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### Overall composition of an E. coli cell

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/orModels 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 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.

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)

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

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

Indicates well mixed

Gas out

Gas phase

Air/gas interface

Liquid phase

Control region

Air in

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.

How kinetics fits into overall design and operation of a process

Industrial Lab

Fermenter

Test

ideas

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

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

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:

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

- 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, 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:

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 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.

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.

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

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 differentialequations
- 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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