Solutions to
Dynamic
Bioprocess Modeling Examples
Chapter 9 of
the Book
As stated
in the Chapter 9, the most difficult and decisive task in any modeling exercise
often lies in the initial analysis of the problem. Most flow situations can be
described using the tanks-in-series concept, in which tanks are connected by
volumetric flow rates having certain concentrations. Therefore, a few selected
examples are provided here, in the hope they will assist the reader in learning
the necessary skills. The actual solutions will be presented in hand-drawn
sketches and hand-written equations to illustrate the modelling approach
recommended.
The four
examples are:
1. The first example is the relatively simple case of a Roman fountain, but which on first sight may appear stunningly complex. In this however, we provide a partial first step towards its solution and leave the reader to set up the relevant balance equations and then to modify the problem accordingly.
2. The second example represents a simple attempt to model the effect of a potentially polluting factory, situated close to a river and feeding into a large lake. Again, a possible starting point is suggested for the reader to develop the solution further.
3. The third example concerns the processes of protein synthesis and secretion within a eukaryotic cell. This problem represents a considerable degree of additional complexity compared to the two previous examples.
4. The fourth example is also of biological nature and concerns the essential components of a human liver, i.e. a sinusoid, hepatocytes and bile canaliculi. This example is to illustrate how one could approach more difficult problems. Here the reader is again asked to modify the problem by introducing interaction between cells.
The Trevi fountain in Rome is famous for its beauty and is a
favorite place for tourists. A simplified sketch is depicted below (Fig. 9.1).
A proposal has been made to “spice it up” for the Italian national day. Towards
this goal, a chemist has been working in the laboratory to find different
combinations of (bio)chemicals to create fountain water colors of the Italian
national flag. Three chemical reagents are employed. Dye A is simply a green
dye and is added into the top basin. Chemical B when added to a water solution
containing A produces a white coloration. Therefore, it is added to the middle
basin. Chemical C when added to a solution of both A and B will turn the water
red. Each of these reactions is very fast. A proposed scheme of operation is
shown in Fig. 9.2 below for the continuous addition of the three chemicals to
each of the fountain basins to give the desired Italian flag colors.
Figure 9.1. Roman fountain with overflowing water falling into the basin below.
Figure 9.2. Roman fountain with overflowing water falling into the basin below.
Before this
scheme can even be considered by the relevant authorities, it is required that
the effect should be studied using a dynamic simulation model. As part of this,
you are required to carry out the following exercises.
Exercise
Solutions to Exercises of Ch 9.1
A lake somewhat similar in shape to Lake Zurich (sketched below)
receives water input via a major river at the upper end of the lake (Fig. 9.3).
It flows through the lake leaves it at the other end as a river through a major
town. The city authorities are somewhat perturbed by a proposal of a new
industrial site at the far end of the lake. It could risk pollution of the lake
and the water passes through the town. Also the
drinking water supply of the city, which relies on reasonably pure water, would
be in danger.
Figure 9.3. Lake with a potentially polluting factory at the river feeding the lake.
Before the relevant authorities can even consider this industrial
development, it is required that the danger of potential pollution be modelled
and simulated.
Exercise
Figure 9.4. Scheme of lake depicted in Figure 9.3.
Solutions to Exercises of Ch 9.2.
A membrane
reactor for the cultivation of mammalian cells attaching to the membrane, e.g.
hepatocytes, shall be constructed. Medium containing necessary nutrients is
circulated through the porous hollow fiber membranes which has the cells
attached on their outer surface. The pumped recirculation stream passes through
an aeration unit with a gas permeable membrane, e.g. a thin-walled silicone
rubber tube. In this way oxygen is transferred into the circulating liquid.
Assume that the residence time in the hollow fiber membrane is short enough to
avoid significant gradients of nutrients except of oxygen. The oxygen
consumption of the cells is a function of the dissolved oxygen concentration,
which has to diffuse through the hollow fiber membrane to eventually reach the
cells.
Exercise
Solutions to Exercises of Ch 9.3.
As can be
read in biochemical textbooks, the protein secretion in eukaryotic cells uses
an exocytotic process. After expression of a gene, the mRNA is transported into
the cytosol. Here it recruits ribosomal subunits, and after initial protein
synthesis, it docks to a receptor at the endoplasmic reticulum. In this way the
protein is synthesized into the lumen of the endoplasmic reticulum (ER). The
protein is matured involving posttranslational modifications, e.g.
glycosylation, and refolding. Proteins travel to the Golgi complex (GC) via
vesicles that are separated from the ER and later fused with the Golgi membrane
system. Proteins are sorted and eventually separated from the GC in secretory
granules, transported to the cell membrane in our case and then fused with
membrane causing a release into the extracellular environment. Various
signaling molecules are involved in these processes. We want to focus first on
some essential steps, i.e. the synthesis of the protein into the ER, its
maturation, transport to the GC, protein sorting and transport to the cell
membrane. Assuming that a large number of such processes are occurring
simultaneously we assume that these processes can be modeled with a continuous,
deterministic model.
