# Team:UC Chile/Mathematical Model

### From 2013.igem.org

# Modeling

How can we prove the advantages that our in vitro Whateversisome system could potentially have? Let’s explore it step by step.

**Regular bacteria vs Bacteria with the Carboxysome**

As a spatial separation, this micro compartment is increasing the encounter probability between the substrate and the enzyme, which maximizes the generation of a product. Bacteria having this synthetic organelle would maximize the output of what can be produced inside the BMC.

With this goal in mind, we have done an estimation of the internal space available. Based on this estimation, we can have an idea of how that encounter would happen and thus, in a future model, how fast metabolic reactions can occur inside it.

For more information about this section, please go to Carboxysome internal space and Efficiency .

After doing these calculations, the next step is to measure the efficiency after getting rid of the bacteria.

**Bacteria with the Carboxysome vs in vitro Carboxysome**

There are many advantages of having an in vitro system like we are proposing here:

- 1. No need of determining the optimal genetic manipulations in order to sustain the survival of the microorganism.
- 2. Some desirable products cannot cross the bacterial membrane, they are highly toxic and also have a low concentration or are co-synthesized with side-by-products.
- 3. Metabolic bypass can be done more easily in vitro. Instead, it can be really difficult to implement exogenous multi-enzyme systems in bacteria without altering other metabolic pathways.
- 4. Metabolic fluxes in a metabolic network need to be in a equilibrium inside the cell. It is necessary to determine which reactions may be altered in order to maximize the yield of the output. Carboxysome doesn’t need this balance.

In order to calculate the different possibilities that we can achieve with our system and to show the potential combinatorial capacity of it, we have reduced the number of enzymatic steps from some of the main natural metabolic pathways in E. coli to create a more direct via of production. Regardless of the possible complications that this would entail in an in vivo system. In the future, we could model all the combinatorial possibilities, just assuming that all the enzymes can go and be functional inside the Carboxysome.

For more information about this section, please go to Carboxysome combinatorial in vitro channeling

**Carboxysome internal space and efficiency**

First at all, we need to identify the available internal space of Carboxysome in order to characterize any enzymatic activity or protein production. We will describe all the procedures to achieve this: Space distribution and volume estimation of the microcompartment, internal occupied space (protein volume estimation), internal available space, RuBisCO reduction and production estimation.

Space distribution and volume estimation of the micro compartment:

Carboxysomes are described and observed as a protein-based shell that has an icosahedral shape with 20 triangular faces, 30 edges and 12 vertices. In previous observations of this structures [1] it was described as a non-regular icosahedron with a diameter of approximate 40-90 nm and an edge of approximate 40-75 nm (Figure 1).To find the best way for modelig the carboxysome, we will consider it as a regular icosahedron. With this considerations, and knowing the volume of an icosahedron, we have the volume of the Carboxysome protein shell as:

Figure 1. Structural size and shape of Carboxysome

Internal occupied volume:

To determinate the available space of the Carboxysome we need to calculate first the space that is already occupied inside it. For this aim, we have considered the two internal proteins: RuBisCO (2000 copies) and Carbonic Anhydrase (100 copies)[1], which their protein structures are fully described. The 3D protein structures were taken to estimate their volumes and using the highest and lowest point described in each coordinate. Also, we made a box volume (Figure 2) as the best volume associated.

RuBisCO protein total volume:

Carbonic Anhydrase protein total volume:

Figure 2. Volumen approximation of a protein structure

**Internal available space**

Considering the volume of the shell and the volume of all the copies of the proteins inside it, we have an approximate available volume of:

Corresponding to the 29.35% of the volume of the Carboxysome.

** The thickness volume of the protein-based shell was not considered in this approximation*

**Further work**

**Protein volume estimation**

Considering that the box estimation of the protein volume over evaluates the actual volume of the protein, we will work on a more exact approximation of this: which it will give us more available space for our desired production or encapsulation.

