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CHNG 2804 Biological Systems

Lecture Outline. As Engineers we need to be able to Measure the rate at which Cells growProducts are formedBy-products are formedPredict the rates Cells growProducts are formedBy-products are formed. Data Analysis. NoiseExacerbated the need to take derivativesNeed to consider experimental errors and the uncertainty of parameters..

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CHNG 2804 Biological Systems

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    1. CHNG 2804 Biological Systems Kinetics and Modelling John Kavanagh

    2. Lecture Outline As Engineers we need to be able to Measure the rate at which Cells grow Products are formed By-products are formed Predict the rates Cells grow Products are formed By-products are formed

    3. Data Analysis Noise Exacerbated the need to take derivatives Need to consider experimental errors and the uncertainty of parameters.

    4. Mathematical Models A mathematical model is a set of relationships between inputs (manipulated variables) and outputs (the fermenter states). In the case of fermentations, the inputs are typically the conditions in the fermenter, such as temperature, media composition and substrate concentration. The outputs are typically biomass, products and by-products. In more complex models intracellular compounds may also be included.

    5. Uses of Models The motivations for modelling fermentations are varied, reasons including; improving understanding, process control, fermenter simulation, optimisation and estimation of conditions within the fermenter. These different motivations for modelling make different types of models appropriate. The intended use of the model dictates its type and complexity. As a general rule, the model should be as simple as possible but be able to represent the broad range of the fermentations behaviour.

    6. Simulations are not Experiments Simulation allows data to be collected at any frequency, and the data is free from experimental noise. However data from a fermentation model should be considered to be subject to the same uncertainty as the fermentation data to which it was fitted.

    7. Models are Approximations To make a complete and all encompassing model, it would be necessary to describe the workings of cells at the atomic level. The interactions within cells are extremely complex. E.coli contains approximately 2000 different proteins, making an exact description of microbial metabolism beyond the capability of any model. Hence, any model will need to be an approximation - the level of detail used in the approximation will be dependent on the data available and the purpose of the model.

    8. Types of Models In constructing a model it is necessary to select its type and complexity. Models can be classified as follows: Structured Unstructured Segregated Unsegregated Cybernetic Black-Box

    10. Commonly made assumptions for Unstructured Models Homogenous cell population Cell composition invariant Perfect mixing Uniform gas concentration

    11. Microbial Growth Many but not all microorganisms grow exponentially Dependent on type of microorganism Dependent on sugar/substrate concentration

    12. Microbial Growth Typical Simplified Batch growth given in microbiology text books.

    13. Growth Rate For most microorganisms the growth rate is dependent on sugar concentration

    14. Growth Equations Biomass formation is most simply described by assuming that the rate of growth is proportional to the number or mass of cells present. This proportionality is termed the specific growth rate and is generally symbolised by m and has units of hr-1. This proportionality is not however constant, and is affected by many factors such as the type of substrate and its concentration. Mycelial fermentations generally dont fit this model well

    15. Growth Equations In the following equations, a number of parameters and variables are common; m Specific growth rate mmax Maximum specific growth rate S Substrate concentration (typically glucose) X Biomass concentration The models all contain additional parameters to express the relationship between the specific growth rate and the substrate and biomass concentrations.

    16. Growth Equations - Monod Monod assumed that at low substrate concentrations the growth rate is first-order with respect to substrate concentration, whilst at higher concentrations the growth rate was independent of the glucose concentration. The parameter km is the substrate concentration at which the growth rate is half of the maximum.

    17. Growth Equations - Monod Monods equation is analogous to the Michaelis-Menten enzyme kinetics, and some authors have tried to give the equation physical meaning, assuming that the cells growth could be taken as limited by a single enzyme reaction. However, Monod arrived at his equation empirically and suggested that any sigmoidal curve could be fitted to the experimental data

    18. Growth Equations - Tessier Tessiers model also predicts a decrease in the specific growth rate with decreasing substrate concentrations. This model also predicts that the specific growth rate reaches a maximum value which is practically independent of substrate concentration. Like Monods model, Tessiers approaches this maximum asymptotically.

    19. Growth Equations - Moser Mosers equation, if l=1, can be derived from Monods simply by dividing both the numerator and the denominator by the substrate concentration. It has a numerical disadvantage, if the substrate concentration goes to zero, as the value of the specific growth rate becomes indeterminate rather than zero. This problem can be overcome by multiplying the numerator and the denominator by Sl.

