Modeling of the CARTEL^TM AND gate

In synthetic biology, modeling can be applied to a wide range of topics, for example, modeling of genetic circuits. In this subfield of mathematical and computational biology, ordinary differential equations (ODEs) are used to describe the transcriptional and translational process and predict the behavior of the desired circuit and also to support, if enough data is available, their further development such as optimizing for a desired output (Chen et al., 1999).

As already stated, the hypoxia response element (HRE), cAMP response element (CRE) and CTLA4 promoter could be utilized for more specific CAR expression. Combining these three enhancers in an AND gate to even further boost the specificity of the CAR expression (Brophy et al., 2014) proved to be quite challenging. Using endogenous systems came with the drawback, we had to generate a knockout or knockdown of either HIF1, VEGFR-2 or TDAG8.

Generating a single knockout or knockdown is challenging enough but generating three to test all possibilities seemed quite impossible in six months. So using modeling to find the best combination of enhancers and knockout was an attractive option instead of choosing a random AND gate design. This was carried out and is described below.

Finding the CARTEL^TM AND gate

The ODEs, that have been used to describe the different AND gate possibilities, consist of a set of parameters and a system of functions and their derivatives. How they can be set up in their simplest form is shown below.

The variables are functions of time t where $f\left(t\right)$ describes the mRNA concentration, $y\left(z\right)$ the transcription function and $z\left(t\right)$ the protein concentration. Rate constants are enlisted in the following table.

This system of ODEs can now be solved via numerical integration, but to obtain constants like $k1$, $k2$, $k3$, experimental data has to be produced that is suitable for a nonlinear regression (Ingalls et al., 2012).

The obtained model is to be compared with new experimental data in order to verify the parameter sets. If necessary, parameters or assumptions have to be corrected.