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.

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.

4. Model construction

5. Components of Modelling

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

32. Recommended lecture