Industrial Microbiology
INDM 4005
Lecture 9
23/02/04
PROCESS ANALYSIS
Lecture 9
(
1)
Kinetics and models

Predictive microbiology
(
2)
Growth kinetics (and product)
(
3)
Models

example, Continuous culture model
Overview
•
Fermentation Kinetics
•
Mathematic models
•
Stoichiometry
•
Chemical kinetics
•
Michaelis menten model
•
The Monod model
•
Yield coefficients
•
Modelling fermentation processes
•
Types of model
INTRODUCTION TO KINETICS and MODELS

PREDICTIVE MICROBIOLOGY
Kinetics and/or
Models describe the process or data
•
Used to make
predictions
•
Enhances
experimental design

cuts down on the
number of experiments (allows process simulation)
Why study Fermentation Kinetics?
•
The overriding factor that propels biotechnology is profit. Without profit,
there would be no money for research and development and
consequently no new products.
•
A biotechnologist seeks to use biological systems to either maximize
profits or maximize the efficiency of resource utilization.
•
The large scale cultivation of cells is central to the production of a large
proportion of commercially important biological products.
•
Not surprisingly, the maximization of profits is closely linked to
optimizing product formation by cellular catalysts; ie. producing the
maximum amount of product in the shortest time at the lowest cost.
Fermentation Kinetics
•
To achieve this objective, cell culture systems must be described
quantitatively.
•
In other words, the kinetics of the process must be known. By
determining the kinetics of the system, it is possible to predict yields
and reaction times and thus permit the correct sizing of a bioreactor.
•
Obviously, reaction kinetics must be determined prior to the
construction of the full scale system. In practice, kinetic data is obtained
in small scale reactors and then used with mass transfer data to scale

up the process.
•
In this lecture, we shall learn how fermentation kinetics are determined
and how they can be applied.
Fermentation Kinetics
•
Quantitative research is based on numerical data, i.e a precise
measurement or determination expressed numerically
•
Considering the complex nature of microbial growth this is a difficult
task
•
Product formation kinetics is also difficult
•
Increased understanding in cellular function has allowed advanced
methods in modeling cellular growth kinetics
•
Mathematical models now describe gene expression, individual
reactions in central pathways, macroscopic models of cell
growth/product formation with simple mathematical expressions
Mathematical models
What are they, why use them?
•
Cell culture systems are extremely complex. There are many inputs and
many outputs.
•
Unlike most chemical systems, the catalysts themselves are self
propagating.
•
To assist in both understanding quantifying cell culture systems,
biotechnologists often use mathematical models.
•
A mathematical model is a mathematical description of a physical
system.
•
A good mathematical model will focus on the important aspects of a
particular process to yield useful results.
Framework for Kinetic models
•
Net result of many biochemical reactions within a
single cell is the conversion of substrates to biomass
and metabolic end

products
Intracellular
biochemical
reactions
Substrates
(Glucose)
Metabolic products
Extracellular
macromolecules
Biomass
constituents
Framework for Kinetic models
•
Conversion of glucose to biomass involves
many reactions
•
Reactions can be structured as follows
(1) Assembly reactions
(2) Polymerisation reactions
(3) Biosynthetic reactions
(4) Fuelling reactions
Overall composition of an
E. coli
cell
Macromolecule
% of total dry
Different kinds
weight
of molecules
Protein
55
1050
RNA
20.5
rRNA
16.7
3
tRNA
3
60
mRNA
0.8
400
DNA
3.1
1
Lipid
9.1
4
Lipopolysaccaride
3.4
1
Peptidoglycan
2.5
1
Glycogen
2.5
1
Metabolic pool
3.9
Data taken from Ingraham et al., (1983)
Control of metabolite levels
•
The number of cellular metabolites is therefore quite large,
but still account for a small percentage of the total biomass
•
Due to en bloc control of individual reaction rates
•
Also high affinity of enzyme to substrate ensures reactants
are at a low concentration
•
Therefore not important to consider kinetics of individual
reactions, reduces complexity
To model a fermentation process, must consider
Bioreactor Performance
e.g.
Flow patterns of liquids and mixing,
Mass Transfer of nutrients and gases
Microbial Kinetics
e.g.
Cell model (growth rate / yields of the individual cell)
and also
Population Models (e.g. mixed populations, competing
microorganisms/ contamination etc.)
Formulating mathematical
models
•
A model is a set of relationships between variables of interest in the
system being studied
•
A set of relationships may be in the form of equations, graphs, or tables
•
The variables of interest depend upon the use to which the model is to
be put
•
For example, a biotechnologist, electrical engineer, mechanical
engineer, an accountant would have different variables of interest
Constructing a mathematical model
To construct a conventional mathematical model we write a set of
equations for each control region
•
1) Balance equations for each extensive property of the system, eg
mass, energy or chemical elements
•
2) Rate equations
1) rate of transfer of mass
2) rates of generation or consumption,
substrate or product across boundaries of the region
•
3) Thermodynamic equations relate thermodynamic properties
(pressure, temperature, density, concentration) within the control region
or across phases
Abstracted physical model of a
batch fermenter
Liquid phase
Control region
Gas phase
Air/gas interface
Air in
Indicates well mixed
Gas out
Mathematical models

