Team:UANL Mty-Mexico/Modeling

We will model the E. cologic system through two different approaches, both of them deterministic:

1. A broad level approach
2. A genetic circuit level approach

The goal of both models is to predict the change in the concentration of arsenic and the biosensor activity after exposure to the E. cologic system.

Broad scale approach

The first approach will describe and predict the behavior of the system in a broad abstraction level. Briefly, empirically obtained kinetic constants will be obtained after fitting data to the following ODE set:

\begin{equation} \frac{d[Asac]}{dt} = Vmax_{1}\bigg(\frac{[Asex]}{K_{1}+[Asex]}\bigg) \end{equation} \begin{equation} \frac{d[Asex]}{dt} = -[Asac] \end{equation} \begin{equation} \frac{dUM}{dt} = Vmax_{2}\bigg(\frac{[Asac]}{K_{2}+[Asac]}\bigg) \end{equation}

Where the variables Asac, Asex and UM represent the arsenic accumulated inside the cells, the extracellular arsenic (i.e. the arsenic that remains in the solution) and Miller units, respectively.

This approach will allow us to use the data from relatively simple characterization experiments and can be extended to take into account total cell volume available and the effect of the silica-binding kinetics on this available cell volume.

Note that the equations that describe the change of extracellular arsenic and Miller units through time are a simple Michaelis-Menten model. Nevertheless, this will not be the only scenario considered; we will look for other models to which our data may fit better.

Genetic circuit level approach

The second approach is divided in four modules, all of which intend to describe different parts of the E. cologic system taking into account molecular kinetic constants and gene expression models for the proteins involved.

The four modules of the genetic circuit level approach are:

1. Arsenic transport – this will be a model that describes the movement of arsenic across the cell membrane, which takes into account the kinetic constants of the interaction with the arsenic transporter.
2. Arsenic accumulation – this model will describe the sequestration of arsenic inside the cell as a result of the interaction with the proteins ArsR and the metalothionein.
3. Biosensor – a classical switch-like model that produces a reporter protein in direct proportion to the concentration of an input, which in this case is arsenic.
4. Silica binding – a model that describes the interaction between the membrane-embedded chimeric protein with the silica binding domain and the silica particles.