**Protein-based shell approximation**

To have an acquire approximation of the internal Carboxysome space, we need to consider the thickness that the protein-based shell has. The actual complication about this is the layout and spatial 3D arrangement of these proteins into the space which could variate the volume.

We need to further work on our Carboxysome formation for the following parts to be done

**RuBisCO reduction**

Since a fully assembled Carboxysome with reduced amount of RuBisCO have been described (3) and considering that, for our project, we are just aiming to get a structural formation and not necessary to retain the native function of the microcompartment. We could reduce the quantities of RuBisCO inside it and with this, as the principal component, the available volume.

**Production estimation**

To calculate the maximal yield that could be obtained in our proof of concept with the -galatosidase, we need to estimate the volume of this enzyme (same method as with RuBisCO and Carbonic Anhidrase) in order to know the quantity of enzymes per micro compartment, and with the approximation of number of Carboxysomes for each bacteria described in E.coli, we can have an approach to the amount and quantity of the production.

**Carboxysome Efficiency**

To calculate the efficiency of using the Carboxysome as a reactor, we could model the diffusion of the product towards the Carboxysome by a mass balance. For this, the kinetics of the reaction, which can be calculated empirically, are needed. This can be compared with the efficiency observed using E. coli.

**Carboxysome combinatorial in vitro channeling**

As a proof of concept, we took some of the essential metabolic pathways on E.coli: Glycolysis, the Pentose Phosphate pathway and Citrate cycle (TCA cycle). This 3 pathways are described as a serial of enzymatic activities leading in different networks with products that can be used immediately, be stored or even initiate another metabolic pathway.

In the first place we quantify the enzymes included in the whole natural metabolic pathways named before (Figure 3.1):

Figure 3: Graphic description of the benefits of using carboxysomes compared with bacteria. Every circle represents an enzyme. The arrows are the compounds, signaling where they come and where they are going. The letter H is the last step in order to get a desirable compound, starting from A. In 1), it is showed a pathway schema on E.coli: There are 7 enzymes involved. C represents product that is used for other processes. G represents an enzyme that does not exist in E. coli. E and F are enzymes from the bacteria. 1) Pathway scheme using Carboxysomes to simulate E.coli enzymes. There are 6 enzymes involved. 3) Pathway scheme using Carboxysomes with bypass. This uses only 5 enzymes.

On a further step we compared the same pathway, this time quantifying just the enzymes that were needed for the final production, ignoring all the productions that the bacteria does naturally towards other pathways, considering this ones as losses (Figure 3.2). This enzymes will be considered 1:1 with the quantities of Carboxysomes that it is needed.

The final comparison is made with the same pathway, but also taking the advantage of producing enzymes that does not produces E.coli, making a bypasses with those, reducing even more the quantity of the enzymes associated as Carboxysomes in comparison with the enzymes required in E.coli (Figure 3.3).

The results of this analysis will open our knowledge of how we can replicate the enzymatic activity of these metabolic pathways through reducing the actual number of enzymes required in Carboxysomes.

**Further work**

As the work done was only a proof of concept, the next step is modeling the entire metabolism of the bacteria.

**References:**

- 1. Bonacci, W., Teng, P. K., Afonso, B., Niederholtmeyer, H., Grob, P., & Silver, P. A. (2011). Modularity of a carbon-fixing protein organelle. Proceedings of the National Academy of Sciences of the United States of America, 109(2), 478-483. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/22184212
- 2. Menon, B. B., Dou, Z., Heinhorst, S., Shively, J. M., & Cannon, G. C. (2008). Halothiobacillus neapolitanus Carboxysomes Sequester Heterologous and Chimeric RubisCO Species. (J. Rutherford, Ed.)PLoS ONE, 3(10), 10. Public Library of Science. Retrieved from http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2570492&tool=pmcentrez&rendertype=abstract
- 3. Meyer, A., Pellaux, R., & Panke, S. (2007). Bioengineering novel in vitro metabolic pathways using synthetic biology. Current opinion in microbiology,10(3), 246-253. Retrieved from http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?dbfrom=pubmed&id=17548240&retmode=ref&cmd=prlinks