Cellular Growth Modelling and Classification
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The document discusses cellular growth modeling and classification, describing various types of models from unstructured and unsegregated to structured and segregated. It also covers the components of model construction including state variables, parameters, equations, and the definition of volumetric and specific rates for microbial growth, death, product formation, and substrate uptake. The classification aims to account for heterogeneity at both the population and intracellular levels in modeling biological systems.
Cellular Growth Modelling and Classification
- 1. Cellular Growth Modelling and Classification Cellular Engineering Author: Guillermo Garibay Benítez University: Instituto Politécnico Nacional Mexico
- 2. What is a model?
- 3. A model describes how the system will behave in response to changes we make in the system or the environment The set of relationships in a model can be a set of mathematical equations, graphs, tables, or unexpressed set of cause/effect relationships. The system studied can be a bioreactor, a single cell, a microbial culture, an immobilised cell, an enzyme, any equipment of unit operations. The variables of interest can be the feed rate, temperature, pH, the rate and mode of agitation, inoculum quality, and operational costs.
- 6. The Control Region Is a space in the system we want to model, chosen by the modeller in such a way that all variables or interest (concentration, temperature, pH, pressure, etc.) are uniform everywhere in the control region. The concentration of a compound, for example, can be constant or it can change with time. We may define several control regions within the system we want to model, in order to acount for heterogeneity.
- 7. Boundaries of the control region: Phase boundaries across wich no exchange takes place Phase boundaries across wich an exchange of mass and/or energy takes place Geometrically defined boundaries within one phase across which exchanges take place
- 8. State Variables These define the state of the process and there is one for each extensive property, for example: Xv viable cell concentration Xd nonviable cell concentration S outlet and bioreactor substrate concentration P outlet and bioreactor product concentration
- 9. Operating variables In these variables the values of which can be set by the operator of the process, for example: D dilution rate F volumetric feed flow rate Si, Xvi, Xdi, Pi inlet conentrations of the four conserved quantities
- 10. Intermediate variables These are all the volumetric rates: rx, rd, rSx, rSm, rSp, and rP Which can all be expressed in terms of the state variables listed before
- 11. Parameters Kinetic parameters: These are constants that are associated with the kinetic rate expressions for the system, such as µmax, KS, kd, mS, α, β, etc. Stoichiometric parameters. These define the stoichiometric relationships in the reactions or biological activity, such as yields: YP/S, YX/S
- 12. Equations Balance equations for each extensive property of the system. Rate equations: rates of reaction, generation of consumption of the individual species within the control region rates of transfer of mass, energy, momentum across the boundaries of the control region. Thermodynamic equations.
- 14. Model Classification in Biological Systems
- 15. Cell populations models classifications Cellular representations which are multicomponent are called structured Single component representations are designed unstructured Considerations of discrete, heterogenous cells constitutes a segregated viewpoint Unsegregated perspective considers average cellular properties
- 16. Unstructured StructuredUnsegregated Most idealized case Cell population treated as one component solute Multicomponent average cell description Segregated Single component, heterogeneous individual cells Multicomponent description of cell-to- cell heterogeneity Balanced growth (approximation ) Balanced growth (approximation ) “average cell” approximation “average cell” approximation Actual situation
- 17. Unsegregated models relies on an average cell description, describes biomass as consisting of several variables (such as NADH, precursors, metabolites, ATP, biomass). Unstructured models use a single variable to describe biomass Segregated models consider individual cells in recognition of the fact that cells in a population –a pure culture- are different, and are most often formulated as a population balance model.
- 18. An unstructured segregated model characterizes cells by one distributed by one distributed property, i.e. cell size or age of individual cells without considering intracellular composition. Structured segregated models considers the distribution of one or more intracellular variables.
