Mathematical modelling is essentiell to understand complex biological systems (Klipp 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 (Klipp 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. Therefor 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 (fig. x). Starting with the electrons which are transported into the system by mediators we calculated the resulting production of all other molecules. The results are shown in figure XXX and listed below.

In theory there are XXXX 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 its survival. The number of consumed electrons per produced isobutanol molecule is therefor 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 * 10^{18 electrons.}

First of all it was our aim to predict the production of isobutanol. Our model should give information about the optimal concentration of each enzyme in the isobutanol production pathway. The next aim was the prediction of isobutanol production in a carbon dioxide fixing cell. The complete system is shown in figure 1. This complex network of reactions was reduced to the system shown in figure 2. This reduced version was used for modelling.

We started our modelling work by reading publications about the isobutanol production pathway (Atsumi et al., 2008 and Atsumi et al., 2010). 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 (REFERENZ EINFÜGEN). All needed V_max and K_M values were colleted from the literature and from databases like KEGG, biocyc and BRENDA (table 1).

The starting concentrations for different metabolites were also taken from the literature and from different databases (table 2).

We implemented the system of differential equations in matlab (link to source code) and created first results (fig. 3).

To improve our prediction we decided to switch from V_max and K_M to k_cat and the amount of the different enzymes (table 3). 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.

The modelling results indicated that the concentration of IlvD is limiting the isobutanol production. This bottle neck could be removed by overexpression of the corresponding coding sequence. This could be achived by using a strong promotor and RBS in front of this coding sequence.

Carbon dioxide fixing reacions

The next model improvement was the addition of some of the carbon fixing reactions and the pathway leading to pyruvate. We used k_cat values for all relevant steps (fig.2 and table 4).