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Chapter 7
transferred to quantitative models, but a major obstacle in applying quantitative models is
estimation of the parameters of the model. In order to do this it is necessary to have precise
measurements of the different variables of the system, and preferably at very different
experimental conditions. Thus, in order to obtain a quantitative description of the biological
system considered (or model simulations) no effort should be spared to obtain reliable data.
Ideally carbon and nitrogen balances should close to within 99%, and the concept of elemental
balancing described in Chapter 3 therefore represents a natural
first step in evaluating
experimental data that are going to be used for quantitative modeling of a given system. A failure
to close the mass balances will lead to inaccuracies in the estimated fluxes in and out of the cell
and in general make it impossible to develop meaningful kinetics. The last ten years have
witnessed a revolution in experimental techniques applied to life sciences. This has made it
possible to measure more variables and at a higher precision, resulting in a far more detailed
modeling of cellular processes. Furthermore, the availability of powerful computers permits
complex numerical problems to be solved with a reasonable computational time. At the present
even complex mathematical models for biological processes can therefore be handled and
experimentally verified. Such detailed (or mechanistic) models are often of little use in the
design of a bioprocess, whereas they serve a purpose in fundamental research on biological
phenomena. In this presentation we will focus on models that are useful for design of
bioprocesses, but we will also shortly treat mathematical models that can give a better insight
into biological processes at the molecular level.
Note 7.1 Mathematical models.
A mathematical model is a set of
relationships
between the
variables
in the system being studied, and
generally it can predict the output variables, “the state of the system”, from the input variables. These
relationships are normally expressed in the form of mathematical equations, but they may also be
specified as logic expressions (or cause/effect relationships) that are used in the operation of a process.
The variables include as inputs any property that is of importance for the process,
e.g.
the agitation rate
in the bioreactor, the feed rate to the bioreactor, pH of the medium, the temperature of the medium, the
concentration of substrates in the feed. Output variables are concentrations in the reactor of substrates,
metabolic products, the biomass concentration, and the state of the biomass - often represented by a set
of key intracellular components. In order to set up a mathematical model it is necessary to specify a
control volume wherein all the variables of interest are taken to be uniform, i.e. there is no variation in
their values throughout the control volume. For fermentation processes the
control volume
is typically
given by the whole bioreactor, but for large bioreactors the medium may be inhomogeneous due to
mixing problems and here it is necessary to divide the bioreactor into several control volumes (see
Chapter 11). When the control volume is the whole bioreactor it may either be of constant volume or it
may change with time depending on the operation of the bioprocess. When the control volume has been
defined a set of balance equations can be specified for the variables of interest. These balance equations
specify how materia] is flowing in and out of the control volume and how material is converted within
the control volume. The conversion of material within the control volume is specified by so-called rate
equations for kinetic expressions), and together with the mass balances these specify the complete model.