We used the following modelling approaches to identify bottlenecks in the constructed pathways and to predict the formation of a desired product in a given time. First of all we created a map of all metabolic reactions which are part of our project (figure 1). This is a usefull tool to give a first overview. Due to the huge amount of components it seems not feasible to create a computational model for all of these reactions at once. Therefore we started our modelling work by carrying out a stochiometric analysis. Afterwards we started a break down of the project in different parts. Dynamic modelling of the metabolites by creating kinetic equations was carried out for the isobutanol production pathway. This was done to identify bottlenecks. Finally we extended the existing model by adding some of the carbon dioxide fixing reactions.

Mathematical modelling is essentiell to understand complex biological systems (Schaber et al., 2009). The analysis of isolated biological components is supplemented by a systems biology approach since ten years ago (Chuang et al., 2010). Mathematical modelling is used to combine biological results (Kherlopian et al., 2008). Modelling is also a way to gain results without carrying out experiments in a laboratory. The behaviour of a system can be simulated to get results which cannot be derived from simply looking at the given system (Schaber et al., 2009). The most important aim of any modelling approach is the reduction of complexity. The given biological reality is often very divers and variable. Therefore it is important to identify the major rules and principles which can describe a system.

We calculated the stoichiometric relations of all substances involved in our complex reaction network (figure 1). The calculation starts with the electrons. They are transported into the system by mediators. We calculated the resulting production of intracellular molecules based on our map of the metabolic system (figure 1). The results are listed below.

The theoretical electron costs of different molecules is listed in table 1. All calculations are based on our pathway map. The housekeeping metabolism of E. coli is ignored. In theory and according to our pathway map there are 42 electrons needed for the production of one molecule isobutanol if CO₂ is used as sole carbon source. Our calculation does not involve the house keeping metabolism of E. coli which consumes lots of energy for the survival of the cell. The number of consumed electrons per produced isobutanol molecule is therefore much higher. The applied electric power can be converted into a number of electrons by the following equation: 1 A = 1 C * s^-1 = 6.2415065 * 10¹⁸ electrons.

After the stoichiometric analysis of the system we decided to use a dynamic model with kinetic equations for the metabolite concentrations. It allows the identification of metabolic or enzymatical bottlenecks. This is our major aim. At least we would like to identify them. Maybe we could even use this information in the next step to modify constructs e.g. exchange a RBS or a promotor sequence. This could be necessary to optimize the different enzyme concentrations. Beside that it was our aim to predict the production of isobutanol per substrate. An improvement of this model could predict the isobutanol production in a carbon dioxide fixing cell. To achieve our aims we broke down the complex system shown in figure 1. It was reduced to the system shown in figure 2. This reduced version was suitable for modelling.

We started our modelling work on the isobutanol pathway by reading publications about the isobutanol production pathway (Atsumi et al., 2008 and Atsumi et al., 2010). Doing that we collected a lot of information. The first modelling approach was a system of differential equations using Michealis-Menten kinetics. This was published as the best approach if reaction kinetics are not known (Breitling et al., 2008; Chubukov et al., 2014). All needed k_cat and K_M values were collected from the literature and from databases like KEGG, biocyc and BRENDA (table 2). Missing values were replaced by estimations. The starting concentrations for different metabolites were also taken from the literature and from different databases (table 3).

We implemented the system of differential equations (figure 3) in matlab (link to source code) and created first results. The predicted changing of the metabolic concentration over the time is shown in figure 4.

The amount of expressed proteins could differ depending on the distance of the coding sequence downstream of the promotor. Different values can be used to simulate the usage of promotors of different strength. This approach also allows the modelling of different growth states. The growth is represented by an increase in the amount of enzyme. Our modelling results indicated that the concentration of IlvC is limiting the isobutanol production. An experimentell verification of this hind is the next logical step. This bottleneck could be removed by overexpression of the corresponding coding sequence. One way to achieve this is the integration of a strong promotor and RBS upstream of this coding sequence. Due to a lack of time we were not able to follow up this lead. It could be a great possibility to improve our isobutanol production.

Carbon dioxide fixing reactions

The next model improvement was the addition of some of the carbon fixing reactions and the pathway leading to pyruvate. We collected k_cat and K_M values for nearly all relevant steps (table 4). They were used for differential equations which describe these additional reactions.