Exercise
This
starting model could be used to build more and more complexity into it, e.g.
synthesis of elements necessary for ribosome recruitment and docking to the ER.
Exercise
9.4.1 Solution
Biosynthesis,
maturation and secretion of proteins is a very complex process and to our
knowledge, there is no mathematical model available describing all relevant
processes. Here, we cannot present a complete solution but rather outline some
steps of a potential approach to a solution of this challenging problem.
Presently,
there is a big effort to apply modelling to the improvement of glycosylation of
proteins, a step most important for high quality therapeutic protein,
especially monoclonal antibodies (e.g. Zhang, Liang; Wang, MingLiang;
Castan, Andreas; et al. (2020) Glycan Residues
Balance Analysis - GReBA: A novel model for the
N-linked glycosylation of IgG produced by CHO cells. Metabol.
Eng. 57: 118-128; Kontoravdi, Cleo; del Val, Ioscani
Jimenez (2018) Computational tools for predicting and controlling the
glycosylation of biopharmaceuticals. Curr. Opin. Chem. Eng. 22:89-97). These models use compartments,
at least representing the cytosol and the Golgy
apparatus.
Here we
start with a scheme (Figure 9.4.1) modified adapted from Klein et al. (2015;
Klein, Tobias; Niklas, Jens; Heinzle, Elmar (2015)
Engineering the supply chain for protein production/secretion in yeasts and
mammalian cells. J. Ind. Microbiol. Biotechnol 42:452-464).
Fig.
9.4.1. Scheme of protein synthesis,
modification and secretion starting with ribosomal protein secretion into the
endoplasmic reticulum (ER) that is linked via vesicle transport to the Golgi
apparatus (G) that produces vesicles transporting the protein to the cellular
membrane allowing secretion of mature proteins (P). The ribosomal protein
synthesis requires the correct ribosomal RNA (mRNA) and various other factors
for protein synthesis.
Solutions to Exercise 9.4.2 to 9.4.5
The liver
is a vital organ primarily in vertebrates and has a wide range of essential
functions. These include glycogen storage for controlling blood glucose level,
detoxification, protein synthesis, and production of biochemicals necessary for
digestion. Liver lobules are its functional units. Lobules consist of sinusoids
that are primarily built of hepatocytes but also non-parenchymal cells. We
consider a sinusoid built of hepatocytes that absorb substances, metabolize
them and partly secret them into bile canaliculi that collect the bile fluid
that is eventually released into the gallbladder via bile ducts (Fig. 9.5).
Here we consider a simplified version of a sinusoid only built of a few
hepatocytes and direct secretion of bile into a bile duct.
Figure 9.5. Simplified liver sinusoid structure consisting of hepatocytes. PV- portal vein, CV – central vein, BD – bile duct.
As shown in
Fig. 9.5, blood enters the sinusoid from the portal vein. Hepatocytes
surrounding the sinusoid import low and high molecular weight compounds from
the blood stream and secrete others into the blood stream. Some compounds are
converted are converted into bile acids and are secreted into bile canaliculi
that collect the bile leading it to the bile duct as shown in Figure 9.5.
Obviously, hepatocyte activities are far more complex in reality but let as
start with a simple model describing essential elements of the sinusoid. In a
first modeling step we simplify the system of Fig. 9.5 even more and
investigate a system having only three hepatocytes, a portal, a middle and one
next to the central vein. This even more simplified scheme with only three
hepatocytes is depicted in Fig. 9.6. Here it is assumed that blood and bile flow
can be modeled by a tanks-in-series model with each tank connected to one
hepatocyte. As shown in Fig. 9.6, compound A enters the lower left sinusoid
element with volume VS1,
some of it flows with enrichment up to the adjoining hepatocyte with volume VH1, where it is
converted to B. Compound B flows up to the bile tank with volume VB1 and is
secreted out in flow FB1.
Transfers of compound A from the sinusoid to hepatocytes and of compound B from
hepatocytes to the bile duct involve active transport rather than just
convective flow. This taken into account by enrichment factors a and b. The rest of A flows to the right into the
next sinusoid with volume VS2.
This example demonstrates very well how the tanks-in-series concept can be used
to model extremely complex flow and reaction networks.
Figure 9.6. Simplified model of a liver sinusoid with compound A absorbed by hepatocytes with an enrichment factor a. Compound A is converted to compound B that is eventually secreted into the bile canaliculi with an enrichment factor b. V – volume, F – flow rate, C – concentration. Indices: S – sinusoid, H – hepatocyte, B – bile.
Exercise