1.3. PROCESS ANALYSIS

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Industrial MicrobiologyINDM 4005Lecture
923/02/04
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PROCESS ANALYSIS

• Lecture 9
• (1) Kinetics and models - Predictive microbiology
• (2) Growth kinetics (and product)
• (3) Models - example, Continuous culture model

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Overview

• Fermentation Kinetics
• Mathematic models
• Stoichiometry
• Chemical kinetics
• Michaelis menten model
• The Monod model
• Yield coefficients
• Modelling fermentation processes
• Types of model

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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)

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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.

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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.

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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

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Mathematical modelsWhat 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.

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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

Metabolic products Extracellular macromolecules
Biomass constituents
Intracellular biochemical reactions
Substrates (Glucose)
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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

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Overall composition of an E. coli cell
Macromolecule of total dry Different
kinds weight of molecules Protein 55 1
050 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)
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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

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• 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.)

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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

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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

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Abstracted physical model of a batch fermenter
Indicates well mixed
Gas out
Gas phase Air/gas interface
Liquid phase Control region
Air in
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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.
• Vmax, Km 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.

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How kinetics fits into overall design and
operation of a process
Industrial Lab Fermenter
Test ideas
Scientific Experimental Engineering judgeme
nt data judgement
Determine model parameters
Validate model
Kinetic and Abstracted stoichiometric physic
al models model
Mathematical model
Use model for control process and economic
studies
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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

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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 (YLG) is 2
  moles of lactate (L) per mole of glucose (G). The
  relationship between lactate formation and
  glucose utilization would be

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Chemical kinetic equations as mathematical models

• 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
• dS kS
• dt
• where S is the concentration of the reactant
  and k is a rate constant.

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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.

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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, Vmax is the
  maximum specific rate and Km 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

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The Monod model and the Michaelis Menten model
The Monod Model looks similar to the Michaelis
Menten equation.
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The Monod model and the Michaelis Menten model

• The parameters µm and Ks are analogous to Vmax
  and Km. 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.

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Monod Model

• Monod's model describes the relationship between
  the specific growth rate and the growth limiting
  substrate concentration as
• where µm 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.

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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.

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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.

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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.

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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.

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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

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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 YX/S -DX/DS
• Product/substrate YP/S -DP/DS
• Product/biomass YP/X DP/DX
• Approximated as constants

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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

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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

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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

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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

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• 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

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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.

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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?

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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

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• 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. )

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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.

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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

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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