# 1.3. PROCESS ANALYSIS

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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

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

Sound theoretical basis

Superior predictive capabilities

Extensible to new culture conditions & cell strains

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

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