parameters,
variables and constraints
•
Differential equations describe rates of change within a system. Many
mathematical models are formulated using differential equations.
•
Each equation contains variables and parameters. The
variables
in the
Michaelis Menten model are [S] and [P]. The values of variables will
change with time.
•
V
max
, K
m
and Y are assumed to not change with time. These expressions
are examples of
parameters
. Parameters are terms which are assumed
to be constant under a given set of conditions. With each different
condition eg. pH or temperature, or a different calatalyst, a different set of
parameters are required.
•
Variables are expressed as concentrations (eg. g.l

1
) rather than as
absolute values (eg. g). This is not obligatory but the use of relative
expressions makes the model more useful when used to scale

up a
process.
Industrial Lab
Fermenter
Scientific
Experimental
Engineering
judgement
data
judgement
Determine
model
parameters
Validate
model
Kinetic and
Abstracted
stoichiometric
physical
models
model
Mathematical model
Use model for control
process and economic studies
Test
ideas
How kinetics fits into overall design and operation of a process
Stoichiometry
•
First step in a quantitative description of cellular growth is
to specify the stoichiometry for those reactions that are to
be considered for analysis
•
Conversion of substrates into products and cellular
materials is represented by chemical equations
Stoichiometric yield coefficients
•
Models describing biochemical reactions use
stoichiometric yield
coefficients
to determine how much product (or biomass) will be
produced from each unit of reactant or substrate utilized.
•
Yield coefficients describe how efficiently a reactant is converted into a
product or biomass. The formation of lactic acid from glucose can be
represented as:
•
The yield of lactate from glucose (Y
LG
) is 2 moles of lactate (L) per
mole of glucose (G). The relationship between lactate formation and
glucose utilization would be:
•
Chemical reactions are similarly simplified. For example, a first order
chemical reaction in which 1 mole of reactant (S) is converted to a
product (P):
S
n P
•
Can be expressed as a differential equation of the form:
d[S]
= k[S]
dt
where [S] is the concentration of the reactant and k is a rate constant.
Chemical kinetic equations as
mathematical models
Chemical kinetic equations as
mathematical models
•
Note that for this reaction, a differential equation describing product
formation is:
where [P] is the concentration of the product and n is the stoichiometric yield
constant describing the relationship between the removal of S and formation of P.
•
Note that as the concentration of S decreases, the concentration of P
increases.
•
By solving this equation, it is possible to predict the values of S and P at
any time.
The Michaelis Menten Model
as a Mathematical Model
•
In enzyme studies, you will have learnt the Michaelis Menten
equation which is a mathematical model describing activity of
many different enzymes:
where [S] is the substrate concentration, V is the rate of substrate removal, V
max
is the maximum specific rate and K
m
is the saturation constant.
•
The Michaelis Menten equation describes the rate of substrate
breakdown by an enzyme and can be written as a differential
equation:
The Monod model and the Michaelis
Menten model
The Monod Model looks similar to the Michaelis Menten equation.
The Monod model and the
Michaelis Menten model
•
The parameters µ
m
and K
s
are analogous to V
max
and K
m
. Both models
predict that only when the concentration of a rate limiting substrate or
nutrient becomes limiting, then the reaction rate will slow.
•
There is however one very distinct difference between the two models.
The Michaelis Menten equation was derived using the
mechanism of enzyme action as a basis.
The Monod Model in contrast is used because it fits the
typical curve shown in previous slide.
•
The Monod Model is therefore classified as an emperical model (based
on experience or observational information and not necessarily on
proven scientific data
)
, while the Michaelis Menten equation is a
mechanistic model.
Monod Model
•
Monod's model describes the relationship between the specific growth
rate and the growth limiting substrate concentration as:
where µ
m
is the maximum specific growth rate and Ks is a saturation
constant.
•
Despite its empirical nature Monod's model is widely used to describe
the growth of many organisms. Basically because it does adequately
describe fermentation kinetics.
•
Model has been modified to describe complex fermentation systems.
A simple mathematical model
of a fermentation process
•
Thus far, we have a model which describes biomass formation:
•
However to complete the model, equations for substrate utilization and
product formation need to be developed.
•
If biomass formation and product formation are assumed to be directly
linked to substrate utilization by yield coefficients, therefore:
•
Note the negative signs used. Substrate concentrations decrease
during a fermentation and thus dS/dt has a negative value. In contrast,
biomass and product concentrations generally increase in value.
Why solve the model?
•
When the model is solved numerically, a number of curves are
obtained.
•
With the model, it is possible for example, to determine the number of
fermentations that can be performed per year and consequently, the
amount of profit that can be made.
Assumptions and constraints
•
Monod model represents a very simple model of cell growth and
product formation. However, fermentation processes are often much
more complex.
•
Modifications to the Monod model, may need to be introduced to
handle more complicated systems. Additional equations would be
required to handle multiple products and multiple organisms.
•
The model has also assumed that product formation is linked to
biomass growth; ie. growth associated. In reality, many commercially
important products are produced in a non