- 19. This classification stands assuming a homogenous reactor environment
- 20. Kinetic Model Structure Model construction starts by defining the stoichometry of the reactions to be considered in the model. N subtrates are taken up by the cells and converted into M metabolic products and Q biomass constituents. The conversions are carried out in J reactions. Since the number of reactions and processes involved in cellular growth is very large, the actual reactions used are typically lumped reactions.
- 21. To describe the stoichometry of the reactions, we introduce stoichometry coefficients for all components in the system: αi for the substrate Si βi for metabolic product Pi γi for biomass constituent Xi
- 22. αji es is the stoichometric coefficient for the ith subtrate in the jth reaction. We introduce stoichometric coefficients for all substrates, metabolic products and biomass constituents in each of the J reactions. Many of the coefficients will be zero, since only a few compounds participate in any given reactions.
- 23. For the substrates Si, the metabolic products Pi and the biomass constituents Xi, the stoichometry for the jth cellular reaction can be specified as: Si = substrates Pi = Metabolic products Xi = biomass constituents
- 24. Is convenient to write the stoichometry in matrix notation: A, B, y Г contain the stoichometric coefficients in the J reactions for substrates, metabolic products, and biomass constituents respectively. Rows represents reactions and columns compounds.
- 25. Definition of Volumetric and Specific rates for basic microbial activities
- 26. Volumetric rate of any biological reaction The extent of any microbial activities, expressed as volumetric rates, depends on the concentration of viable biomass Xv in the control volume
- 27. Specific rate Specific rates are usually defined for growth, product formation and substrate uptake:
- 28. For growth: Units of rx are (kg live biomass) m-3 h-1 Units of µ are (kg live biomass) (kg live biomass)-1 or simply h-1 Here, with xv we denote the concentration of living cells as opposed to dead, and we make the distinction that growth is a biological activity performed by living cells, Specific growth rate
- 29. For death Units of rd are (kg dead biomass) m-3 h-1 Units of kd are (kg dead biomass) (kg live biomass)- 1h-1 We use the living cell concentration since the process of dying is performed by living cells only.
- 30. For product formation Units of rp are (kg product) m-3 h-1 Units of qp are (kg product) (kg live biomass)-1h-1
- 31. For substrate uptake Units of rs are (kg substrate)m-3h-1 Units of qs are (kg substrate)(kg live biomasss)-1h-1
Editor's Notes
- #16: It is a common practice to formulate the growth medium so that all components but one are present at sufficiently high concentrations that changes in their concentrations do not significantly high concentrations that changes in their concentrations do not significantly affect overall rates. A single component becomes the rate-limiting nutrient, and we need consider only the concentration of this one component when analyzing the effects of medium composition on cell growht kinetics. Occasionally, it is necessary to include other medium components, such as an inhibitory product which accumulates in the medium in order to obtain a suitable description of cell kinetics.
- #17: This classification approaches to microbial systems according to the number of components used in the cellular representation and wether or not the cells are viewed as a heterogeneous collection of discrete entities, as they really are, or instead as some kind of average cell which becomes almost the same conceptually as a component in solution. - Cellular representations which are multicomponent are called structured - Single component representations are designed unstructured - Considerations of discrete, heterogenous cells constitutes a segregated viewpoint - Unsegregated perspective considers average cellular properties
- #25: Con este tipo de planteamiento como el de la formula, un gran número de coeficientes estequiométricos se vuelve cero y se puede volver engorroso especificar coeficientes estequiométricos para todos los compuestos y todas las reacciones consideradas en el modelo. Sin embargo, la ventaja de utilizar planteamientos en forma de matrices facilita los análisis ya que pueden realizarse en simulaciones de computadoras.
- #29: Here, with xv we denote the concentration of living cells as opposed to dead, and we make the distinction that growth is a biological activity performed by living cells
- #31: Here we assume that product formation is performed by the living cells, and hence biomass concentration is used in the definition of the specific product formation rate. In some cases, product formation may due to dead cells, for example, if the product is formed as a result of dead cells autolysing.