.

Start the MATLAB-based tool and enter your growth model on the screen (you will be
guided through the use of the tool).

Determine initial concentration values for each member of the model.

Choose a cultivation time and enter its value on the screen.

Click “Run Process”. This will start the simulation and produce graphical outputs of
glucose, acetic acid, biomass and the specific rates.

Examine the curve of the specific growth rate.

Try different initial concentrations for glucose and biomass and comment on their
effects.
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REFERENCES
Augusto, E. F. P., Barral, M. F., & Piccoli, R. A. M. (2007). Mathematical models for growth
and product synthesis in animal cell culture. In Animal cell technology from
biopharmaceuticals to gene therapy. New York [u.a.]: Taylor & Francis.
Bailey, J. E. (1998). Mathematical Modeling and Analysis in Biochemical Engineering: Past
Accomplishments and Future Opportunities. Biotechnology Progress, 14(1), 8–20.
doi:10.1021/bp9701269
Bailey, J. E., & Ollis, D. F. (1986). Biochemical Engineering Fundamentals (2 Sub.).
McGraw-Hill Science/Engineering/Math.
Dubach, A. C., & Maerkl, H. (1992). Application of an extended kalman filter method for
monitoring high density cultivation of Escherichia coli. Journal of Fermentation and
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Engasser, J. M., Goergen, J. L., & Marc, A. (1998). Modelling. In Cell and Tissue Culture:
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Luedeking, R., & Piret, E. L. (1959). A kinetic study of the lactic acid fermentation. Batch
process at controlled pH. Journal of Biochemical and Microbiological Technology and
Engineering, 1(4), 393–412. doi:10.1002/jbmte.390010406
Monod, J. (1949). The Growth of Bacterial Cultures. Annual Review of Microbiology, 3(1),
371–394. doi:10.1146/annurev.mi.03.100149.002103
Pirt, S. J. (1965). The Maintenance Energy of Bacteria in Growing Cultures. Proceedings of
the Royal Society of London. Series B, Biological Sciences, 163(991), 224–231.
Sezonov, G., Joseleau-Petit, D., & D’Ari, R. (2007). Escherichia coli Physiology in LuriaBertani Broth. Journal of Bacteriology, 189(23), 8746–8749. doi:10.1128/JB.0136807
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