growth associated manner.
•
The model assumes that biomass and product formation can be
represented by averaged yield coefficients.
•
These assumptions may sometimes be an oversimplification and such
a model would give unrealistic results.
Kinetic Models
•
The basis of kinetic modelling is to express functional relationships
between the forward reaction rates and the levels of substrates,
metabolic products, biomass constituents, intracellular metabolites and
/ or biomass concentration
•
Models vary with degree of complexity
•
Structured models
Model divides cell mass into components (by molecule or by element)
and predicts how these components change as a result of growth.
These models are very complex and not used very often.
•
Unstructured models
Models presume balanced growth where cell components do not
change with time. Much less complex and much more commonly used.
Valid for batch growth during exponential growth phase and also for
continuous culture during steady state growth.
Bioreactor Modeling Terminology
•
Structured vs. unstructured
–
Structured
–
“detailed” intracellular description
–
Unstructured

“simple” intracellular description
•
Segregated vs. unsegregated
–
Segregated
–
differentiate individual cells
–
Unsegregated
–
treat all cells as equivalent
Unstructured Growth Models
•
General characteristics
–
Simple description of cell growth & product formation rates
–
No attempt to model intracellular events
•
Specific growth rate
•
Yield coefficients
–
Biomass/substrate: Y
X/S
=

DX/DS
–
Product/substrate: Y
P/S
=

DP/DS
–
Product/biomass: Y
P/X
= DP/DX
–
Approximated as constants
(g/L)
ion
concentrat
mass
cell
1
X
dt
dX
X
Structured Metabolic Models
•
General characteristics
–
Mechanistic description of cell growth & product formation
rates
–
Detailed modeling of intracellular reactions
•
Advantages
–
Sound theoretical basis
–
Superior predictive capabilities
–
Extensible to new culture conditions & cell strains
•
Disadvantages
–
Requires detailed knowledge of cellular metabolism
–
Experimentally intensive
–
Difficult to formulate
Deterministic v Stochastic modelling
•
Deterministic
Pertaining to a process, model, simulation or
variable whose outcome, result, or value does
not depend upon chance
•
Stochastic
Applied to processes that have random
characteristics
Model types
1)
STOCHASTIC

considers individual cells (example

the
distribution of plasmids within the individual cells in a culture)
2)
DETERMINISTIC