    20. Growth Equations - Contois The Contois equation is similar to that of Monod, however it assumes that the specific growth rate decreases with both decreasing substrate and increasing biomass concentrations. At low biomass and high substrate concentrations, the specific growth rate asymptotically approaches its maximum value. It would be better to include the effect of biomass/lack of nutrients separately as this could cause trouble with fed batch fermentations.

    21. Growth Equations - Blackman m = mmax for Cs > mmaxB m = Cs/B for Cs <= mmaxB Blackmans equation is a piece-wise linear approximation of Monods model.

    22. Growth Equations - Logistic m = kx(1-bx) The logistic equation assumes that the growth rate decreases as the biomass concentration increases. Hence, the effect of decreasing nutrient concentration is included implicitly. This can be a problem if extra nutrients are fed!

    23. Growth Equations - Haldane Haldanes model incorporates substrate inhibition into Monods equation. That is, the Haldane model can describe a decrease in the specific growth rate due to an excessive substrate concentration This can occur during batch yeast fermentations if high sugar concentrations are used (>150g/L)

    24. Growth Equations All of these equations predict that the growth rate decreases as the substrate concentration decreases. A number of authors have looked at these different types of models and found few differences between them. Some equations have the advantage that they can be integrated analytically. However as the growth equation is generally part of a much larger model, this is of little use. Also necessary to include by-product inhibition terms.

    25. Determination of mmax and km Method 1 Starting from Monod Model if S>>km can assume Linearise/Determine m from log plot. Reasonable approximation in the initial stages of fermentations if: Substrate concentration large compared to cell concentration

    26. Determination of mmax and km Method 2 Determine m (dX/dt) from experimental data (noise is aproblem here) Graph m vs S mmax is the maximum value of m Km is the the value of S at which m = mmax Method 3 Construct model and use a minimisation routine to determine parameters Requires a lot of noise free data for complex models

    27. Similarities between Enzyme Kinetics and Microbial Growth Enzymes [E] react with substrates [S] to form products. Experimentally it was observed by Brown (1902) that the action of Invertase on Sucrose to produce Glucose and Fructose was first order with respect to sucrose concentrations at low concentrations and zero order with respect to sucrose concentrations at higher concentrations An explanation was formalised by Henri (1903) and more completely by Michaelis and Menten (1913)

    28. Enzyme kinetics

    29. Michaelis Menten Kinetics

    30. Michaelis Menten Kinetics

    31. Lineweaver Burk Plot

    32. Eadie Hofstee Plot

    33. Hanes Plot

    34. Substrate Uptake Equation below typical for batch system Substrate (sugar) uptake is generally assumed to be linearly related to Cell numbers Maintenance Energy (m) Cell growth Yield Coefficient (YX/S) Product formation - Yield Coefficient (YP/S) For simulations need to ensure that sugar concentration is not < 0

    35. Parameter Determination Can determine the yield coefficient by the amount of biomass (or product) formed divided by the amount of sugar consumed. This can be a problem if biomass and product are formed simultaneously.

    36. Product Formation Product formation is generally proportional to cell numbers/mass Can be complicated by factors such as Inhibitory By-Products Substrate concentrations Dissolved Oxygen concentrations

    37. By Product Inhibition The production of By-products such as Acetate (E.coli) Ethanol (Yeast) Have been found to reduce growth and production rates Effect on growth and production can be different

    38. By Product Inhibition Typical Inhibition Model Pmax is the concentration at which growth/production stops, n is typically 1-2. Can cause problems numerically when running simulation due to fractional indices of negative numbers. Can also cause errors in product formation parameters by exaggerating the effect of low by-product concentrations

    39. By Product Inhibition Red Xs data Blue Line Model n=1 Green line Model n~2 Without extra data hard to tell if n=1 over predicts both by product tolerance uninhibited production rate

    40. Running Models Need to solve several ODEs Biomass Substrate Product By-product Oxygen Software Matlab, Simulink Excel (Eulers method, need small step size, simple models)

    41. Review Kinetics of cell growth Types of models that can be used Some of the pitfalls when modelling Types of Software that can be use to solve model equations.

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