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.

 


 

9.1      Modeling a Roman Fountain

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

  1. Derive the basic material balance equations for the systems, defining all constants, parameters and initial variable values. It is required that the model be as general as possible.
  2. What assumptions, initial values and unknown parameter values apply? What uncertainties in your modeling may yet apply?

 

 

Solutions to Exercises of Ch 9.1


 

9.2      Modeling a Lake 

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

  1. The lake appears in the form of three distinct regions separated by two land peninsulas penetrating out into the lake. This suggests a dynamic analysis could be made using a simple three tank model, as in the Roman Fountain example. Derive the material balances of this lake using the scheme of Figure 9.4.

Figure 9.4. Scheme of lake depicted in Figure 9.3.

 

  1. There is an alternative plan to build the factory at the central part of this lake. How could this be taken into account in a model? Sketch a corresponding block diagram and redefine the streams! How do the balance equations change?
  2. Local fishermen report a strong backward flow from the central to the upper part of the lake. How would this effect the model?
  3. Town records have been found, giving sufficient differences in actual depth and width of the lake. How can data be included in your original model?
  4. Critics of the scheme point out that the lake has sediment, plant life and fish life which require consideration. If suitable information were available, how could such effects be incorporated into your model?

Solutions to Exercises of Ch 9.2.

9.3      Modeling a Mammalian Cell Recirculation Reactor with External Aeration.

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

 

  1. Draw the system with the essential elements so far. Draw a block diagram and the streams connecting the elements. What type of streams are these?
  2. Assume that the oxygen gradients in the hollow fiber membrane and inside the aeration tube can be modeled by a tanks-in-series approximation. Use three tanks for the sake of simplicity. Draw a block diagram for both units including the cells consuming oxygen.
  3. Write balance equations for this system.
  4. Identify ways to describe the oxygen transfer through the hollow fiber and gassing tube walls!
  5. Convert the reactor into a continuous reactor by adding a feed and a purge stream. How does this change the balance equations?

 

 

Solutions to Exercises of Ch 9.3.

9.4      Modeling Protein Synthesis and Secretion in a Eukaryotic Cell

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

  1. Consult a biochemistry or cell biology textbook and draw a sketch of the system, first starting with protein synthesis into the ER.
  2. Identify essential fluid elements of this system and sketch a matching block diagram.
  3. Identify streams and add them to the sketch.
  4. Assume that the protein synthesis rate is influenced by a molecule in the cytosol with varying concentration. What is the influence of this additional process to the system?
  5. Include a simple model of transcription producing the mRNA of the protein of interest. Take into account that mRNA synthesis is taking place in another compartment, the nucleus, and that it has to be transported to the cytosol before building the complex with the ribosome that is attached to the ER. Draw a sketch first and identify corresponding additional flow terms.

 

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

9.5      Modeling a Liver Sinusoid

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

 

  1. Derive the material balances of one combination of one sinusoid and bile element connected to the corresponding hepatocyte.
  2. Assume that the conversion of A into B in the hepatocyte depends on dissolved oxygen levels. These levels are influenced by oxygen uptake by the hepatocytes causing a gradient along the sinusoid. How can this be included in the model? How would the structure of the model (block diagram) change? How can the kinetics of the oxygen dependent conversion of A be modeled?
  3. Assume that because of an anatomic change, the central vein blood circulates back into the adjacent sinusoidal fluid element. The concentration of A is different in sinusoidal element 3 and in the central vein. Redraw the system and redefine those balance equations that are to be changed.
  4. Because of an injury or other disease, the flow out of the bile canaliculi is mechanically blocked. Since the hepatocyte continuously pumps into the bile canaliculi, the volume of the canaliculi and the pressure will increase until a limiting value is reached. At this point, the flow from the hepatocytes into the canaliculi will be completely blocked. What kind of changes in the model would be necessary?

 

 

Solutions to Exercises of Ch 9.5