considers cell mass, can be;
•
(i) distributed

cell mass part of the culture
•
(ii) segregated

separate phase (e.g. model of mass transfer)
•
(iii) structured

total biomass considered as sum of two or
more components (e.g. series of enzyme reactions)
•
(iv) unstructured
Deterministic v Stochastic modelling
•
In a description of cellular kinetics macroscopic (designating a size
scale very much larger than that of atoms and molecules)
balances
are normally used, i.e the rates of the cellular reactions are functions
of average concentrations of the intracellular components
•
Many cellular processes are stochastic in nature so assigning
deterministic descriptions to them is incorrect
•
However the application of macroscopic or (deterministic) description
is convenient and represents a typical engineering approximation for
describing the kinetics in an average cell in a population of cells
Thus kinetics must be expressed at different levels
•
1
.
Molecular
or
enzyme
level
i
.
e
.
rate
of
a
single
enzyme
reaction
•
2
.
Macromolecular
or
cellular
components
i
.
e
.
RNA
or
ribosome
synthesis,
plasmid
segregation
.
•
3
.
Cellular
level
i
.
e
.
substrate
uptake,
biomass
production
.
•
4
.
Population
level
(Logistic
/
Gompertz
Eqs)
i
.
e
.
competition
between
two
cultures
.
•
5
.
Process
level
i
.
e
.
amount
of
product
produced
after
fermentation
and
efficiency
of
recovery
linked
to
cost,
length
of
lag
phase,
secondary
vs
primary
metabolite
Some limitations to above treatment of
kinetics
•
The
growth
kinetics
above
generally
refer
to
exponential
rates

not
always
applicable
to
microbial
systems
e
.
g
.
hyphae
.
•
Also
exponential
growth
is
the
major
process
in
the
fermenter
(most
of
the
production
phase
re
cell
growth)
however
in
other
areas
such
as
shelf

life
predictions
other
phases
may
dominate
(for
example
the
lag
phase

if
cells
are
damaged
during
food
"preservation")
.
Thus
in
this
latter
case
kinetics
must
concentrate
on
other
phases
of
the
growth
curve

the
concept
of
the
logistic
or
gompertz
equations
become
important
.
•
Equally
in
the
case
of
secondary
metabolites
the
concept
of
trophophase
and
idiophase
must
be
considered
re
kinetic
treatments
.
In
this
case
the
focus
is
on
the
effect
of
m
on
product
formation
.
Kinetics Of Product Formation
•
Product formation can be independent of growth rate and
thus is only influenced by the amount of biomass present.
•
The kinetic treatment is usually simplified to calculation of
yield. Effectively the amount of product parallels the amount
of biomass e.g. ethanol produced by yeast.
•
However to obtain a clear picture one must consider
amount of product produced as a function of the amount of
biomass present (or the amount of substrate consumed) but
also as a function of time.
•
For example 50% yield of product per unit substrate in 6
hours or 90% yield in 6 years !!

which is more efficient to
the industrialist?
Summary of Models
•
Cyclical

involves formulation of a hypothesis, then experimental design
followed by experiments and analysis of results, which ideally should
further advance the original hypothesis. Thus the cycle is repeated etc.
•
Models are
•
set of hypotheses based on mathematical relations between
measurable quantities within the system
•
used to (a) correlate data, (b) predict performance
•
generated by a combination of processes ranging from well established
principles to educated guesses
•
tested by (a) comparison of predicted vs observed results (b) curve
fitting

analysis of patterns
Summary of model development
•
Simplification of system

identify factors having an effect on overall
behaviour.
It is the foundation of project design, management and
monitoring; and it is the first part of a complete project plan
=
CONCEPTUAL MODEL
•
Correlate performance data

empirical mathematical relationships
(black

box)
=
EMPIRICAL MODEL
•
Support relationships with theory

more fundamental approach
=
MECHANISTIC MODEL
(example model of penicillin ferm. )
Modelling fermentation systems
•
Mathematical modelling of fermentation processes has been an
intensely researched aspect of biotechnology.
•
Using models helps us to better understand the complex processes.
They allow us to systematically analyze these systems and identify
important variables and parameters.
•
Many complex models have been developed to describe complex
fermentation systems. Unfortunately, more often than not, complex
models are not used in the design process.
•
Firstly because they take a long time to develop and secondly
because they use parameters which cannot be determined.
Summary
•
Mathematical models of fermentation systems are generally based on
the model which relates the specific growth rate and substrate
utilization
•
Numerical methods are available and used for solving differential
equations
•
When applied to fermentation models, the computer programs used to
implement these methods will show how biomass, substrate and
product concentrations vary with time
•
One important piece of information that mathematical models of
fermentation systems can provide is the time that the fermentation
takes
•
This information is important in determining the required scale of the
process and the potential costs and profits
Conclusions
•
Fermentation kinetics are determined through mathematical
models to quantify rate of change in a fermentation process
•
Mathematical models must be formulated, constructed and solved
to yield meaningful data
•
Kinetic modelling can be as complex or as simple as you make it
•
Models normally relate to exponential bacterial growth
•
Main model types include stochastic and deterministic
•
Modelling of fermentations enables process operators to determine
the time it takes to produce a specified product
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